// Numbas version: exam_results_page_options {"percentPass": 0, "type": "exam", "duration": 0, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}, "allowPause": true}, "name": "Blathnaid's copy of Brad's copy of Maria's copy of mathcentre: Probability distributions", "shuffleQuestions": false, "navigation": {"preventleave": false, "reverse": true, "showresultspage": "never", "showfrontpage": false, "onleave": {"action": "none", "message": ""}, "allowregen": true, "browse": true}, "pickQuestions": 0, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "\n \t\t

6 questions on standard statistical distributions.

\n \t\t

Binomial, Poisson, Normal, Uniform, Exponential.

\n \t\t", "notes": ""}, "questions": [], "allQuestions": true, "showQuestionGroupNames": false, "question_groups": [{"name": "", "questions": [{"name": "BS3.1", "extensions": [], "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/"}], "tags": ["Binomial Distribution", "Binomial distribution", "binomial distribution", "poisson distribution", "Poisson distribution", "random variables", "sc", "statistical distributions", "statistics"], "metadata": {"description": "

Given descriptions of  3 random variables, decide whether or not each is from a Poisson or Binomial distribution.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\n

Which of the following random variables could be modelled with a binomial distribution and which could be modelled with a Poisson distribution?

\n

You will lose 1 mark for every incorrect answer. The minimum mark is 0.

#### a)

\n

1. $X \\sim \\operatorname{Poisson}(\\var{thismany})$, so $\\lambda = \\var{thismany}$.

\n

2. The expectation is given by $\\operatorname{E}[X]=\\lambda=\\var{thismany}$

\n

3. $\\operatorname{stdev}(X)=\\sqrt{\\lambda}=\\sqrt{\\var{thismany}}=\\var{sd}$ to 3 decimal places.

\n

#### b)

\n

1. \$\\begin{eqnarray*}\\operatorname{P}(X = \\var{thisnumber}) &=& \\frac{e ^ { -\\var{thismany}}\\var{thismany} ^ {\\var{thisnumber}}} {\\var{thisnumber}!}\\\\& =& \\var{prob1} \\end{eqnarray*} \$ to 3 decimal places.

\n

2. If an employee receives a warning then he or she must have sold less than {number1}.

\n

Hence we need to find:

\n

\$\\begin{eqnarray*}\\operatorname{P}(X < \\var{number1})& =& \\simplify[all,!collectNumbers]{P(X = 0) + P(X = 1) + {v}*P(X = 2)}\\\\& =& \\simplify[all,!collectNumbers]{e ^ { -thismany} + {thismany} * e ^ { -thismany} + {v} * (({thismany} ^ 2 * e ^ { -thismany}) / 2)} \\\\&=& \\var{prob2} \\end{eqnarray*} \$

\n

to 3 decimal places.

\n

", "rulesets": {}, "parts": [{"prompt": "

Assuming a Poisson distribution for $X$, {descX}, write down the value of $\\lambda$.

\n

$X \\sim \\operatorname{Poisson}(\\lambda)$

\n

$\\lambda =$ [[0]]

\n

Find $\\operatorname{E}[X]$ the expected {descX}.

\n

$\\operatorname{E}[X]=$ [[1]]

\n

Find the standard deviation for {what}.

\n

Standard deviation = [[2]] (to 3 decimal places).

", "marks": 0, "gaps": [{"allowFractions": false, "marks": 0.25, "maxValue": "thismany", "minValue": "thismany", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.25, "maxValue": "thismany", "minValue": "thismany", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.5, "maxValue": "sd+tol", "minValue": "sd-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "

Find the probability that {this} $\\var{thisnumber}$ {things}

\n

$\\operatorname{P}(X=\\var{thisnumber})=$ [[0]] (to 3 decimal places).

\n

\n

Find the probability that {thisaswell}

\n

Probability = [[1]] (to 3 decimal places).

", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "prob1+tol", "minValue": "prob1-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "prob2+tol", "minValue": "prob2-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\n

{pre} $\\var{thismany}$.

