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rebelmaths

Printing errors in the work produced by a particular film occur randomly at an average rate of p per page.

i.What is the probability that a one page document will contain x1 printing error(s)?

ii.If a n page document is printed, calculate the probability of having more than x2 errors. Assume a Poisson distribution.

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Please give your answer to at least 3 decimal places.

Printing errors in the work produced by a particular film occur randomly at an average rate of $\\var{l}$ per page.

The Poisson distribution formula: $P(r)=\\frac{\\lambda^re^{-\\lambda}}{r!}$ or $P(r)=e^{-\\lambda}\\left[\\frac{\\lambda^r}{r!}\\right]$

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Part (a)

Remember that for a Poisson random variable:
\\begin{align}
\\operatorname{P}(X=x)&=\\dfrac{\\lambda^x\\times e^{-\\lambda}}{x!}\\\\
\\end{align}

1.\\[ \\begin{eqnarray*}\\operatorname{P}(X = \\var{x}) &=& \\frac{\\var{l} ^ {\\var{x}}e ^ { -\\var{l}}} {\\var{x}!}\\\\& =& \\var{answer1} \\end{eqnarray*} \\] to 3 decimal places.

Part (b)

2. For a $\\var{n}$ page document, $\\lambda=\\var{n} \\times \\var{l} = \\var{l2}$

The probability of having less than $\\var{y}$ errors is given by:

$P(X < \\var{y}) = P(X=0) + P(X=1) +P(X=2)$

where

$P(X=0) =\\frac{\\var{l2}^{0}e^{-\\var{l2}}}{0!}=\\var{pr0}$

$P(X=1) =\\frac{\\var{l2}^{1}e^{-\\var{l2}}}{1!}=\\var{pr1}$

$P(X=1) =\\frac{\\var{l2}^{2}e^{-\\var{l2}}}{2!}=\\var{pr2}$

Hence

$P(X < \\var{y})$ = $\\var{pr0}+\\var{pr1}+\\var{pr2}=\\var{answer2}$

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What is the probability that a one page document will contain exactly $\\var{x}$ printing error(s)?

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If a $\\var{n}$ page document is printed, calculate the probability of having less than $\\var{y}$ errors.

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