Page 57 - Probability, Random Variables and Random Processes

CHAP. 21 RANDOM VARIABLES

(b) If P(H) = p, then P(T) = 1 - p. Since the tosses are independent, we have

2.3. An information source generates symbols at random from. a four-letter alphabet (a, b, c, d} with
probabilities P(a) = f, P(b) = $, and P(c) = P(d) = i. A coding scheme encodes these symbols
into binary codes as follows:

Let X be the r.v. denoting the length of the code, that is, the number of binary symbols (bits).
(a) What is the range of X?
(b) Assuming that the generations of symbols are independent, find the probabilities P(X = I),
P(X = 2), P(X = 3), and P(X > 3).
(a) TherangeofXisR, = {1,2, 3).
(b) P(X = 1) = P[{a)] = P(a) =
P(X = 2) = P[(b)] = P(b) = $
P(X = 3) = P[(c, d)] = P(c) + P(d) = $
P(X > 3) = P(%) = 0

2.4. Consider the experiment of throwing a dart onto a circular plate with unit radius. Let X be the
r.v. representing the distance of the point where the dart lands from the origin of the plate.
Assume that the dart always lands on the plate and that the dart is equally likely to land
anywhere on the plate.
(a) What is the range of X?
1.
(b) Find (i) P(X < a) and (ii) P(a < X < b), where a < b I
(a) The range of X is R, = (x: 0 I x < 1).
(b) (i) (X < a) denotes that the point is inside the circle of radius a. Since the dart is equally likely to fall
anywhere on the plate, we have (Fig. 2-10)

(ii) (a < X < b) denotes the event that the point is inside the annular ring with inner radius a and
outer radius b. Thus, from Fig. 2-10, we have

DISTRIBUTION FUNCTION
2.5. Verify Eq. (2.6).
Let x, < x,. Then (X 5 x,) is a subset of (X I x,); that is, (X I x,) c (X I x,). Then, by Eq. (1.27),
we have