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5-13

(a) Show that the moment generating function is r 1 x
f 1x2 1 x2 e , x
0
1r2
t
pe
M X 1t2
t
1 11 p2e (a) Show that the moment generating function of X is
(b) Use M X (t) to find the mean and variance of X. t r
M X 1t2 a1 b
S5-16. The chi-squared random variable with k degrees of
freedom has moment generating function M X (t) (1 2t) k 2 .
Suppose that X 1 and X 2 are independent chi-squared random (b) Find the mean and variance of X.
variables with k 1 and k 2 degrees of freedom, respectively.
S5-21. Let X 1 , X 2 , . . . , X r be independent exponential ran-
What is the distribution of Y X 1 X 2 ?
dom variables with parameter .
S5-17. A continuous random variable X has the following (a) Find the moment generating function of Y X 1
probability distribution: X 2 p X r .
(b) What is the distribution of the random variable Y?
f 1x2 4xe 2x , x
0 [Hint: Use the results of Exercise S5-20].
S5-22. Suppose that X i has a normal distribution with mean
2
(a) Find the moment generating function for X. i and variance i , i 1, 2. Let X 1 and X 2 be independent.
(b) Find the mean and variance of X. (a) Find the moment generating function of Y X 1 X 2 .
S5-18. The continuous uniform random variable X has den- (b) What is the distribution of the random variable Y?
sity function S5-23. Show that the moment generating function of the
chi-squared random variable with k degrees of freedom is
1 M X (t) (1 2t) k 2 . Show that the mean and variance of this
f 1x2 , x
random variable are k and 2k, respectively.
S5-24. Continuation of Exercise S5-20.
tX
(a) Show that the moment generating function is (a) Show that by expanding e in a power series and taking
expectations term by term we may write the moment gen-
e t e t erating function as
M X 1t 2
t1 2
tX
M X 1t 2 E 1e 2
2
(b) Use M X (t) to find the mean and variance of X. t p
1 ¿ 1 t ¿ 2
S5-19. A random variable X has the exponential distribution r 2!
t p
¿ r
f 1x2 e x , x
0 r!
r
Thus, the coefficient of t r! in this expansion is ¿ r , the rth
(a) Show that the moment generating function of X is
origin moment.
(b) Continuation of Exercise S5-20. Write the power series
t 1
M X 1t 2 a1 b expansion for M X (t), the gamma random variable.

(c) Continuation of Exercise S5-20. Find ¿ 1 and ¿ 2 using the
results of parts (a) and (b). Does this approach give the
(b) Find the mean and variance of X. same answers that you found for the mean and variance of
S5-20. A random variable X has the gamma distribution the gamma random variable in Exercise S5-20?

5-10 CHEBYSHEV’S INEQUALITY (CD ONLY)

In Chapter 3 we showed that if X is a normal random variable with mean and standard
deviation , P( 1.96 < X < 1.96 ) 0.95. This result relates the probability of a
normal random variable to the magnitude of the standard deviation. An interesting, similar re-
sult that applies to any discrete or continuous random variable was developed by the mathe-
matician Chebyshev in 1867.