Joint Probability Distributions

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Joint Probability Distributions

Dr. Jerrell T. Stracener, SAE Fellow

Leadership in Engineering

EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS

Systems Engineering ProgramDepartment of Engineering Management, Information and Systems

Stracener_EMIS 7370/STAT 5340_Sum 08_07.15.08


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The function p(x, y) is a joint probability mass function of the discrete random variables X and Y if

1. p(x, y) 0 for all (x, y)

2.

3. P(X = x, Y = y) = p(x, y)

For any region A in the xy plane,

P[(X, Y) A] =

x y

yxp 1),(

y)p(x,

A

Joint Probability Mass Function


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Two refills for a ballpoint pen are selected at randomfrom a box that contains 3 blue refills, 2 red refillsand 3 green refills. If X is the number of blue refillsand Y is the number of red refills selected, find

a. The joint probability mass function p(x, y), and

b. P[(X, Y) A], where A is the region {(x, y)|x + y 1}

Example


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The possible pairs of values (x, y) are (0, 0), (0, 1),(1, 0), (1, 1), (0, 2), and (2, 0), where p(0, 1), forexample represents the probability that a red and agreen refill are selected. The total number of equallylikely ways of selecting any 2 refills from the 8 is:

The number of ways of selecting 1 red from 2 redrefills and 1 green from 3 green refills is

28!6!2

!8

2

8

61

3

1

2

Example - Solution


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Hence, p(0, 1) = 6/28 = 3/14. Similar calculations yield the probabilities for the other cases, which are presented in the following table. Note that the probabilities sum to 1.

X0 1 2 Row Totals

0

y 1

2

Column Totals

28

3

14

3

28

1

28

9

14

3

28

3

14

5

28

15

28

3

28

15

7

3

28

1

1

Example - Solution


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for x = 0, 1, 2; y = 0, 1, 2; 0 x + y 2.

b. P[(X, Y) A] = P(X + Y 1)

= p(0, 0) + p(0, 1) + p(1, 0)

2

8

yx2

3

y

2

x

3

yx,p

28

9

14

3

28

3

14

9

Example - Solution


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The function f(x, y) is a joint probability density function of the continuous random variables X and Y if

1. f(x, y) 0 for all (x, y)

2.

3. P[(X, Y) A] =

For any region A in the xy plane.

1y)dxdyf(x,

dxdyyxf ),(A

Joint Density Functions


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The marginal probability mass functions of x alone and of Y alone are

and

for the discrete case.

y

yxpxg ),( x

yxpyh ),(

Marginal Distributions


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The marginal probability density functions of x alone and of Y alone are

and

for the continuous case.

dxdyyxfxg

),( dxdyyxfyh

),(

Marginal Distributions


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Let X and Y be two discrete random variables, with joint probability mass function p(x,y) and marginal probability mass functions m(x) and n(y). The conditional probability mass function of the random variable Y, given that X = x, is

, m(x) > 0

Similarly, the conditional probability mass function of the random variable X, given that Y = y, is

, n(y) > 0

xm

yx,px|y l

yn

yx,py|x l

Conditional Probability Distributions


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Let X and Y be two discrete random variables, with joint probability mass function p(x, y) and marginal probability mass functions m(x) and n(y), respectively. The random variables X and Y are said to be statistically independent if and only if

for all (x, y) within their range.

ynxmyx,p

Statistical Independence


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Let X1, X2, …, Xn be n discrete random variables, with joint

probability mass functions p(x1, x2, …, xn) and marginal

probability mass functions p1(x1), p2(x2), …, pn(xn),

respectively. The random variables X1, X2, …, Xn are said to

be mutually statistically independent if and only if

p(x1, x2, …, xn) = p1(x1),p2(x2), …, pn(xn)

for all (x1, x2, …, xn) within their range.

