// Numbas version: exam_results_page_options
{"name": "Week 7 Quiz", "metadata": {"description": "", "licence": "None specified"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-shuffled", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], [], [], []], "questions": [{"name": "Accessibility and Commnication", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Does accessibility imply communication?</p>", "advice": "<p>Communication requires accessibility in both directions</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// Reveal answers, score and advice\nthis.display.revealAnswer();\nthis.display.showScore();\nNumbas.controls.getAdvice();", "order": "before"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["False", "True"], "matrix": ["1", 0], "distractors": ["", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Ergodic Markov chain", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Which of the following are required for an Ergodic Markov chain?</p>", "advice": "<p>An Ergodic Markov chain is aperiodic (period = 1) and irreducible (Only one communicating class)</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// Reveal answers, score and advice\nthis.display.revealAnswer();\nthis.display.showScore();\nNumbas.controls.getAdvice();", "order": "before"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["Aperiodic", "Irreducible", "Have only one periodic recurrent class", "Have at least two recurrent classes"], "matrix": ["1", "1", "-1", "-1"], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Finite Markov chain states", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Which of the following states can a finite Markov chain have?</p>", "advice": "<p>All recurrent states in a finite Markov chain will be positive recurrent, since the expected number of steps to return to a recurrent state in a finite Markov chain is finite.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// Reveal answers, score and advice\nthis.display.revealAnswer();\nthis.display.showScore();\nNumbas.controls.getAdvice();", "order": "before"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["Positive recurrent", "Transient", "Null recurrent"], "matrix": ["1", "1", "-1"], "distractors": ["", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Identify Classes", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Identify the classes for the Markov chain with the following transitional probability matrix:</p>\n<p>$\\mathbf{P}= \\begin{bmatrix}\\var{a1} &amp; \\var{a2} &amp; \\var{a3} &amp; \\var{a4} &amp; \\var{a5} \\\\ \\var{b1} &amp; \\var{b2} &amp; \\var{b3} &amp; \\var{b4} &amp; \\var{b5} \\\\ \\var{c1} &amp; \\var{c2} &amp; \\var{c3} &amp; \\var{c4} &amp; \\var{c5} \\\\ \\var{d1} &amp; \\var{d2} &amp; \\var{d3} &amp; \\var{d4} &amp; \\var{d5} \\\\ \\var{e1} &amp; \\var{e2} &amp; \\var{e3} &amp; \\var{e4} &amp; \\var{e5} \\\\ \\end{bmatrix} $</p>", "advice": "<p>Periodicity is determined by the GCD of steps taken to return to the state.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a1": {"name": "a1", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 0,\n  Dec=3, 1/random(3..6))", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 1/random(2..4),\n  Dec=3, 1/random(3..6))", "description": "", "templateType": "anything", "can_override": false}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 1-a1-a2,\n  Dec=3, 1-a1-a2)", "description": "", "templateType": "anything", "can_override": false}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 0,\n  Dec=3, 0)", "description": "", "templateType": "anything", "can_override": false}, "a5": {"name": "a5", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1-a1-a2-a3-a4,\n  Dec=2, 0,\n  Dec=3, 0)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 1/random(3..6),\n  Dec=3, 1/random(3..6))", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 0,\n  Dec=3, 1/random(3..6))", "description": "", "templateType": "anything", "can_override": false}, "b3": {"name": "b3", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 1 -b1 -b2,\n  Dec=3, 1 -b1 -b2)", "description": "", "templateType": "anything", "can_override": false}, "b4": {"name": "b4", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 0,\n  Dec=3, 0)", "description": "", "templateType": "anything", "can_override": false}, "b5": {"name": "b5", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1-b1-b2-b3-b4,\n  Dec=2, 0,\n  Dec=3, 0)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 1/random(3..6),\n  Dec=3, 1/random(3..6))", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 1-c1-c3,\n  Dec=3, 1-c1-c3)", "description": "", "templateType": "anything", "can_override": false}, "c3": {"name": "c3", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 0,\n  Dec=3, 1/random(3..6))", "description": "", "templateType": "anything", "can_override": false}, "c4": {"name": "c4", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 0,\n  Dec=3, 0)", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 