Answer to Question #204613 in Statistics and Probability for ahmed anwar

Question

The number of people arriving at a post office in a small town follows a Poisson distribution. On average, five people arrive at the post office in one minute. Find the probability that in a given minute, 

A) more than two people arrive at the post office

B) less than eleven people arrive at the post office

c) between six and ten people arrive at the post office

Answer 1

Let $X=$ the number of people arriving at a post office: $X\sim Po(\lambda).$

$$P(X=x)=\dfrac{e^{-\lambda}\cdot\lambda^x}{x!}$$

Given $\lambda=5$

A)

$$P(X>2)=1-P(X=0)-P(X=1)-P(X=2)$$

$$=1-\dfrac{e^{-5}\cdot5^0}{0!}-\dfrac{e^{-5}\cdot5^1}{1!}-\dfrac{e^{-5}\cdot5^2}{2!}$$

$$\approx0.875348$$

B)

$$P(X<11)=P(X=0)-P(X=1)-P(X=2)$$

$$+P(X=3)+P(X=4)+P(X=5)$$

$$+P(X=6)+P(X=7)+P(X=8)$$

$$+P(X=9)+P(X=10)$$

$$\approx0.986305$$

C)

$$P(6\leq X\leq10)=P(X=6)+P(X=7)$$

$$+P(X=8)+P(X=9)+P(X=10)$$

$$\approx0.370344$$