Statistically Independent


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A candy company distributed boxes of chocolates with a mixture of creams, toffees, and nuts coated in both light and dark chocolate. For a randomly selected box, let X and Y, respectively, be the proportions of the light and dark chocolates that are creams and suppose that the joint density function is

a) Verify whether

b) Find P[(X,Y) A], where A is the region {(x,y) | 0<x<½,

¼<y<½}.

elsewhere,0

10,10),32(),( 5

2 yxyxyxf

Example

1y)dxdyf(x,


14

a)

15

3

5

2

5

3

5

2

5

6

5

2

5

6

5

2

)32(5

2),(

1

0

21

0

1

0

1

0

2

1

0

1

0

yydy

y

dyxyx

dxdyyxdxdyyxf

x

x

Example – Solution


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3D plotting for example problem


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b)

160

13

16

3

4

1

4

3

2

1

10

1

10

3

105

3

10

1

5

6

5

2

)32(5

2

),0(),(

21

41

21

41

21

41

21

21

41

21

2

0

2

0

21

41

21

yydy

y

dyxyx

dxdyyx

YXPAYXP

x

x



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Show that the column and row totals of the following table give the marginal distribution of X alone andof Y alone.

X0 1 2 Row Totals

0

y 1

2

Column Totals

28

3

14

3

28

1

28

9

14

3

28

3

14

5

28

15

28

3

28

15

7

3

28

1

1

Example


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For the random variable X, we see that

28

300

28

3

)2,2()1,2()0,2(),2()2(2

28

150

14

3

28

9

)2,1()1,1()0,1(),1()1(1

14

5

28

1

14

3

28

3

)2,0()1,0()0,0(),0()0(0

2

0

2

0

2

0

fffyfgXP

fffyfgXP

fffyfgXP

y

y

y



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Find g(x) and h(y) for the joint density function ofthe previous example :

elsewhere,0

10,10),32(),( 5

2 yxyxyxf

Example


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By definition,

For 0x 1, and g(x)=0 elsewhere. Similarly,

For 0 y 1, and h(y)=0 elsewhere.

5

34

10

6

5

4)32(

5

2),()(

1

0

21

0

xyxydyyxdyyxfxg

y

y

5

)31(4)32(

5

2),()(

1

0

ydxyxdxyxfyh



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Let X and Y be two continuous random variables, with joint probability density function f(x,y) and marginal probability density functions g(x) and h(y). The conditional probability density function of the random variable Y, given that X = x, is

, g(x) > 0

Similarly, the conditional probability density function of the random variable X, given that Y = y, is

, h(y) > 0

xgyxf

xyf,

|

yhyxf

yxf,

|

Conditional Probability Distribution


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Given the joint density function

Find g(x), h(y), f(x|y), and evaluate P(¼<X<½|Y=1/3).

elsewhere,0

10,20,4

)31(),(

2

yxyx

yxf

Example


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By definition,

and

therefore,

and

2x0 ,2444

)31(),()(

1

0

31

0

2

xxyxydy

yxdyyxfxg

y

y

1y0 ,2

31

8

3

84

)31(),()(

22

0

2222

0

2

yyxxdy

yxdxyxfyh

x

x

2x0 ,22/)31(

4/)31(

)(

),()|(

2

2

x

y

yx

yh

yxfyxf

.64

3

23

1|

2

1

4

1 2

1

4

1

dx

xYXP



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Let X and Y be two continuous random variables, with joint

probability density function f(x, y) and marginal probability

density functions g(x) and h(y), respectively. The random

variables X and Y are said to be statistically independent if

and only if

for all (x, y) within their range.

yhxgyxf ,

Statistical Independence


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Let X1, X2, …, Xn be n continuous random variables, with joint probability density functions f(x1, x2, …, xn) and marginal probability functions f1(x1),f2(x2), …, fn(xn), respectively. The random variables X1, X2, …, Xn are said to be mutually statistically independent if and only if

f(x1, x2, …, xn) = f1(x1),f2(x2), …, fn(xn)

for all (x1, x2, …, xn) within their range.

Statistically Independent


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Two refills for a ballpoint pen are selected at randomfrom a box that contains 3 blue refills, 2 red refillsand 3 green refills. If X is the number of blue refillsand Y is the number of red refills selected, Show that the random variables are not statistically independent.

Example


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Let us consider the point (0,1). From the following table:

0 1 2 Row Totals

0

y 1

2

Column Totals

28

3

14

3

28

1

28

9

14

3

28

3

14

5

28

15

28

3

28

15

7

3

28

1

1


X


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We find the three probabilities f(0,1), g(0),and h(1) to be

Clearly, and therefore X and Y arenot statistically independent.

2

0

2

0

17

30

14

3

14

3)1,()1(

,14

5

28

1

14

3

28

3),0()0(

,14

3)1,0(

y

y

xfh

yfg

f

)1()0()1,0( hgf


Joint Probability Distributions

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