// Numbas version: exam_results_page_options {"name": "Maria's copy of Binomial (practice of formula)", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Maria's copy of Binomial (practice of formula)", "metadata": {"description": "

$X \\sim \\operatorname{Binomial}(n,p)$. Find $P(X=a)$, $P(X \\leq b)$, $E[X],\\;\\operatorname{Var}(X)$.

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rebelmaths

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Compute $P(X=\\var{x1})=\\;\\;$[[0]] (to 3 decimal places).

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Compute $P(X\\le\\var{x2})=\\;\\;$[[0]] (to 3 decimal places).

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Answer the following questions on the Binomial Distribution.

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Suppose \$X \\sim \\operatorname{Binomial}(\\var{n},\\var{p}),\$

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that is $n=\\var{n}$ and $p=\\var{p}$.

a)
\$\\simplify[std,!otherNumbers]{P(X = {x1}) = {n}! / ({n -x1}! * {x1}!) * {p} ^ {x1} * (1 -{p}) ^ {n -x1}} = \\var{ans1}\$

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to 3 decimal places.

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b)

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We have:

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\$\\begin{eqnarray*} F_X (\\var{x2}) &=& P(X \\le \\var{x2}) =\\simplify[std]{ P(X = 0) + P(X = 1) + P(X = 2) + {v3} * P(X = 3) + {v4} * P(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{eqnarray*} \$
to 3 decimal places.

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", "showQuestionGroupNames": false, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}