Fundamentals of Probability and Statistics for Engineers

Suppose that voltage source v in the circuit is a deterministic constant.

(a) Find the pdf of current I, where I v/R, passing through the circuit.

(b) Find the pdf of power W, where W I^2 R, dissipated in the resistor.

5.16 The independent random variables X 1 and X 2 are uniformly and identically
distributed, with pdfs

and similarly for X 2 .LetY X 1 X 2.

(a) Determine the pdf of Y by using Equation (5.56).

(b) Determine the pdf of Y by using the method of characteristic functions devel-

oped in Section 4.5.

5.17 Two random variables, T 1 and T 2 , are independent and exponentially distributed
according to

(a) Determine the pdf of T T 1 T 2.

(b) Determine mT and

5.18 A discrete random variable X has a binomial distribution with parameters (n,p). Its
probability mass function (pmf) has the form

Show that, if X 1 and X 2 are independent and have binomial distributions with

parameters (n 1 ,p)and(n 2 , p), respectively, the sum Y X 1 X 2 has a binomial

distribution with parameters (n 1 n 2 ,p).

5.19 Consider the sum of two independent random variables X 1 and X 2 where X 1 is
discrete, taking values a and b with probabilities P(X 1 a) p, and P(X 1 b)
q (p q 1), and X 2 is continuous with pdf fX 2 (x 2 ).
(a) Show that Y X 1 X 2 is a continuous random variable with pdf

where fY 1 (y) and fY 2 (y) are, respectively, the pdfs of Y 1 aX 2 ,andY 2 bX 2

at y.

(b) Plot fY (y) by letting a 0, b 1, p^13 ,q^23 ,and

Functions of Random Variables 157

ˆ

ˆ

fX 1 …x 1 †ˆ

1

2

; for 

1 x 1 1;

0 ; elsewhere;

8

<

:

ˆ‡

fT 1 …t 1 †ˆ

2e^2 t^1 ; fort 1  0 ;

0 ; elsewhere;



fT 2 …t 2 †ˆ

2e^2 t^2 ; fort 2  0 ;

0 ; elsewhere:



ˆ

^2 T

pX…k†ˆ

n

k



pk… 1

 p†n^ k; kˆ 0 ; 1 ; 2 ;...;n:

ˆ‡

‡

ˆˆ ˆˆ

‡ˆ

ˆ‡

fY…y†ˆpfY 1 …y†‡qfY 2 …y†;

ˆ‡ ˆ‡

ˆˆˆˆ

fX 2 …x 2 †ˆ

1

… 2 †^1 =^2

exp

x^22

2



; 

1 <x 2 < 1

: