Page 62 - Probability, Random Variables and Random Processes

RANDOM VARIABLES [CHAP 2

Fig. 2-15

2.13. (a) Verify that the function p(x) defined by
x =o, 1, 2, ...
otherwise
is a pmf of a discrete r.v. X.
(b) Find (i) P(X = 2), (ii) P(X I 2), (iii) P(X 2 1).
(a) It is clear that 0 5 p(x) < 1 and

Thus, p(x) satisfies all properties of the pmf [Eqs. (2.15) to (2.17)] of a discrete r.v. X.
(b) (i) By definition (2.14),
P(X = 2) = p(2) = $($)2 =
(ii) By Eq. (2.1 8),

(iii) By Eq. (l.25),

2.14. Consider the experiment of tossing an honest coin repeatedly (Prob. 1.35). Let the r.v. X denote
the number of tosses required until the first head appears.
(a) Find and sketch the pmf p,(x) and the cdf F,(x) of X.
(b) Find (i) P(l < X s 4), (ii) P(X > 4).
(a) From the result of Prob. 1.35, the pmf of X is given by

Then by Eq. (2.1 8),