Q.6.25 - A First Course in Probability

Question

Suppose that 106 people arrive at a service station at times that are independent random variables, each of which is uniformly distributed over (0, 106). Let N denote the number that arrive in the first hour. Find an approximation for P{N = i}. 

Step-by-Step Solution

Answer

The approximation of P {N = i} is $P (N=i)=\frac{1}{i!}{e}^{-1}$

Step 1: Content Introduction

The likelihood or probability of an occasion is the extent of times the occasion occurs out of an enormous number of preliminaries.

Step 2: Content Explanation

The number of people that comes to station within the first hour has exact binomial distribution with parameters $n= {10}^6$ and the probability of success $p=\frac{1}{{10}^6}$. As we have the possible hours are from 0 to 10⁶ and we are looking for first hour $N~ Binom ({10}^6, {10}^{-6})$

But since the size of population is very big and the probability of success is very small, we can use poisson approximation in form $N\approx Pois.({10}^6. {10}^{-6})=Pois \left(1\right)$.

Hence, $P (N=i) =\frac{1}{i!}{e}^{-1}$