0,\n  Dec=3, 0)", "description": "", "templateType": "anything", "can_override": false}, "d2": {"name": "d2", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 0,\n  Dec=3, 0)", "description": "", "templateType": "anything", "can_override": false}, "d3": {"name": "d3", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 1/2,\n  Dec=3, 0)", "description": "", "templateType": "anything", "can_override": false}, "d4": {"name": "d4", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 1/random(3,4),\n  Dec=3, 0)", "description": "", "templateType": "anything", "can_override": false}, "d5": {"name": "d5", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1-d1-d2-d3-d4,\n  Dec=2, 1-d3-d4,\n  Dec=3, 1)", "description": "", "templateType": "anything", "can_override": false}, "e1": {"name": "e1", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 0,\n  Dec=3, 0)", "description": "", "templateType": "anything", "can_override": false}, "e2": {"name": "e2", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 0,\n  Dec=3, 0)", "description": "", "templateType": "anything", "can_override": false}, "e3": {"name": "e3", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 0,\n  Dec=3, 0)", "description": "", "templateType": "anything", "can_override": false}, "e4": {"name": "e4", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1/random(5..10),\n  Dec=2, 1/random(2,3),\n  Dec=3, 1)", "description": "", "templateType": "anything", "can_override": false}, "e5": {"name": "e5", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1-e1-e2-e3-e4,\n  Dec=2, 1-e4,\n  Dec=3, 0)", "description": "", "templateType": "anything", "can_override": false}, "Dec": {"name": "Dec", "group": "Ungrouped variables", "definition": "random(1 .. 3#1)", "description": "<p>Chooses from the 3 answers.</p>", "templateType": "randrange", "can_override": false}, "c5": {"name": "c5", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1-c1-c2-c3-c4,\n  Dec=2, 0,\n  Dec=3, 0)", "description": "", "templateType": "anything", "can_override": false}, "ans1": {"name": "ans1", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 1,\n  Dec=2, 0,\n  Dec=3, 0)", "description": "", "templateType": "anything", "can_override": false}, "ans3": {"name": "ans3", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 0,\n  Dec=2, 0,\n  Dec=3, 1)", "description": "", "templateType": "anything", "can_override": false}, "ans2": {"name": "ans2", "group": "Ungrouped variables", "definition": "switch(\n  Dec=1, 0,\n  Dec=2, 1,\n  Dec=3, 0)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a1", "a2", "a3", "a4", "a5", "b1", "b2", "b3", "b4", "b5", "c1", "c2", "c3", "c4", "c5", "d1", "d2", "d3", "d4", "d5", "e1", "e2", "e3", "e4", "e5", "Dec", "ans1", "ans2", "ans3"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// Reveal answers, score and advice\nthis.display.revealAnswer();\nthis.display.showScore();\nNumbas.controls.getAdvice();", "order": "before"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["$\\{A, B, C, D, E\\}$ - aperiodic, recurrent", "$\\{A, B, C\\}$ - aperiodic, recurrent, $\\{D, E\\}$ - transient", "$\\{A, B, C\\}$ - aperiodic, recurrent, $\\{D, E\\}$ - period = 2, recurrent"], "matrix": ["{ans1}", "{ans2}", "{ans3}"], "distractors": ["", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Irreducible periodic Markov chain", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>An irreducible periodic Markov chain will <strong>not</strong> have a limiting distribution?</p>", "advice": "<p>While this will have a unique stationary distribution, the periodicity will destroy the existence of a limiting distribution.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// Reveal answers, score and advice\nthis.display.revealAnswer();\nthis.display.showScore();\nNumbas.controls.getAdvice();", "order": "before"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["True", "False"], "matrix": ["1", 0], "distractors": ["", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Limiting Ergodic Markov chain", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Drovandi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/9840/"}, {"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>A Markov chain must be Ergodic for the limiting distribution to exist and to have a unique stationary distribution?</p>", "advice": "<p>It is possible to have a Markov chain that has a unique stationary and limiting distribution and is NOT Ergodic.</p>\n<p>An example is a finite Markov chain that has 1 recurrent aperiodic class, and 1 or more transient classes.</p>\n<p>Such a Markov chain is not irreducible as it has more than 1 class (so not Ergodic), but it still will have a unique stationary and limiting distribution.