Question

The number of customers per day at a sales counter, Y , has been observed for a long period of time and found to have mean 20 and standard deviation 2. The probability distribution of Y is not known. What can be said about the probability that, tomorrow, Y will be greater than 16 but less than 24?

Answer 1

Solution

Let Y : number of customer per day. 

$\implies E(Y)=20\hspace{2mm}, \delta(X)=2$

Probaility that tomorrow $16< x<24$

by using Tchebysheff's theorem,  $k>0$

$\implies P(|X-E(X)|< k\delta)\geq 1-\frac{1}{k^2}$

or  $P(E(X)-k \delta < X < k\delta+E(X))\geq 1-\frac{1}{k^2}$

Choose  $k=2\implies P(16< X< 24)\geq 1-\frac{1}{4}=\frac{3}{4}$

 

Therefore. $\frac{3}{4} \leq P(16< X< 24)\leq 1$