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Suppose $p$ is the number of cars per minute passing through a certain road junction between $5$ PM and $6$ PM, and $p$ has a Poisson distribution with mean $3$. What is the probability of observing fewer than $3$ cars during any given minute in this interval?

1. $\dfrac{8}{(2e^{3})}$
2. $\dfrac{9}{(2e^{3})}$
3. $\dfrac{17}{(2e^{3})}$
4. $\dfrac{26}{(2e^{3})}$

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Poisson Probability Density Function (with mean $\lambda$) =$\dfrac{ \lambda^{k}} { \left(e^{\lambda}k!\right)}$,

We have to sum the probability density function for $k = 0,1$ and $2$ and $\lambda = 3$(thus finding the cumulative mass function)

=$\left(\dfrac{1}{e^{3}}\right) + \left(\dfrac{3}{e^{3}}\right) + \left(\dfrac{9}{2e^{3}}\right)$

=$\dfrac{17}{\left(2e^{3}\right)}$

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As poisson distribution is for discrete random variables, isn't it supposed to be called as "probability mass function" instead of "probability density function"? check this.

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Yes, probability mass function is right.

Let X be a random variable which indicates number of cars per minute.

Given E(X) (i.e) expected number of cars per minute (i.e) average number of cars per minute                                                         (i.e) mean number of cars per minute = 3

probability of number of cars being 0 or 1 or 2 per minute

= P(X = 0) + P(X = 1) + P(X = 2)

= [e-3 (3)0 / 0!] + [e-3 (3)1 / 1!] + [e-3 (3)2 / 2!]

= 17 / 2e

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