Question

A random variable x assumes values 1, 2, 3, ... n with equal probabilities. If the ratio of variance of x to expected value of x is equal to 4, then the value of n is

• A. 50
• B. 30
• C. 25
• D. 35

Answer 1

Answer: (C)

25

Answer 2

For a discrete uniform distribution on {1, 2, …, n}:

$$E(X)=\frac{n+1}{2}, \quad Var(X)=\frac{n^2-1}{12}$$

The ratio is:

$$\frac{Var(X)}{E(X)} = \frac{\frac{n^2-1}{12}}{\frac{n+1}{2}} = \frac{n^2-1}{6(n+1)} = \frac{(n-1)(n+1)}{6(n+1)} = \frac{n-1}{6}$$

Setting the ratio equal to 4:

$$\frac{n-1}{6}=4 \quad \Rightarrow \quad n-1=24 \quad \Rightarrow \quad n=25$$