// Numbas version: exam_results_page_options {"name": "Maths Support: Probability density functions", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "preventleave": false, "browse": true, "showfrontpage": false, "showresultspage": "never"}, "duration": 0.0, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution 4.0 International"}, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "shufflequestions": false, "questions": [], "percentpass": 75.0, "allQuestions": true, "pickQuestions": 0, "type": "exam", "feedback": {"showtotalmark": true, "advicethreshold": 0.0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "IS2.1", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "ungrouped_variables": ["a", "valk", "xl", "p", "pval", "xu"], "tags": ["CDF", "PDF", "Probability", "cdf", "continuous random variables", "cumulative distribution functions", "density function", "distribution function", "distribution functions", "integration", "pdf", "probability", "probability density function", "random variables", "statistics"], "preamble": {"css": "", "js": ""}, "advice": "

a)

Note that in order for $f_X(x)$ to be a pdf it must satisfy two important conditions:

1. $f_X(x) \\ge 0$ in the range $\\var{xl} \\le x \\le \\var{xu}$

2. The area under the curve given by $f_X(x)$ is $1$ and this implies that:
\\[\\int_{\\var{xl}}^{\\var{xu}}f_X(x)\\;dx = 1\\] as the value of the function is $0$ outside this range.

We first check condition 2. and then check that condition 1. is satisfied.

Note that \\[\\int kx\\;dx = k\\frac{x^2}{2}\\] on forgetting the constant of integration.

Hence \\[\\begin{eqnarray*} \\int_{\\var{xl}}^{\\var{xu}}kx\\;dx &=&\\frac{k}{2}(\\var{xu}^2-\\var{xl}^2)\\\\ &=&\\frac{k}{2}\\times \\var{xu^2-xl^2} \\end{eqnarray*} \\]

But we must have this last value equal to $1$ hence:
\\[ \\frac{k}{2}\\times \\var{xu^2-xl^2}=1 \\Rightarrow k = \\simplify[std]{2/{xu^2-xl^2}}\\]

Hence the pdf is:
\\[f_X(x) = \\simplify[std]{2/{xu^2-xl^2}x}\\;\\;\\;\\;\\;\\var{xl} \\le x \\le \\var{xu}\\]

We have to check condition 1. that the function $f_X(x)$ is positive for $\\var{xl} \\le x \\le \\var{xu} $ – but this is clear from
the definition of $f_X(x)$ and the value of $k$.

b)

If $F_X(x)$ is the distribution function of the distribution given by $f_X(x)$ then:

$F_X(x) = 0\\;\\;\\;x \\lt \\var{xl},\\;\\;\\;\\;F_X(x)=1\\;\\;\\;x \\ge \\var{xu}$

and for $\\var{xl} \\le x \\le \\var{xu}$:

\\[\\begin{eqnarray*} F_X(x)&=&\\int_{-\\infty}^x f_X(x)\\;dx=\\simplify[std]{2/{xu^2-xl^2}}\\int_{\\var{xl}}^x x\\;dx\\\\ &=&\\simplify[std]{2/{xu^2-xl^2}}\\times\\frac{\\left(x^2-\\var{xl}^2\\right)}{2}\\\\ &=&\\frac{x^2-\\var{xl^2}}{\\var{xu^2-xl^2}} \\end{eqnarray*} \\]

c)

We have
\\[\\begin{eqnarray*} P\\left(X \\lt \\simplify[std]{{xl+xu}/2}\\right)&=&F_X\\left(\\simplify[std]{{(xl+xu)}/2}\\right)\\\\ &=& \\frac{1}{\\var{xu^2-xl^2}}\\left(\\simplify[std]{({(xl+xu)}/{2})^2-{xl}^2}\\right)\\\\ &=&\\simplify{{3*xl+xu-4*a}/{4*(xu+xl-2*a)}} \\end{eqnarray*} \\]

$f_X(x) = \\left \\{ \\begin{array}{l} \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\end{array} \\right .$	$kx$	$\\var{xl} \\leq x \\leq \\var{xu},$

$0,$	$\\textrm{otherwise.}$

\n \n \n \n

What value of $k$ makes $f_X(x)$ into the pdf of a distribution?

\n \n \n \n

Input your answer here as a fraction and not as a decimal.

\n \n \n \n

$k=\\;\\;$[[0]]

\n \n \n ", "marks": 0, "gaps": [{"notallowed": {"message": "

input as a fraction and not a decimal.

