Page 60 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 21 RANDOM VARIABLES

2.11. Find the values of constants a and b such that

is a valid cdf.
To satisfy property 1 of FX(x) [0 I FX(x) 5 11, we must ha.ve 0 5 a 5 1 and b > 0. Since b > 0, pro-
perty 3 of FXfx) [Fx(~) = 1) is satisfied. It is seen that property 4 of FX(x) [F,(-m) = O] is also satisfied.
For 0 5 a I 1 and b > 0, F(x) is sketched in Fig. 2-14. From Fig. 2-14, we see that F(x) is a nondecreasing
function and continuous on the right, and properties 2 and 5 of t7,(x) are satisfied. Hence, we conclude that
F(x) given is a valid cdf if 0 5 a 5 1 and b > 0. Note that if a = 0, then the r.v. X is a discrete r.v.; if a = 1,
then X is a continuous r.v.; and if 0 < a < 1, then X is a mixed r.v.

0
Fig. 2-14

DISCRETE RANDOM VARIABLES AND PMF'S
2.12. Suppose a discrete r.v. X has the following pmfs:
PXW = 4 P X = $ px(3) = i
~
(a) Find and sketch the cdf F,(x) of the r.v. X.
(b) Find (i) P(X _< I), (ii) P(l < X _< 3), (iii) P(l I X I
3).
(a) By Eq. (2.1 8), we obtain

which is sketched in Fig. 2-15.
(b) (i) By Eq. (2.1 2), we see that
P(X < I)= Fx(l-)=0
(ii) By Eq. (2.10),
P(l < X I 3) = Fx(3) - Fx(l) = - 4 = 2
(iii) By Eq. (2.64),
P(l I X I 3) = P(X = 1) + Fx(3) - Fx(l) = 3 + 4 - 3 = 3