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04. Important Random Variable

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Five computers construct a network server (1) A computer is down at 150th hour. (2) ... PowerPoint Presentation Author: Kuo Last modified by: Kuo Created Date: – PowerPoint PPT presentation

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Title: 04. Important Random Variable


1
04. Important Random Variable
  • Independent random variable
  • Mean and variance
  • ??? 2009/03/23

2
Outline
  • Review
  • Affect independence
  • Independent
  • random variable
  • Important
  • random variable
  • Continuous
  • random variable

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
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)

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
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) ?

7
Bernoulli
  • pX(k) p, 1-p
  • EX S x pX(x)
  • var(X) EX2 (EX)2

8
Binomial
  • pX(k) Cnk pk (1 p)n k
  • EX
  • EX EX1 EXn
  • EX S k Cnk pk (1 p)n k
  • var(X)

9
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

10
Poisson
  • pX(k) e??k / k!
  • EX
  • var(X)

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
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 ?

13
Continuous Random Variable
  • Uniform
  • fX(x) , a?x?b
  • EX
  • var(X)

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
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.
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