How to set up problem involving Poisson RV

Question

Consider an example where customers entering a store is a Poisson random variable with $\lambda=15$. How do you find the probability that 100 or fewer people will walk into the store in any five-day period? I have set up the pmf and I know how to use it, but I have no idea where to start with this problem. If $x_i$ represents the number of people entering on each day $i$ then $x_1+x_2+x_3+x_4+x_5\leq100$. Any assistance would be appreciated in setting up this problem.

Answer 1

We assume that the parameter $15$ refers to the number entering in a day.

Let $Y$ be the number of people entering in a $5$ day period. Then $Y$ has Poisson distribution parameter $(5)(15)$.

Now we can find an explicit formula for the probability that $Y\le 100$.

Unfortunately, it is a long sum.

For an approximation, we can, since the parameter $\lambda=75$ is large, approximate the probability that $Y\le 100$ by the probability that a normally distributed random variable $W$ with the same mean and variance as $Y$ is $\le 100$. We may want to use a continuity correction: we will likely get a somewhat better approximation by finding $\Pr(W\le 100.5)$.