\n

{something} $\\var{number1}$ {else}

\n

\n ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"pre": {"definition": "\"The mean number of sales per day at a telecommunications centre is \"", "templateType": "anything", "group": "Ungrouped variables", "name": "pre", "description": ""}, "what": {"definition": "\"daily sales.\"", "templateType": "anything", "group": "Ungrouped variables", "name": "what", "description": ""}, "this": {"definition": "\"a randomly selected employee makes exactly \"", "templateType": "anything", "group": "Ungrouped variables", "name": "this", "description": ""}, "things": {"definition": "\"sales.\"", "templateType": "anything", "group": "Ungrouped variables", "name": "things", "description": ""}, "prob1": {"definition": "precround(tprob1,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob1", "description": ""}, "v": {"definition": "if(number1=2,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "v", "description": ""}, "descx": {"definition": "\"the number of sales per day\"", "templateType": "anything", "group": "Ungrouped variables", "name": "descx", "description": ""}, "else": {"definition": "\"per day.\"", "templateType": "anything", "group": "Ungrouped variables", "name": "else", "description": ""}, "thismany": {"definition": "random(5..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "thismany", "description": ""}, "something": {"definition": "\"Employees receive a warning if they make less than \"", "templateType": "anything", "group": "Ungrouped variables", "name": "something", "description": ""}, "tol": {"definition": "0.001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "number1": {"definition": "if(thismany<8,2, 3)", "templateType": "anything", "group": "Ungrouped variables", "name": "number1", "description": ""}, "tprob1": {"definition": "(thismany^thisnumber)*e^(-thismany)/fact(thisnumber)", "templateType": "anything", "group": "Ungrouped variables", "name": "tprob1", "description": ""}, "tprob2": {"definition": "if(number1=2,e^(-thismany)*(1+thismany),e^(-thismany)*(1+thismany+thismany^2/2))", "templateType": "anything", "group": "Ungrouped variables", "name": "tprob2", "description": ""}, "prob2": {"definition": "precround(tprob2,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob2", "description": ""}, "thisnumber": {"definition": "if(thismany<8,thismany-1, random(3..7))", "templateType": "anything", "group": "Ungrouped variables", "name": "thisnumber", "description": ""}, "thisaswell": {"definition": "\"a randomly selected employee receives a warning.\"", "templateType": "anything", "group": "Ungrouped variables", "name": "thisaswell", "description": ""}, "sd": {"definition": "precround(sqrt(thismany),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "sd", "description": ""}}, "metadata": {"notes": "\n \t\t

31/12/2012:

\n \t\t

Can be configured to other applications using the string variables supplied. Hence added tag sc.

\n \t\t

Not as yet properly tested.

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

Application of the Poisson distribution given expected number of events per interval.

\n \t\t

Finding probabilities using the Poisson distribution.

\n \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "BS3.3", "extensions": [], "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": ["pre", "descx1", "something", "thisnumber", "what", "things", "descx", "tol", "prob", "thisaswell", "else", "thismany", "number1", "post", "prob2", "prob1", "thatnumber", "this", "v", "tprob1", "tprob2", "sd"], "tags": ["Binomial Distribution", "Binomial distribution", "Probability", "binomial distribution", "expectation", "expected number", "probabilities", "probability", "sc", "standard deviation", "statistical distributions", "statistics"], "preamble": {"css": "", "js": ""}, "advice": "

#### a)

\n

1. $X \\sim \\operatorname{bin}(\\var{number1},\\var{prob})$, so $n= \\var{number1},\\;\\;p=\\var{prob}$.

\n

2. The expectation is given by $\\operatorname{E}[X]=n\\times p=\\var{number1}\\times \\var{prob}=\\var{number1*prob}$

\n

3. $\\operatorname{stdev}(X)=\\sqrt{n\\times p \\times (1-p)}=\\sqrt{\\var{number1}\\times \\var{prob} \\times \\var{1-prob}}=\\var{sd}$ to 3 decimal places.

