Page 159 - Schaum's Outlines - Probability, Random Variables And Random Processes

FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS [CHAP. 4

Hence, Var(X) = E(X2) - [E(X)I2 = A2 + A - A2 = 1

4.48. Let X be an exponential r.v. with parameter 1.
Find the moment generating function of X.
Find the mean and variance of X.
By definition (4.40) and Eq. (2.48),

The first two derivatives of Mx(t) are

Thus, by Eq. (4.42),

2 1
Hence, Var(X) = E(X2) - [E(X)J2 = - -

4.49. Find the moment generating function of the standard normal r.v. X = N(0; 1) and calculate the
first three moments of X.
By definition (4.40) and Eq. (2.52),

Combining the exponents and completing the square, that is,

we obtain

since the integrand is the pdf of N(t; 1).
Differentiating M,(t) with respect to t three times, we have
Mi(t) = tet2I2 Mi(t) = (t2 + l)et2I2 ~~(~)(t)
= (t3 + 3t)et2I2
Thus, by Eq. (4.42),
E(X) = MgO) = 0 E(X2) = MgO) = 1 E(X3) = MX(3)(0) = 0

4.50. Let Y = ax + b. Let Mx(t) be the moment generating function of X. Show that the moment
generating function of Y is given by
My(t) = etbMx(at)
By Eqs. (4.40) and (4.105),