Page 78 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 21 RANDOM VARIABLES

CONDITIONAL DISTRIBUTIONS
2.49. Let X be a Poisson r.v. with parameter 2. Find the conditional pmf of X given B = (X is even).

From Eq. (2.40), the pdf of X is
;Ik
px(k) = e-A - k = 0, 1, ...
k !
Then the probability of event B is

Let A = {X is odd). Then the probability of event A is

Now

Ak Ak
7-
a, (-A)k - ,-A -A - ,-21
f e-Ak!=eu- e -
1 E - k=odd
k = even k = 0
Hence, adding Eqs. (2.101) and (2.1 02), we obtain

Now, by Eq. (2.62), the pmf of X given B is

If k is even, (X = k) c B and (X = k) n B = (X = k). If k is odd, (X = k) n B = fZI. Hence,
P(X = k) 2e-9''
k even
P(B) (1+eT2")k!
P*(k I B) =
k odd

b)
2.50. Show that the conditional cdf and pdf of X given the event B = (a < X I are as follows:
x 5: a
10