Applied Statistics and Probability for Engineers

5-13

(a) Show that the moment generating function is

(b) Use MX(t) to find the mean and variance of X.

S5-16. The chi-squared random variable with kdegrees of

freedom has moment generating function MX(t) (1 2 t)k^2.

Suppose that X 1 and X 2 are independent chi-squared random

variables with k 1 and k 2 degrees of freedom, respectively.

What is the distribution of YX 1 X 2?

S5-17. A continuous random variable Xhas the following

probability distribution:

(a) Find the moment generating function for X.

(b) Find the mean and variance of X.

S5-18. The continuous uniform random variable Xhas den-

sity function

(a) Show that the moment generating function is

(b) Use MX(t) to find the mean and variance of X.

S5-19. A random variable Xhas the exponential distribution

(a) Show that the moment generating function of Xis

(b) Find the mean and variance of X.

S5-20. A random variable Xhas the gamma distribution

MX 1 t 2 a 1 

t



b

 1

f 1 x 2 ex, x
0

MX 1 t 2 

et^ et^ 

t 1  2

f 1 x 2 

1



, x

f 1 x 2  4 xe^2 x, x
0

MX 1 t 2 

pe t

1  11 p 2 e t (a) Show that the moment generating function of Xis

(b) Find the mean and variance of X.

S5-21. Let X 1 , X 2 ,... , Xrbe independent exponential ran-

dom variables with parameter .

(a) Find the moment generating function of YX 1 

X 2 Xr.

(b) What is the distribution of the random variable Y?

[Hint:Use the results of Exercise S5-20].

S5-22. Suppose that Xihas a normal distribution with mean

iand variance Let X 1 and X 2 be independent.

(a) Find the moment generating function of YX 1 X 2.

(b) What is the distribution of the random variable Y?

S5-23. Show that the moment generating function of the

chi-squared random variable with kdegrees of freedom is

MX(t) (1  2 t)k^2. Show that the mean and variance of this

random variable are kand 2k, respectively.

S5-24. Continuation of Exercise S5-20.

(a) Show that by expanding etXin a power series and taking

expectations term by term we may write the moment gen-

erating function as

Thus, the coefficient of trr! in this expansion is the rth

origin moment.

(b) Continuation of Exercise S5-20. Write the power series

expansion for MX(t), the gamma random variable.

(c) Continuation of Exercise S5-20. Find and using the

results of parts (a) and (b). Does this approach give the

same answers that you found for the mean and variance of

the gamma random variable in Exercise S5-20?

¿ 1 ¿ 2

¿r ,

¿r

t r

r!

p

 1 ¿ 1 t¿ 2

t 2

2!

p

MX 1 t 2 E 1 etX 2

^2 i, i1, 2.

p

MX 1 t 2 a 1 

t



b

r

f 1 x 2 



 1 r 2

1  x 2 r^1 ex, x
0

5-10 CHEBYSHEV’S INEQUALITY (CD ONLY)

In Chapter 3 we showed that if Xis 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.

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