Random Variables and Probability Distributions

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Random Variables and Probability Distributions

• Chapter 3
• Probability and Statistics
• for engineering and scientists (6th edition)

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3.1 Random Variables

• Def.
• Random variable is a real-valued function
• defined on the sample space.

Ex 1 ) Toss two fair coins. Let X
the of heads. element x HH 2
HT 1 TH 1 TT 0
sample space
real numbers
X
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3.1 Random Variables (cont)
Ex 2 ) Toss two fair dice. Let Y sum of
the two numbers. element y (1 , 1) 2
(1 , 2) , (2 , 1) 3 (1 , 3) , (2 , 2) ,
(3 , 1) 4 (6 , 6) 12 Ex 3 )
Flipping a coin until a head appears. Ex 4 )
Length of time between failures.
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3.1 Random Variables (cont)

• Types of Random Variables
• Discrete R.V. count data , finite or countable
  (set of possible outcomes)
• Continuous R.V. measured data, infinitely many
  values

Ex ) of IE majors at POSTECH in 2002 of
traffic accidents per year in Pohang weight of
grain produced per acre height of people over
20
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3.2 Discrete Probability Distributions

• probability mass function (pmf)
• Properties 1) 2)
  3)

Ex 1 ) Toss two fair coins. X heads
element x probability HH 2 1/4
HT 1 1/4 TH 1 1/4 TT 0 1/4
x 0 1 2
f(x) P(X x ) ¼ ½ ¼
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3.2 Discrete Probability Distributions (cont)

• (Cumulative) distribution function (cdf or df)
• Properties 1) 2)

for
3) 4) F(x) is nondecreasing
Ex 2) Toss two fair coins. X
heads x f(x) F(x) 0
1/4 1/4 1 1/2
3/4 2 1/4 1
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3.3 Continuous Probability Distributions

• P( X x ) 0 Why?
• Calculate probabilities based on a range of
  values.
• Probability Density Function (pdf) f(x)
• Properties

Eg)

1)
2)
3)
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3.3 Continuous Probability Distributions (cont)
Ex 1) (6 on p. 61) X shelf life of a
prescribed medicine.
, x gt 0
f(x)
0 , otherwise
f(x)
a ) P ( X gt 200 )
b ) P ( 80 ltX lt 120 )
x
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3.3 Continuous Probability Distributions (cont)

• Cumulative distribution function (cdf) F(x)
• Properties

for
1)
2)
Ex 2) (20 on p. 62)
, 2 lt x lt 5
1) F(x)
2) P( 3 lt X lt 4)
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3.5 Joint Probability Distributions

• Record outcomes of several R.V.s.
• eg ) P( X x, Y y) , P( a lt X lt b, c lt Y lt d)
• Discrete Case
• For any region A in the XY plane,

- Joint Probability Mass Function (or
joint pmf) if for all
1)
2)
3)
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3.5 Joint Probability Distributions (cont)

• Marginal Distributions
• Conditional Distributions

, g(x) gt 0
, h(y) gt 0
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3.5 Joint Probability Distributions (cont)
Ex 1) Quiz Scores X score on the first
quiz. Y score on the second quiz
a)
b)
c)
d)
e)
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3.5 Joint Probability Distributions (cont)

• Continuous Case
• Marginal
• Conditional

- Joint density function (Joint pdf)

1)
2)
3)
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3.5 Joint Probability Distributions (cont)
Ex 2) (3.14 on p. 76) ,
, 0 , elsewhere

• Find
• P0ltXlt1, 0ltYlt1/2
• g(x) and h(y)
• f(xy)

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3.5 Joint Probability Distributions (cont)
- Statistical Independence X Y are
independent if and only if f(x,y) g(x)h(y)
Generally, are
mutually statistically independent if and only
if
Ex 3) (3.14 on p. 76) X and Y are independent ?
Ex 4) Two balls are selected at random from a
box. (3.8 on p.70)

• X blue balls Y red balls
• Joint pmf of X and Y ?
• Are they independent ?