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04. Important Random Variable

- Independent random variable
- Mean and variance
- ??? 2009/03/23

Outline

- Review
- Affect independence
- Independent
- random variable
- Important
- random variable
- Continuous
- random variable

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)

Independent Random Variable

- pXA(x) pX(x)
- pX,Y(x,y) pX(x) pY(y)
- pX,Y,Z(x, y, z) pX(x) pY(y) pZ(z)
- EXY EX EY
- var(XY) var(X) var(Y)

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?

Important Random Variable

- Bernoulli
- pX(k) p, 1-p
- Binomial
- pX(k) Cnk pk (1 p)n k
- Geometric
- pX(k) (1 p)k-1 p
- Poisson
- pX(k) e??k / k!

- EX p
- var(X) p(1-p)
- EX np
- var(X) np(1-p)
- EX 1/p
- var(X) (1-p)/p2
- EX ?
- var(X) ?

Bernoulli

- pX(k) p, 1-p
- EX S x pX(x)
- var(X) EX2 (EX)2

Binomial

- pX(k) Cnk pk (1 p)n k
- EX
- EX EX1 EXn
- EX S k Cnk pk (1 p)n k
- var(X)

Geometric

- pX(k) (1 p)k-1 p
- EX
- EX P(X1)EXX1 P(Xgt1)EXXgt1
- EX Sk(1 p)k-1 p
- var(X) EX2 (EX)2

Poisson

- pX(k) e??k / k!
- EX
- var(X)

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.

Example 3 (Geometric)

- One child each family in China!
- If 1st child is a boy, parents have no more

child. - If 1st child is a girl, parents have another 2nd

child. - Parents wont give birth to more babies until a

boy is born. - The number of boys The number of girls ?

Continuous Random Variable

- Uniform

- fX(x) , a?x?b
- EX
- var(X)

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

Example 4 (PDF)

- Computers lifetime is a random variable (unit

hour). - Five computers construct a network server
- (1) A computer is down at 150th hour.
- (2) A computer is down before 150th hour.
- (3) A computer is down before 200th hour.
- (4) A server is crash before 700th hour.