\n

#### b)

\n

1. \$\\begin{eqnarray*}\\operatorname{P}(X = \\var{thisnumber}) &=& \\dbinom{\\var{number1}}{\\var{thisnumber}}\\times\\var{prob}^{\\var{thisnumber}}\\times(1-\\var{prob})^{\\var{number1-thisnumber}}\\\\& =& \\var{comb(number1,thisnumber)} \\times\\var{prob}^{\\var{thisnumber}}\\times\\var{1-prob}^{\\var{number1-thisnumber}}\\\\&=&\\var{prob1}\\end{eqnarray*} \$ to 3 decimal places.

\n

2.

\n

\$\\begin{eqnarray*}\\operatorname{P}(X \\leq \\var{thatnumber})& =& \\simplify[all,!collectNumbers]{P(X = 0) + P(X = 1) + {v}*P(X = 2)}\\\\& =& \\simplify[zeroFactor,zeroTerm,unitFactor]{{1 -prob} ^ {number1}+ {number1} *{prob} *{1 -prob} ^ {number1 -1} + {v} * ({number1} * {number1 -1}/2)* {prob} ^ 2 *( {1 -prob} ^ {number1 -2})}\\\\& =& \\var{prob2}\\end{eqnarray*} \$

\n

to 3 decimal places.

\n

", "rulesets": {}, "parts": [{"prompt": "

Assuming a binomial distribution for $X$ , {descX}, write down the values of $n$ and $p$.

\n

$X \\sim \\operatorname{bin}(n,p)$

\n

$n=$ [[0]]        $p=$ [[1]]

\n

Find $\\operatorname{E}[X]$ the expected {descX1}

\n

$\\operatorname{E}[X]=$ [[2]]

\n

Find the standard deviation for the {descX1}

\n

Standard deviation = [[3]] (to 3 decimal places).

", "marks": 0, "gaps": [{"allowFractions": false, "marks": 0.25, "maxValue": "number1", "minValue": "number1", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.25, "maxValue": "prob", "minValue": "prob", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 0.5, "maxValue": "number1*thismany/100", "minValue": "number1*thismany/100", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "sd+tol", "minValue": "sd-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "

Find the probability that {this} $\\var{thisnumber}$ {things}

\n

$\\operatorname{P}(X=\\var{thisnumber})=$ [[0]] (to 3 decimal places).

\n

\n

Find the probability that {thisaswell} {thatnumber} {things}

\n

Probability = [[1]] (to 3 decimal places).

", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "prob1+tol", "minValue": "prob1-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "prob2+tol", "minValue": "prob2-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\n

