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QUESTION

 A new car of a certain model may be assumed to have $X$ minor
   faults where $X$ has a Poisson distribution with mean $\mu$. A
   report is sent to the manufacturer listing the faults for each
   car which has at least one fault. Write down the probability
   function of $Y,$ the number of faults listed on a randomly chosen
   report card and find $E(Y).$ Given $E(Y)=2.5$ find $\mu$ correct to
   1 decimal place.

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ANSWER

$X  \sim P(\mu )\ \ \ \ p(x)=e^{-\mu}\frac{\mu ^x}{x!},\
x=0,1,2\ldots\\
   P(X=0)=p(0)=e^{-\mu }\ \ P(X\neq 0)=1-e^{-\mu}$

   \begin{eqnarray*}
    P(Y=y)&=&P(X=y|X \neq 0)\\&=&\frac{P(X=y \textrm{ and } X\neq
    0)}{P(X \neq 0)}\\
    &=&\frac{P(X=y)}{P(X \neq
    0)}\\&=&\frac{e^{-\mu}\mu^y}{y!(1-e^{-\mu})},\ \ y=1,2,\ldots
   \end{eqnarray*}

   \begin{eqnarray*}
    E(Y)&=&\sum_{y=1}^\infty\frac{ye^{-\mu}\mu^y}{y!(1-e^{-\mu})}\\
    &=&\frac{1}{1-e^{-\mu}}\sum_{y=0}^\infty\frac{e^{-\mu}\mu^yy}{y!}\\
    &=&\frac{\mu}{1-e^{-\mu}}\\
   \end{eqnarray*}

   Since the summation is $E(x).$
   Given $E(Y)=2.5=\frac{\mu}{1-e^{-\mu}}$, we know that $\mu<2.5$
   and we can use trial and error to obtain $\mu \approx 2.2$

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