Given the value of $k$ found in the first part, determine and input the distribution function $F_X(x)$

\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n

$F_X(x) = \\left \\{ \\begin{array}{l} \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\end{array} \\right .$	[[0]]	$x \\lt \\var{xl},$

[[1]]	$\\var{xl} \\leq x \\leq \\var{xu},$

[[2]]	$x \\gt \\var{xu}.$

\n ", "marks": 0, "gaps": [{"expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "scripts": {}, "answer": "0", "marks": 0.5, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"notallowed": {"message": "

input numbers as fractions or integers and not as decimals

", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "(((x + ( - {xl})) * (x + {(xl + ( - (2 * a)))})) / {((xu + ( - xl)) * (xu + xl + ( - (2 * a))))})", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "scripts": {}, "answer": "1", "marks": 0.5, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n \n \n

Find and input as a fraction not a decimal:

\n \n \n \n

$P\\left(X \\lt \\simplify[std]{{xl+xu}/2}\\right) = \\phantom{{}}$[[0]]

\n \n \n ", "marks": 0, "gaps": [{"notallowed": {"message": "

input as a fraction or integer and not as a decimal

A random variable $X$ has a probability density function (PDF) given by:

", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "0", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "valk": {"definition": "precround(2/(p*(xu+xl-2*a)),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "valk", "description": ""}, "xl": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "xl", "description": ""}, "p": {"definition": "(xu-xl)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "pval": {"definition": "precround((3*xl+xu-4*a)/(4*(xu+xl)-2*a),2)", "templateType": "anything", "group": "Ungrouped variables", "name": "pval", "description": ""}, "xu": {"definition": "xl+random(1..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "xu", "description": ""}}, "metadata": {"notes": "\n \t\t

8/07/2012:

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Added tags.

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Checked calculations, OK.

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23/07/2012:

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Added description.

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1/08/2012:

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Added tags.

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Question appears to be working correctly.

\n \t\t", "description": "

The random variable $X$ has a PDF which involves a parameter $k$. Find the value of $k$. Find the distribution function $F_X(x)$ and $P(X \\lt a)$.

", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "IS2.2", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "ungrouped_variables": ["a", "ux", "xl", "i1", "lo", "j1", "up", "exans", "p", "u", "t", "tol", "ans", "cval", "lx", "xu"], "tags": ["CFD", "continuous random variables", "cr1", "cumulative distribution functions", "density functions", "distribution function", "distribution functions", "integration", "PDF", "pdf", "probabilities", "probability density function", "random variables", "statistics", "tested1"], "advice": "

a)

Note that in order for $f_X(x)$ to be a pdf it must satisfy two important conditions:

1. $f_X(x) \\ge 0$ in the range $\\var{xl} \\le x \\le \\var{xu}$

2. The area under the curve given by $f_X(x)$ is $1$ and this implies that:
\\[\\int_{\\var{xl}}^{\\var{xu}}f_X(x)\\;dx = 1\\] as the value of the function is $0$ outside this range.

We first check condition 2. and then check that condition 1. is satisfied.

Hence \\[\\begin{eqnarray*} \\int_{-\\infty}^{\\infty}f_X(x)\\;dx=\\int_{\\var{xl}}^{\\var{xu}}f_X(x)\\;dx&=&\\int_{\\var{xl}}^{\\simplify[std]{{xu+xl}/2}}cx\\;dx+\\int_{\\simplify[std]{{xu+xl}/2}}^{\\var{xu}}c(\\var{xu}-x)\\;dx\\\\ &=&\\simplify[std]{{(xu-xl)^2}/8}c+\\simplify[std]{{(xu-xl)^2}/8}c=\\simplify[std]{{(xu-xl)^2}/{4}}c \\end{eqnarray*} \\]
But this has to equal $1$ and so \\[c=\\simplify[std]{4/{p^2}}\\]

Hence with this value of $c$ we see that condition 2. is satisfied i.e.

\\[\\int_{-\\infty}^{\\infty}f_X(x)\\;dx=1\\]

Condition 1. is clearly satisfied as $c \\gt 0$.

b)

We must have $F_X(x)=0,\\;\\;\\;x \\le \\var{xl},\\;\\;\\;\\textrm{and}\\;\\;\\;F_X(x)=1,\\;\\;\\;x \\ge \\var{xu}$

Apart from that we find an expression for $F_X(x)$ in each of the ranges $[\\var{xl},\\simplify[std]{{xl+xu}/2}],\\;\\;\\;[\\simplify[std]{{xl+xu}/2},\\var{xu}]$

