Compute $\\operatorname{P}(Y=\\var{y1})= $ [[0]]
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", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{p}", "minValue": "{p}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{p}", "minValue": "{p}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Find:
\nEnter your answers to the following questions to $3$ decimal places.
\nSuppose $Y \\sim \\operatorname{Poisson}(\\var{p})$.
", "tags": ["checked2015", "cr1", "discrete random variable", "distributions", "expectation of poisson distribution", "MAS1604", "MAS2304", "poisson distribution", "Poisson distribution", "Probability", "probability", "random variables", "statistics", "tested1", "variance of poisson distribution"], "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": "Numbas.jme.display.texOps['prob'] = function(thing,texArgs) {\n return '\\\\operatorname{P}\\\\left( '+texArgs.join(', ')+' \\\\right)';\n}"}, "type": "question", "metadata": {"notes": "7/07/2012:
\nAdded tags.
\nCannot access the poisson functions in the stats extension and will not run. Issue posted.
\nSet new tolerance variable tol=0.001 for first two questions.
\nCannot check calculations at present under Test Run.
\n22/07/2012:
\nAdded description.
\nCorrected typo in formula for Poisson Distribution.
\nCan now test as stats extension ticked.
\nChecked calculation.
\n31/07/2012:
\nQuestion appears to be working correctly.
\n20/12/2012:
\nChecked calculations agains standard tables, OK. Rounding, OK. Added cr1 tag. Do not use the jstats poissoncdf function! The poissonpdf is OK.
\nAdded tested1 tag.
", "licence": "Creative Commons Attribution 4.0 International", "description": "$Y \\sim \\operatorname{Poisson}(p)$. Find $P(Y=a)$, $P(Y \\gt b)$, $E[Y],\\;\\operatorname{Var}(Y)$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "If $Y \\sim \\operatorname{Poisson}(\\lambda)$ then
\n\\[P(Y=y)=\\frac{\\lambda^y\\;e^{-\\lambda}}{y!}\\]
\nHence
\n\\[ \\operatorname{P}(Y=\\var{y1}) = \\simplify[std,!otherNumbers]{({p}^{y1}*e^{-p}) / {y1}!} = \\var{tans} \\approx \\var{ans} \\]
\nto 3 decimal places.
\n\\begin{align}
\\operatorname{P} (Y \\gt \\var{y2}) &= 1-P(Y \\le \\var{y2}) = \\simplify[std]{ 1-prob(Y = 0) - prob(Y = 1) - prob(Y = 2) - {v3} * prob(Y = 3) - {v4} * prob(Y = 4)} \\\\
&= \\simplify[unitFactor,zeroTerm,zeroFactor]{1 -Exp( -{p}) -({p} * Exp( -{p})) -(({p} ^ 2 * Exp( -{p})) / 2) -{ v3} * (({p} ^ 3 * Exp( -{p})) / 6) -{ v4} * (({p} ^ 4 * Exp( -{p})) / 24)} \\\\
&= \\var{tans2} \\\\
&\\approx \\var{ans2}
\\end{align}
to 3 decimal places.
\nFor the Poisson distribution $\\operatorname{Poisson}(\\lambda)$ we have:
\n\\[ \\operatorname{E}[Y] = \\operatorname{Var}(Y) = \\lambda \\]
\nHence in this case:
\n\\begin{align}
\\operatorname{E}[Y] &= \\var{p} \\\\
\\operatorname{Var}(Y) &= \\var{p}
\\end{align}