Compute $\\operatorname{P}(X=\\var{x1}) = $ [[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "ans2", "maxValue": "ans2", "precision": "3", "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": false, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "Compute $F_X(\\var{x2}) = \\operatorname{P}(X\\le\\var{x2})=$ [[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{n*p}", "minValue": "{n*p}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "{n*p*(1-p)}", "minValue": "{n*p*(1-p)}", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Find:
\nEnter your answers to the following questions to $3$ decimal places.
\nSuppose $X \\sim \\operatorname{Binomial}(\\var{n},\\var{p})$
", "tags": ["binomial distribution", "Binomial distribution", "Binomial Distribution", "CDF", "cdf", "CDF of binomial distribution", "checked2015", "cr1", "cumulative density function", "Discrete random variables.", "distributions", "Expectation of binomial distribution", "MAS1604", "MAS2304", "probability", "Probability", "random variables", "statistics", "tested1", "variance of binomial 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 stats extension at present, so question does not run. Issue posted.
\nSet new tolerance variable tol=0.001 for first two answers.
\nCalculation to be tested under Test Run.
\n22/07/2012:
\nNow runs after stats extension box ticked.
\nAdded description.
\nChecked calculation.
\n31/07/2012:
\nAdded tags.
\nQuestion appears to be working correctly.
\n20/12/2012:
\nRounding seems to be OK. Added cr1 tag. Replaced sum of pdf values by built in binomialcdf function from jstats.
\nChecked calculation, OK. Added tested1 tag.
", "licence": "Creative Commons Attribution 4.0 International", "description": "$X \\sim \\operatorname{Binomial}(n,p)$. Find $P(X=a)$, $P(X \\leq b)$, $E[X],\\;\\operatorname{Var}(X)$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\\[ \\simplify[std,!otherNumbers]{prob(X = {x1}) = {n}! / ({n -x1}! * {x1}!) * {p} ^ {x1} * (1 -{p}) ^ {n -x1}} = \\var{ans1}\\]
\nto 3 decimal places.
\nWe have:
\n\\begin{align}
F_X (\\var{x2}) &= \\operatorname{P}(X \\le \\var{x2}) = \\simplify[std]{ prob(X = 0) + prob(X = 1) + prob(X = 2) + {v3} * prob(X = 3) + {v4} * prob(X = 4)} \\\\
&= \\simplify[unitFactor,zeroTerm,zeroFactor]{(1 -{p}) ^ {n} + {n} * (1 -{p}) ^ {n -1} * {p} + {(n * (n -1)) / 2} * (1 -{p}) ^ {n -2} * {p} ^ 2 + {v3} * {comb(n , 3)} * (1 -{p}) ^ {n -3} * {p} ^ 3 + {v4} * {comb(n , 4)} * (1 -{p}) ^ {n -4} * {p} ^ 4} \\\\
&= \\var{ans2}
\\end{align}
to 3 decimal places.
\nFor the binomial distribution $\\operatorname{Binomial}(n,p)$ we have:
\n\\begin{align}
\\operatorname{E}[X] &= np \\\\
\\operatorname{Var}(X) &= np(1-p)
\\end{align}
Hence in this case:
\n\\begin{align}
\\operatorname{E}[X] &= \\var{n} \\times \\var{p} = \\var{n*p} \\\\
\\operatorname{Var}(X) &= \\var{n} \\times \\var{p} \\times \\var{(1-p)} = \\var{n*p*(1-p)}
\\end{align}