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
fX 1
x 1
1
2
; for
1 x 1 1;
0 ; elsewhere;
8
<
:
fT 1
t 1
2e^2 t^1 ; fort 1 0 ;
0 ; elsewhere;
fT 2
t 2
2e^2 t^2 ; fort 2 0 ;
0 ; elsewhere:
^2 T
pX
k
n
k
pk
1
pn^ k; k 0 ; 1 ; 2 ;...;n:
fY
ypfY 1
yqfY 2
y;
fX 2
x 2
1
2 ^1 =^2
exp
x^22
2
;
1 <x 2 < 1
: