SECTION 6.3 Parameters and Statistics 371
Next, continuing to assume that X anY are independent random
variables, we proceed to computeE(X+Y). We have
E(X+Y) =
∫∞
−∞x(fX∗fY)(x)dx
=∫∞
−∞x∫∞
−∞fX(t)fY(x−t)dtdx
=∫∞
−∞fX(t)∫∞
−∞xfY(x−t)dtdx
=∫∞
−∞fX(t)∫∞
−∞(t+x)fY(x)dxdt
=∫∞
−∞fX(t)(t+E(Y))dt
= E(X) +E(Y).Exercises
- Compute the mean and the variance of the random variablerand.
(Recall thatrandhas density function
f(x) =
1 if 0≤x≤ 1 ,
0 otherwise.)- Compute the mean and the variance of the random variable
√
rand.
(Recall that√
randhas density functionf(x) =
2 x if 0≤x≤ 1 ,
0 otherwise.)- Compute the mean and the variance of the random variablerand^2.
(See Exercise 7 on page 356.) - Compute the mean and the variance of the random variable having
density function given in Exercise 6 on page 355.