{pre} $\\var{thismany}$ {post}

\n

{something} $\\var{number1}$ {else}

\n

\n \n ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"pre": {"definition": "' '", "templateType": "anything", "group": "Ungrouped variables", "name": "pre", "description": ""}, "descx1": {"definition": "\"number of chocolate chip cookies in our sample:\"", "templateType": "anything", "group": "Ungrouped variables", "name": "descx1", "description": ""}, "something": {"definition": "''", "templateType": "anything", "group": "Ungrouped variables", "name": "something", "description": ""}, "thisnumber": {"definition": "if(number1<6,random(2..3), if(number1<8,random(2..4),random(3..6)))", "templateType": "anything", "group": "Ungrouped variables", "name": "thisnumber", "description": ""}, "what": {"definition": "\"daily sales.\"", "templateType": "anything", "group": "Ungrouped variables", "name": "what", "description": ""}, "things": {"definition": "\"chocolate chip cookies.\"", "templateType": "anything", "group": "Ungrouped variables", "name": "things", "description": ""}, "descx": {"definition": "\"the number of chocolate chip cookies\"", "templateType": "anything", "group": "Ungrouped variables", "name": "descx", "description": ""}, "tol": {"definition": "0.001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "prob": {"definition": "thismany/100", "templateType": "anything", "group": "Ungrouped variables", "name": "prob", "description": ""}, "thisaswell": {"definition": "\"our selection contains no more than \"", "templateType": "anything", "group": "Ungrouped variables", "name": "thisaswell", "description": ""}, "else": {"definition": "\"biscuits are selected at random.\"", "templateType": "anything", "group": "Ungrouped variables", "name": "else", "description": ""}, "thismany": {"definition": "random(15..20)", "templateType": "anything", "group": "Ungrouped variables", "name": "thismany", "description": ""}, "number1": {"definition": "random(5..12)", "templateType": "anything", "group": "Ungrouped variables", "name": "number1", "description": ""}, "post": {"definition": "\"% of biscuits made by a baker are chocolate chip cookies.\"", "templateType": "anything", "group": "Ungrouped variables", "name": "post", "description": ""}, "prob2": {"definition": "precround(tprob2,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob2", "description": ""}, "prob1": {"definition": "precround(tprob1,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob1", "description": ""}, "thatnumber": {"definition": "random(1,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "thatnumber", "description": ""}, "this": {"definition": "\"our selection contains exactly \"", "templateType": "anything", "group": "Ungrouped variables", "name": "this", "description": ""}, "v": {"definition": "if(thatnumber=1,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "v", "description": ""}, "tprob1": {"definition": "comb(number1,thisnumber)*prob^thisnumber*(1-prob)^(number1-thisnumber)", "templateType": "anything", "group": "Ungrouped variables", "name": "tprob1", "description": ""}, "tprob2": {"definition": "if(thatnumber=2,(1-prob)^number1+number1*prob*(1-prob)^(number1-1)+number1*(number1-1)*prob^2*(1-prob)^(number1-2)/2,(1-prob)^number1+number1*prob*(1-prob)^(number1-1))", "templateType": "anything", "group": "Ungrouped variables", "name": "tprob2", "description": ""}, "sd": {"definition": "precround(sqrt(number1*prob*(1-prob)),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "sd", "description": ""}}, "metadata": {"notes": "\n \t\t \t\t \t\t

31/12/2012:

\n \t\t \t\t \t\t

Can be configured to other applications using the string variables supplied. Hence added tag sc.

\n \t\t \t\t \t\t

Not as yet properly tested.

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

Application of the binomial distribution given probabilities of success of an event.

\n \t\t \t\t

Finding probabilities using the binomial distribution.

\n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "BS3.4", "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": ["units1", "upper", "lower", "p1", "m", "amount", "p", "s", "stuff", "tol", "prob2", "prob1"], "tags": ["Normal distribution", "continuous random variable", "mean", "normal distribution", "normal tables", "probabilities", "random variable", "sc", "standard deviation", "statistical distributions", "statistics", "z-scores"], "preamble": {"css": "", "js": ""}, "advice": "\n

1. Converting to $\\operatorname{N}(0,1)$

\n

$\\simplify[all,!collectNumbers]{P(X < {lower}) = P(Z < ({lower} -{m}) / {s}) = P(Z < {lower-m}/{s}) = 1 -P(Z < {m-lower}/{s})} = 1 -\\var{p} = \\var{precround(1 -p,4)}$ to 4 decimal places.

\n

2. Converting to $\\operatorname{N}(0,1)$

\n

$\\simplify[all,!collectNumbers]{P(X > {upper}) = P(Z > ({upper} -{m}) / {s}) = P(Z > {upper-m}/{s}) = 1 -P(Z < {upper-m}/{s})} = 1-\\var{p1} = \\var{precround(1 -p1,4)}$ to 4 decimal places.

\n ", "rulesets": {}, "parts": [{"prompt": "

Find the probability that in a particular week the {amount} is less than {lower} {units1}:

\n

Probability = [[0]] (to 4  decimal places)

\n

Find the probability that in a particular week the {amount} is greater than {upper} {units1}:

\n

Probability = [[1]] (to 4  decimal places)

", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "prob1+tol", "minValue": "prob1-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "prob2+tol", "minValue": "prob2-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\n

The {amount}, $X$, of {stuff}  is normally distributed with mean {m}k and standard deviation {s}{units1}.

