4.2 Variances of random variables

A useful further characteristic to consider is the degree of dispersion in the distribution, i.e. the spread of the possible values of X. Example :There are two batch of bulbs, the average lifetime is E(X)=1000 hours.

Definition 4.2-P 76 Let X be a r.v. and. The variance of X, denoted by D(X) or Var(X), is defined by 1) If X is discrete with pmf p k, then 2) If X is continuous with pdf f(x), then

(1) we often use variance to consider the degree of dispersion in the distribution of r.v. X. If the value of D(X) is large, it means the degree of dispersion of X is large. (2)D(X) ≥0. (3) Standard deviation 标准差 : Notes

Example 4.6-P76

Proof Theorem 4.2

Example 4.7,4.8-P78

Example Suppose the pmf of X is P(X=k) = p(1 - p) k-1, k=1, 2, …, where 0<p<1, Find Var(X). Solution Let q=1 - p ， then

Find Var(X). Example Suppose the pdf of X is Solution

Theorem 4.3-P79 Y=g(X) 1) If X is discrete with pmf p k, then 2) If X is continuous with pdf f(x), then

Example 4.9,4.10-P79

Proof Properties -P80 (1) If C is a constant, then (2) Suppose X is a r.v., C is a constant, then Proof

(3) Suppose X and Y are independent, D(X), D(Y) exist, then Proof

Generally, suppose X 1,X 2,…,X n mutually Independent, then (4)

distribution If X ~ B(1, p) ， then D(X) = p(1 - p) ； 2. Binomial distribution If X ~ B(n, p) ， then D(X) = n p(1 - p)=npq ； 3. Poisson distribution If X ~ P(λ) ， then D(X) = λ ；

∵ E(X) =λ ， then since So

4. Uniform distribution Suppose X ∼ U(a, b) ， then Since E(X)=(a+b)/2 ， so

5. Exponential distribution –P81

6. Normal distribution Suppose X ∼ N( ,  2 ) ， then

Homework: P89: 3,8,10