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The number of coins that Josh spots when walking to work is a Poission random variable with mean $6$. Each coin is equally likely to be a penny, a nickel ($5$ cent), a dime ($10$ cent) or a quarter ($25$ cent). Josh ignores the pennies but picks up the other coins. Find the probability that Josh picks up exactly $25$ cents on this way ?

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The numbers of each type of coin found are i.i.d., and each is Poisson with mean $3/2$. If $N$, $D$ and $Q$ are the numbers of nickels, dimes, and quarters, respectively, then Josh picks up exactly 25 cents iff $(N,D,Q)$ is in the set $\{(5,0,0),(3,1,0),(1,2,0),(0,0,1)\}$. Since $$ {\Bbb P}(N=n,D=d,Q=q)=e^{-9/2} \frac{(\frac32)^{n+d+q}}{n! d! q!}, $$ the probability is $$ \frac{5241}{1280} e^{-9/2}\approx 0.045486. $$

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  • $\begingroup$ I don't know the proof of this "The numbers of each type of coin found are i.i.d., and each is Poisson with mean $3/2$". Can you elaborate this ? $\endgroup$ – RIchard Williams Feb 13 '13 at 0:58
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    $\begingroup$ If $X$ is Poisson with parameter $\lambda$, and $(Y_1, \dots, Y_k)$ is multinomial with $X$ trials and outcome probabilities $p_1$, $\dots$, $p_k$, then $Y_1$, $\dots$, $Y_k$ are independent and Poisson with parameters $\lambda p_1$, $\dots$, $\lambda p_k$. To prove this, you can just compute the joint probability that $(Y_1,\dots,Y_k)=(a_1,\dots,a_k)$ in both cases. $\endgroup$ – David Moews Feb 13 '13 at 1:13
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    $\begingroup$ You can also think about a Poisson process with rate $\lambda$ where, after each event occurs, you flip a many-sided coin with probability $p_i$ of landing on side $i$ to place it in one of $k$ subcategories. This is equivalent to $k$ Poisson processes, one for each subcategory, with rates $\lambda p_1$, $\dots$, $\lambda p_k$. $\endgroup$ – David Moews Feb 13 '13 at 1:15
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Hints:

  • the number of each coin is Poisson distributed with parameter $\frac{6}{4}= 1.5$

  • Ignoring the pennies you can turn this into "The number of coins that Josh spots when walking to work is a Poission random variable with mean 4.5. Each coin is equally likely to be a nickel (5 cent), a dime (10 cent) or a quarter (25 cent). Josh picks up the coins. Find the probability that Josh picks up exactly 25 cents on this way ?"

  • Which possible ways are of making 25 cents from nickels, dimes and quarters?

  • What are the probabilities of those being the coins Josh finds?

  • The sum of those probabilities?

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