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// Reveal answers, score and advice\nthis.display.revealAnswer();\nthis.display.showScore();\nNumbas.controls.getAdvice();", "order": "before"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["False", "True"], "matrix": ["1", 0], "distractors": ["", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Number of Stationary distributions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Drovandi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/9840/"}, {"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Do the following Markov chains have a unique stationary distribution?</p>", "advice": "<table style=\"border-style: solid;\" border=\"border-style: solid;\">\n<tbody>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n</tbody>\n</table>\n<p>Will have a unqiue stationary distribution $\\pi^T = [0.25, 0.25, 0.25, 0.25]$</p>\n<table style=\"border-style: solid;\" border=\"border-style: solid;\">\n<tbody>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n</tr>\n</tbody>\n</table>\n<p>Will not have a unique stationary distribution two examples are $\\pi^T = [0.5, 0.5, 0, 0]$ and $\\pi^T = [0, 0, 0.5, 0.5]$</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// Reveal answers, score and advice\nthis.display.revealAnswer();\nthis.display.showScore();\nNumbas.controls.getAdvice();", "order": "before"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<table style=\"border-style: solid;\" border=\"border-style: solid;\">\n<tbody>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n</tbody>\n</table>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["True", "False"], "matrix": ["1", "0"], "distractors": ["", ""]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// Reveal answers, score and advice\nthis.display.revealAnswer();\nthis.display.showScore();\nNumbas.controls.getAdvice();", "order": "before"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<table style=\"border-style: solid;\" border=\"border-style: solid;\">\n<tbody>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n</tr>\n</tbody>\n</table>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["True", "False"], "matrix": [0, "1"], "distractors": ["", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Stationary distribution requirements", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Which of the following are a requirement for a unique stationary distribution to exist?</p>", "advice": "<p>A unique stationary distribution will exist so long as there is only one positive recurrent class.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// Reveal answers, score and advice\nthis.display.revealAnswer();\nthis.display.showScore();\nNumbas.controls.getAdvice();", "order": "before"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["Only one recurrent class", "Aperiodic", "Irreducible", "At least two recurrent classes"], "matrix": ["1", "-1", "-1", "-1"], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Limiting distribution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Chris Drovandi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/9840/"}, {"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Which of the following Markov chains has a limiting distribution?</p>", "advice": "<table style=\"border-style: solid;\" border=\"border-style: solid;\">\n<tbody>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n</tbody>\n</table>\n<p>Will not have a limiting distribution as it is periodic.</p>\n<table style=\"border-style: solid;\" border=\"border-style: solid;\">\n<tbody>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n</tr>\n</tbody>\n</table>\n<p>Will not have a unique stationary distribution and so will not have a limiting distribution.</p>\n<table style=\"border-style: solid;\" border=\"border-style: solid;\">\n<tbody>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n</tr>\n</tbody>\n</table>\n<p>Will have a limiting distribution $\\pi^T = [0.25, 0.25, 0.25, 0.25]$.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "// Reveal answers, score and advice\nthis.display.revealAnswer();\nthis.display.showScore();\nNumbas.controls.getAdvice();", "order": "before"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<table style=\"border-style: solid;\" border=\"border-style: solid;\">\n<tbody>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n</tbody>\n</table>", "<table style=\"border-style: solid;\" border=\"border-style: solid;\">\n<tbody>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$1$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n</tr>\n</tbody>\n</table>", "<table style=\"border-style: solid;\" border=\"border-style: solid;\">\n<tbody>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n</tr>\n<tr>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0$</td>\n<td style=\"width: 16px; text-align: center;\">$0.5$</td>\n</tr>\n</tbody>\n</table>"], "matrix": ["0", 0, "1"], "distractors": ["", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "allowPrinting": true, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": []}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "adam bretherton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18177/"}], "extensions": [], "custom_part_types": [], "resources": []}