1. P robability 1 04. Important Random Variable Independent random variable Mean and variance 郭俊利 2009/03/23

2. Probability 2 Outline Review Affect independence Independent random variable Important random variable Continuous random variable 1.5, 2.3, 2.4, 2.7 3.1

3. Probability 3 Example 1 (Affect Independence) Two unfair coins, A and B: P(H | coin A) = 0.9, P(H | coin B) = 0.1 choose either coin with equal probability 1)Once we know it is coin A, are future tosses independent? 2)If we do not know which coin it is, are future tosses independent? P(toss 1 and toss 2 = H) = 3)Compare: P(toss 11 = H) = P(toss 11 = H | first 10 tosses are heads) = 4) Other: P(toss 5 times, 2 Hs shows) = P(above | first 10 tosses are heads) =

4. Probability 4 Independent Random Variable p X|A (x) = p X (x) p X,Y (x,y) = p X (x) p Y (y) p X,Y,Z (x, y, z) = p X (x) p Y (y) p Z (z) E[XY] = E[X] E[Y] var(X+Y) = var(X) + var(Y)

5. Probability 5 Example 2 (Independence) Two tosses of a fair coin X is the number of heads A is the number of even heads X and A are independent?

6. Probability 6 Important Random Variable Bernoulli p X (k) = p, 1-p Binomial p X (k) = C n k p k (1 – p) n – k Geometric p X (k) = (1 – p) k-1 p Poisson p X (k) = e –λ λ k / k! E[X] = p var(X) = p(1-p) E[X] = np var(X) = np(1-p) E[X] = 1/p var(X) = (1-p)/p 2 E[X] = λ var(X) = λ

7. Probability 7 Bernoulli p X (k) = p, 1-p E[X] = Σ x p X (x) = var(X) = E[X 2 ] – (E[X]) 2

8. Probability 8 Binomial p X (k) = C n k p k (1 – p) n – k E[X] E[X] = E[X 1 ] + … + E[X n ] E[X] = Σ k C n k p k (1 – p) n – k var(X) =

9. Probability 9 Geometric p X (k) = (1 – p) k-1 p E[X] E[X] = P(X=1)E[X|X=1] + P(X>1)E[X|X>1] E[X] = Σk(1 – p) k-1 p var(X) = E[X 2 ] – (E[X]) 2 =

10. Probability 10 Poisson p X (k) = e –λ λ k / k! E[X] = var(X) =

11. Probability 11 Example 2 (Binomial + Independence) Alice passes through four traffic lights on her way. (1) What is the PMF? (2) How many red lights Alice encounters? (3) From (2), find the variance.

12. Probability 12 Example 3 (Geometric) One child each family in China! If 1 st child is a boy, parents have no more child. If 1 st child is a girl, parents have another 2 nd child. Parents won ’ t give birth to more babies until a boy is born. The number of boys = The number of girls ?

13. Probability 13 Continuous Random Variable Uniform f X (x) =, a ≦ x ≦ b E[X] = var(X) =

14. Probability 14 Probability Density Function The random variable is a real-valued function of the outcome of the experiment.  Discrete Probability mass function  General = Continuous Probability density function

15. Probability 15 Example 4 (PDF) Computer ’ s lifetime is a random variable (unit: hour). Five computers construct a network server (1) A computer is down at 150 th hour. (2) A computer is down before 150 th hour. (3) A computer is down before 200 th hour. (4) A server is crash before 700 th hour. f(x) = 0, x ≦ 100 100 / x 2, x > 100 {