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