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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.