Question

If X is a continuous random variable with pdf f(x) on the interval [a,b] then show that a<E(X)<b.

Answer 1

Answer:

Given that:

If X is a continuous random variable with pdf f(x) on the interval [a,b] then show that a<E(X)<b.

From the Properties of expectaion of a random variable, we know that if X & Y be two random variables follows the same pdf or distribution. Then,

(I) if X >Y ⇒ E(X) > E(Y) and

(ii) if X < Y ⇒ E(X) < E(Y)

Let us consider X> a ⇒ E(X) > E(a)=a---------(1)

(since, expection of a constant is constant)

Let us consider X < b ⇒ E(X) < E(b)=b---------(2)

From equation (1) & (2), one can say that a < E(X) < b