1. $x \\in [\\var{xl},\\simplify[std]{{xl+xu}/2}]$

We have:\\[\\begin{eqnarray*} f_X(x) &=& \\simplify[std]{{4}/{p^2}} x \\\\ \\Rightarrow F_X(x)&=& \\int_{-\\infty}^xf_X(x)\\;dx \\\\ &=& \\simplify[std]{{4}/{p^2}}\\int_{\\var{xl}}^x x\\;dx\\\\ &=&\\simplify[std]{{2}/{p^2}}x^2 \\end{eqnarray*} \\]

2. $x \\in [\\simplify[std]{{xl+xu}/2},\\var{xu}]$

We have:\\[\\begin{eqnarray*} f_X(x) &=& \\simplify[std]{{4}/{p^2}}(\\var{xu}-x) \\\\ \\Rightarrow F_X(x)&=& \\int_{-\\infty}^xf_X(x)\\;dx = \\int_{\\var{xl}}^{\\simplify[std]{{xl+xu}/2}}f_X(x)\\;dx+\\int_{\\simplify[std]{{xl+xu}/2}}^x f_X(x)\\;dx\\\\ &=& \\simplify[std]{{4}/{p^2}}\\int_{\\var{xl}}^{\\simplify[std]{{xl+xu}/2}} x\\;dx + \\simplify[std]{{4}/{p^2}}\\int_{\\simplify[std]{{xl+xu}/2}}^x(\\var{xu}-x)\\;dx\\\\ &=&1-\\simplify[std]{{2}/{p^2}}(\\var{xu}-x)^2 \\end{eqnarray*} \\]

c)

Using the distribution function we have just found we have that:

$P(\\var{lx} \\lt X \\lt \\var{ux}) = F_X(\\var{ux})-F_X(\\var{lx})$

But $\\var{ux}$ is in the range between $\\var{(xl+xu)/2}$ and $\\var{xu}$ and the distribution function is given by:

\\[F_X(x)=1-\\simplify[std]{{2}/{p^2}}(\\var{xu}-x)^2\\]

Hence $F_X(\\var{ux})=1-\\simplify[std]{{2}/{p^2}}(\\var{xu}-\\var{ux})^2 = \\var{up}$ to 5 decimal places.

Similarly, $\\var{lx}$ is in the range between $\\var{xl}$ and $\\var{(xl+xu)/2}$ and the distribution function is given by:

\\[F_X(x)=\\simplify[std]{{2}/{p^2}}x^2\\]

Hence $F_X(\\var{lx})=\\simplify[std]{{2}/{p^2}}(\\var{lx})^2 = \\var{lo}$ to 5 decimal places.

Hence $P(\\var{lx} \\lt X \\lt \\var{ux}) = F_X(\\var{ux})-F_X(\\var{lx})=\\var{up}-\\var{lo}=\\var{up-lo}=\\var{ans}$ to 2 decimal places.

", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n

$f_X(x) = \\left \\{ \\begin{array}{l} \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\\\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}}\\end{array} \\right .$	$0$	$ x \\leq \\var{xl},$

$cx$	$\\var{xl} \\lt x \\leq \\simplify[std]{{xu+xl}/2},$

$c(\\var{xu}-x)$	$\\simplify[std]{{xu+xl}/2} \\lt x \\leq \\var{xu},$

$0$	$x \\gt \\var{xu}.$

\n \n \n \n

What value of $c$ makes $f_X(x)$ into the pdf of a distribution?

\n \n \n \n

Input your answer here as a fraction and not as a decimal.

\n \n \n \n

$c=\\;\\;$[[0]]

\n \n \n \n ", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "

input as a fraction and not a decimal

Given the value of $c$ found in the first part, determine and input the distribution function $F_X(x)$

\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n

$F_X(x) = \\left \\{ \\begin{array}{l} \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}} \\\\ \\phantom{{.}}\\\\ \\phantom{{.}}\\\\ \\phantom{{.}} \\end{array} \\right .$	[[0]]	$x \\leq \\var{xl},$

[[1]]	$\\var{xl} \\lt x \\leq \\simplify[std]{{xu+xl}/2},$

[[2]]	$\\simplify[std]{{xu+xl}/2} \\lt x \\leq \\var{xu},$

[[3]]	$ x \\gt \\var{xu}.$

\n \n \n \n

Input all numbers as fractions or integers in the above formulae.

\n \n \n \n ", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "0", "marks": 0.5, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"notallowed": {"message": "

input numbers as fractions or integers and not as decimals

", "showStrings": false, "strings": ["."], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "(1 + ( - (({2} / {(p ^ 2)}) * ((x + ( - {xu})) ^ 2))))", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "1", "marks": 0.5, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "\n \n \n

Also, using the distribution function above find:

\n \n \n \n

$P(\\var{lx} \\lt X \\lt \\var{ux})=\\;\\;$[[0]]

\n \n \n \n

(input your answer to $2$ decimal places).

