Page 70 - Probability, Random Variables and Random Processes

RANDOM VARIABLES [CHAP 2

ox2 = Var(X) = +[(- 2)2 + (0)2 + (2)'] =
Note that the variance of X is a measure of the spread of a distribution about its mean.

2.26. Let a r.v. X denote the outcome of throwing a fair die. Find the mean and variance of X
Since the die is fair, the pmf of X is
px(x) = px(k) = i k = 1, 2, . . . , 6
By Eqs. (2.26) and (2.29), the mean and variance of X are
PX= E(X)= +(I + 2 + 3 + 4 + 5 + 6) = g = 3.5
2
ax2 = 4[(1 - T) + (2 - $)2 + (3 - $)' + (4 - 4)' + (5 - 4)' + (6 - $)2] = 35 12
7
Alternatively, the variance of X can be found as follows:
E(X2) = i(12 + 2'+ 32 +4'+ 52 + 62)= 9
Hence, by Eq. (2.31),
ox2 = E(X2) - [E(X)12 = - (5)' = 35
12
2.27. Find the mean and variance of the geometric r.v. X defined by Eq. (2.67) (Prob. 2.15).

To find the mean and variance of a geometric r.v. X, we need the following results about the sum of a
geometric series and its first and second derivatives. Let

Then

By Eqs. (2.26) and (2.67), and letting q = 1 - p, the mean of X is given by

where Eq. (2.83) is used with a = p and r = q.
To find the variance of X, we first find E[X(X - I)]. Now,

where Eq. (2.84) is used with a = pq and r = q.
Since E[X(X - I)] = E(X2 - X) = E(X2) - E(X), we have

Then by Eq. (2.31), the variance of X is

2.28. Let X be a binomial r.v. with parameters (n, p). Verify Eqs. (2.38) and (2.39).