Question

f(x) = 3e −3x x > 0

0 otherwise.

Find the expected value and variance of the random variable.

Answer 1

Expected Value = 0.3333

Variance = 0.1111

EXPLANATION:

(a)

Expected value is given by:

between the limts 0 to .

Applying limts, we get:

(ii)

                          (1)

Taking the constant out, we get:

Put

u = - 3x.                                              (2)

Applying Integration by Parts, we get:

                     (3)

Substituting (2), equation (3) becomes:

                       (4)

Substituting (4), equation (1) becomes:

,

       between the limits 0 to .

Applying limts, we get:

So,

Var(X) = E(X²) - (E(X))² = 0.2222 - 0.3333² = 0.1111