1. 4.2 Variances of random variables

2. 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.

3. 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

4. (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

5. Example 4.6-P76

6. Proof Theorem 4.2

7. Example 4.7,4.8-P78

8. 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

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

11. 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

12. Example 4.9,4.10-P79

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

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

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

16. 1. 0-1 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 ；

17. ∵ E(X) =λ ， then since So

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

19. 5. Exponential distribution –P81

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

21. Homework: P89: 3,8,10