\n

i.e.   \$X \\sim \\operatorname{N}(\\var{m},\\var{s}^2)\$

\n

\n ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"units1": {"definition": "\"k Wh\"", "templateType": "anything", "group": "Ungrouped variables", "name": "units1", "description": ""}, "upper": {"definition": "random(m+0.5s..m+1.5*s#5)", "templateType": "anything", "group": "Ungrouped variables", "name": "upper", "description": ""}, "lower": {"definition": "random(m-1.5*s..m-0.5s#5)", "templateType": "anything", "group": "Ungrouped variables", "name": "lower", "description": ""}, "p1": {"definition": "normalcdf((upper-m)/s,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "p1", "description": ""}, "m": {"definition": "random(750..1250#50)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "s": {"definition": "random(60..100#10)", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": ""}, "p": {"definition": "normalcdf((m-lower)/s,0,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "amount": {"definition": "\"electricity consumption\"", "templateType": "anything", "group": "Ungrouped variables", "name": "amount", "description": ""}, "stuff": {"definition": "\"a frozen foods warehouse each week in the summer months \"", "templateType": "anything", "group": "Ungrouped variables", "name": "stuff", "description": ""}, "tol": {"definition": "0.0001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}, "prob2": {"definition": "precround(1-normalcdf(upper,m,s),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob2", "description": ""}, "prob1": {"definition": "precround(normalcdf(lower,m,s),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "prob1", "description": ""}}, "metadata": {"notes": "\n \t\t

1/1/2012:

\n \t\t

Can be configured to other applications using the string variables suppplied. Included tag sc.

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

Given a random variable $X$  normally distributed as $\\operatorname{N}(m,\\sigma^2)$ find probabilities $P(X \\gt a),\\; a \\gt m;\\;\\;P(X \\lt b),\\;b \\lt m$.

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "BS3.5", "extensions": [], "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": ["upper", "lower", "ans1", "ans2", "ans3", "thisdis", "t", "tol"], "tags": ["continuous distributions", "expectation", "probabilities", "sc", "statistical distributions", "statistics", "uniform distribution", "uniformly distributed", "variance"], "preamble": {"css": "", "js": ""}, "advice": "

#### a)

\n

For a uniform distribution \$X \\sim \\operatorname{U}(\\var{lower},\\var{upper})\$ we have:

\n

$\\displaystyle \\operatorname{E}[X] = \\frac{\\var{lower}+\\var{upper}}{2}=\\var{ans1}$m

\n

$\\displaystyle \\operatorname{Var}[X] = \\frac{(\\var{upper}-\\var{lower})^2}{12}=\\frac{(\\var{upper-lower})^2}{12}=\\var{ans2}$ to 3 decimal places.

\n

#### b)

\n

$\\displaystyle P(X \\le \\var{thisdis}\\textrm{km})=\\frac{\\var{thisdis}\\times 1000 -\\var{lower}}{\\var{upper}-\\var{lower}}=\\var{ans3}$ to 3 decimal places.

", "rulesets": {}, "parts": [{"prompt": "

Find $\\operatorname{E}[X]$, the expected distance in metres of the new supermarket from the town centre:

\n

$\\operatorname{E}[X]=$ [[0]]m (to 3 decimal places).

\n

Also find the variance $\\operatorname{Var}(X)$:

\n

$\\operatorname{Var}(X)=$ [[1]] (to 3 decimal places).

\n

", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "ans1", "minValue": "ans1", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "ans2+tol", "minValue": "ans2-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "

Find the probability that the supermarket opens within $\\var{thisdis}$ kilometres of the town centre.

\n

$P(X \\le \\var{thisdis}\\textrm{km})=$ [[0]] (to 3 decimal places).

", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "ans3+tol", "minValue": "ans3-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\n

A new supermarket plans to open somewhere on the outskirts of a town. In fact, $X$, the distance of a new supermarket from the town centre is Uniformly distributed between $\\var{lower}$ metres and $\\var{upper}$ metres i.e.

