Page 174 - Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 157

Suppose that voltage source v in the circuit is a deterministic constant.
(a) Find the pdf of current I, where I ˆ v/R, passing through the circuit.
2
(b) Find the pdf of power W , where W ˆ I R, dissipated in the resistor.
5.16 The independent random variables X 1 and X 2 are uniformly and identically
distributed, with pdfs
8
1
< ; for
1 x 1 1;
f x 1 †ˆ 2
X 1
0; elsewhere;
:
and similarly for X 2 . Let Y ˆ 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

2e
2t 1 ; for t 1 0;
f t 1 †ˆ
T 1
0; elsewhere;

2e
2t 2 ; for t 2 0;
f t 2 †ˆ
T 2
0; elsewhere:
(a) Determine the pdf of T ˆ T 1
T 2 .
2
(b) Determine m T and :
T
5.18 A discrete random variable X has a binomial distribution with parameters (n, p). Its
probability mass function (pmf) has the form
n n
k

k
p k†ˆ p 1
p† ; k ˆ 0; 1; 2; ... ; n:
X
k
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 ˆ 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 ˆ a) ˆ p, and P(X 1 ˆ b) ˆ
q
q (p ‡ ˆ 1), and X 2 is continuous with pdf f X 2 (x 2 ).
(a) Show that Y ˆ X 1 ‡ X 2 is a continuous random variable with pdf
f y†ˆ pf Y 1 y†‡ qf Y 2 y†;
Y
where f (y) and f (y) are, respectively, the pdfs of Y 1 ˆ a ‡ X 2 , and Y 2 ˆ b ‡
Y 1 Y 2 X 2
at y.
(b) Plot f (y) by letting a ˆ 0, b ˆ 1, p ˆ 1 3 , q ˆ 2 3 ,and
Y
1
x 2
f x 2 †ˆ exp 2 ;
1 < x 2 < 1
X 2 1=2 2
2 †

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