\n \n \n \n ", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "ans+tol", "minValue": "ans-tol", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "

A random variable $X$ has a probability density function (PDF) given by:

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "0", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "ux": {"definition": "j1/2", "templateType": "anything", "group": "Ungrouped variables", "name": "ux", "description": ""}, "xl": {"definition": "0", "templateType": "anything", "group": "Ungrouped variables", "name": "xl", "description": ""}, "i1": {"definition": "round((t+(100-t)*(xu-1))/100)", "templateType": "anything", "group": "Ungrouped variables", "name": "i1", "description": ""}, "lo": {"definition": "precround((2/p^2)*(lx-xl)^2,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "lo", "description": ""}, "j1": {"definition": "round((u*(xu+1)+(100-u)*(2*xu-1))/100)", "templateType": "anything", "group": "Ungrouped variables", "name": "j1", "description": ""}, "up": {"definition": "precround(1-(2/p^2)*(ux-xu)^2,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "up", "description": ""}, "exans": {"definition": "up-lo", "templateType": "anything", "group": "Ungrouped variables", "name": "exans", "description": ""}, "p": {"definition": "(xu-xl)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "u": {"definition": "random(0..100)", "templateType": "anything", "group": "Ungrouped variables", "name": "u", "description": ""}, "t": {"definition": "random(0..100)", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "tol": {"definition": "0.01", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "ans": {"definition": "precround(exans,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}, "cval": {"definition": "precround(4/p^2,4)", "templateType": "anything", "group": "Ungrouped variables", "name": "cval", "description": ""}, "lx": {"definition": "i1/2", "templateType": "anything", "group": "Ungrouped variables", "name": "lx", "description": ""}, "xu": {"definition": "xl+random(4,6,8,12,14)", "templateType": "anything", "group": "Ungrouped variables", "name": "xu", "description": ""}}, "metadata": {"notes": "\n \t\t

8/07/2012:

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Added tags.

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Checked calculations, OK.

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Set tolerance via new variable tol=0.01 for last question.

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23/07/2012:

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Added description.

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1/08/2012:

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In the Advice section, moved \\Rightarrow to the beginning of the line instead of the end of the previous line.

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Question appears to be working correctly.

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21/12/2012:

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Checked calculations, OK. Added tag tested1.

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Checked rounding, OK. Added tag cr1.

\n \t\t", "description": "

The random variable $X$ has a PDF which involves a parameter $c$. Find the value of $c$. Find the distribution function $F_X(x)$ and $P(a \\lt X \\lt b)$.

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "IS2.3", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "functions": {}, "ungrouped_variables": ["a", "f1", "c", "b", "d", "f", "tol1", "m", "pd1", "n", "pc", "s", "r", "u", "t", "pd", "tol", "n1", "s1", "gc"], "tags": ["PDF", "Probability", "conditional probability", "cr1", "density function", "diagram", "integration", "pdf", "probabilities", "probability", "probability density function", "query", "statistics", "tested1"], "preamble": {"css": "", "js": ""}, "advice": "

a)

We have:

\\[\\begin{eqnarray*} P(X \\gt \\var{c})&=&\\int_{\\var{c}}^{\\var{b}}f(x)\\;dx\\\\ &=&\\frac{1}{\\var{n1}}\\int_{\\var{c}}^{\\var{b}}\\simplify[std]{({f1}-{s1}x)}\\;dx\\\\ &=&\\frac{1}{\\var{n1}}\\left(\\simplify[std,!otherNumbers]{{f1}({b}-{c})-{s1}({b}^2-{c}^2)/2}\\right)\\\\ &=&\\simplify[std]{{2*f1 * (b -c) + s1 * (c ^ 2 -(b ^ 2))} / {2*n1} }\\\\ &=& \\var{pc} \\end{eqnarray*} \\] to 4 decimal places.

b)

\\[\\begin{eqnarray*} P(X \\gt \\var{d} | X \\gt \\var{c})&=&\\frac{P(X \\gt \\var{d}\\;\\;\\textrm{and}\\;\\; X \\gt \\var{c})}{P(X \\gt \\var{c})}\\\\ &=&\\frac{P(X \\gt \\var{d})}{P(X \\gt \\var{c})} \\end{eqnarray*} \\] as $\\var{d} \\gt \\var{c}$.

But,
\\[\\begin{eqnarray*} P(X \\gt \\var{d})&=&\\int_{\\var{d}}^{\\var{b}}f(x)\\;dx\\\\ &=&\\frac{1}{\\var{n1}}\\int_{\\var{d}}^{\\var{b}}\\simplify[std]{({f1}-{s1}x)}\\;dx\\\\ &=&\\frac{1}{\\var{n1}}\\left(\\simplify[std,!otherNumbers]{{f1}({b}-{d})-{s1}({b}^2-{d}^2)/2}\\right)\\\\ &=&\\simplify[std]{{2*f1 * (b -d) + s1 * (d ^ 2 -(b ^ 2))} / {2*n1} }\\\\ &=& \\var{pd1} \\end{eqnarray*} \\] to 4 decimal places.

Hence \\[\\begin{eqnarray*} P(X \\gt \\var{d} | X \\gt \\var{c})&=&\\frac{P(X \\gt \\var{d})}{P(X \\gt \\var{c})}\\\\ &=&\\frac{\\var{pd1}}{\\var{pc}}\\\\ &=&\\var{pd} \\end{eqnarray*} \\] to 2 decimal places.

", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n \n \n

$P(X \\gt \\var{c})=\\;\\;$[[0]]

\n \n \n \n

Input to 4 decimal places.

\n \n \n ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "{pc+tol}", "minValue": "{pc-tol}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n

$P(X \\gt \\var{d} | X \\gt \\var{c})=\\;\\;$[[0]]

Input to 2 decimal places.

\n ", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "{pd+tol1}", "minValue": "{pd-tol1}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\n

Given the pdf

\\[\\begin{eqnarray*} f(x)&=&\\simplify[std]{({f1}-{s1}x)/{n1}}\\;\\;\\var{a} \\leq x \\leq \\var{b}\\\\ f(x)&=&0\\;\\;\\;\\textrm{otherwise} \\end{eqnarray*} \\]
find the following probabilities:

\n ", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(0..2)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "f1": {"definition": "f/gc", "templateType": "anything", "group": "Ungrouped variables", "name": "f1", "description": ""}, "c": {"definition": "a+round((t*1+(100-t)*((r-1)))/100)/2", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "a+r", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "a+round((u*2*(r-1)+(100-u)*((r+1)))/100)/2", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "f": {"definition": "b * (2 * n -m) -(a * m)", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "description": ""}, "tol1": {"definition": "0", "templateType": "anything", "group": "Ungrouped variables", "name": "tol1", "description": ""}, "m": {"definition": "random(1,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "pd1": {"definition": "precround((n*r^2+s*(d^2-a^2)-f*(d-a))/(n*r^2),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "pd1", "description": ""}, "n": {"definition": "m+s", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "pc": {"definition": "precround((n*r^2+s*(c^2-a^2)-f*(c-a))/(n*r^2),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "pc", "description": ""}, "s": {"definition": "random(1,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "r": {"definition": "random(4,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}, "u": {"definition": "random(0..100)", "templateType": "anything", "group": "Ungrouped variables", "name": "u", "description": ""}, "t": {"definition": "random(0..100)", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "pd": {"definition": "precround((n*r^2+s*(d^2-a^2)-f*(d-a))/(n*r^2)/pc,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "pd", "description": ""}, "tol": {"definition": "0", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "n1": {"definition": "n*r^2/gc", "templateType": "anything", "group": "Ungrouped variables", "name": "n1", "description": ""}, "s1": {"definition": "2*s/gc", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "gc": {"definition": "gcd(n*r^2,gcd(f,2*s))", "templateType": "anything", "group": "Ungrouped variables", "name": "gc", "description": ""}}, "metadata": {"notes": "\n \t\t

8/07/2012:

\n \t\t

Added tags.

\n \t\t

Set new tolerance variables, tol=0 for first question and tol1=0 for second question. Aslo included statement that second question is to be entered to 2 dps.

\n \t\t

There is an image to be included in the Advice. This needs to be done.

\n \t\t

Checked calculations, OK.

\n \t\t

23/07/2012:

\n \t\t

Added description.

\n \t\t

1/08/2012:

\n \t\t

Question appears to be working correctly.

\n \t\t

21/12/2012:

\n \t\t

Checked calculation. Added tag tested1.

\n \t\t

Added query and diagram tags re possible inclusion of a diagram - which could be dynamic?

\n \t\t

Checked rounding, OK. Added tag cr1.

\n \t\t", "description": "

Given the pdf $f(x)=\\frac{a-bx}{c},\\;r \\leq x \\leq s,\\;f(x)=0$ else, find $P(X \\gt p)$, $P(X \\gt q | X \\gt t)$.

", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": ["stats"], "custom_part_types": [], "resources": []}