\n

\$X \\sim \\operatorname{U}(\\var{lower},\\var{upper})\$

\n ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"upper": {"definition": "lower+random(300..500#50)", "templateType": "anything", "group": "Ungrouped variables", "name": "upper", "description": ""}, "lower": {"definition": "random(500..1000#50)", "templateType": "anything", "group": "Ungrouped variables", "name": "lower", "description": ""}, "ans1": {"definition": "(upper+lower)/2", "templateType": "anything", "group": "Ungrouped variables", "name": "ans1", "description": ""}, "ans2": {"definition": "precround((upper-lower)^2/12,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans2", "description": ""}, "ans3": {"definition": "precround((thisdis*1000-lower)/(upper-lower),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans3", "description": ""}, "thisdis": {"definition": "precround((t*lower+(100-t)*upper)/100000,2)", "templateType": "anything", "group": "Ungrouped variables", "name": "thisdis", "description": ""}, "t": {"definition": "random(20..80)", "templateType": "anything", "group": "Ungrouped variables", "name": "t", "description": ""}, "tol": {"definition": "0.001", "templateType": "anything", "group": "Ungrouped variables", "name": "tol", "description": ""}}, "metadata": {"notes": "

1/01/2013:

\n

Although this application is fixed, it could be made into a \"scenario\" based question by introducing string variables, so added tag sc.

\n

25/01/2013:

\n

Included missed out request to calculate to 3 decimal places.

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

Exercise using a given uniform distribution $X$, calculating the expectation and variance. Also finding $P(X \\le a)$ for a given value $a$.

\n \t\t

\n \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "BS3.6", "extensions": [], "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": ["that", "this", "ans1", "ans2", "ans3", "period", "ra", "tol", "tans3", "thistime"], "tags": ["Probability", "continuous distributions", "distributions", "expectation", "exponential distribution", "probability", "sc", "statistical distributions", "statistics", "variance"], "preamble": {"css": "", "js": ""}, "advice": "

If $X \\sim \\operatorname{exp}(\\lambda)$ then $\\displaystyle \\operatorname{E}[X] =\\frac{1}{\\lambda}$ and  $\\displaystyle \\operatorname{Var}(X)=\\frac{1}{\\lambda^2}$.

\n

Also $P(X \\lt a)=1-e^{-\\lambda a}$.

\n

#### a)

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If $X \\sim \\operatorname{exp}(\\var{ra})$ then:

\n

$\\displaystyle \\operatorname{E}[X] =\\frac{1}{\\lambda}=\\frac{1}{\\var{ra}}=\\var{ans1}$ to 3 decimal places.

\n

$\\displaystyle \\operatorname{Var}(X) =\\frac{1}{\\lambda^2}=\\frac{1}{\\var{ra}^2}=\\var{ans2}$ to 3 decimal places.

\n

#### b)

\n

$P(X \\lt \\var{thistime}) = 1 -(e ^ {-\\var{ ra} \\times \\var{thistime}}) = 1 -(e ^ { -\\var{ra * thistime}}) = \\var{ans3}$ to 3 decimal places.

", "rulesets": {}, "parts": [{"prompt": "

Find $\\operatorname{E}[X]$ between {this}:

\n

$\\operatorname{E}[X]=$ [[0]]{period} (enter as a decimal correct to 3 decimal places).

\n

Find $\\operatorname{Var}(X)$:

\n

$\\operatorname{Var}(X)=$ [[1]] (enter as a decimal correct to 3 decimal places).

", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "ans1+tol", "minValue": "ans1-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "ans2+tol", "minValue": "ans2-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "

Find the probability that the time between {that} is less than $\\var{thistime}$ {period}:

\n

$P(X \\lt \\var{thistime})=$ [[0]](enter as a decimal correct to 3 decimal places)

", "marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "maxValue": "ans3+tol", "minValue": "ans3-tol", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\n

The time,  in {period} between {this} follows an exponential distribution with rate $\\var{ra}$ i.e.

\n

\$X \\sim \\operatorname{exp}(\\var{ra})\$

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1/01/2013:

\n \t\t \t\t

This question can be changed to other applications via string variables. Added tag sc.

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

Question on the exponential distribution involving a time intervals and arrivals application, finding expectation and variance. Also finding the probability that a time interval between arrivals is less than a given period. All parameters and times randomised.

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