1 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 Program Department of Engineering Management, Information and Systems
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1 Joint Probability Distributions Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS.
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1
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
2
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
3
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
4
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
5
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
6
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
7
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
8
The marginal probability mass functions of x alone and of Y alone are
and
for the discrete case.
y
yxpxg ),( x
yxpyh ),(
Marginal Distributions
9
The marginal probability density functions of x alone and of Y alone are
and
for the continuous case.
dxdyyxfxg
),( dxdyyxfyh
),(
Marginal Distributions
10
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
11
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
12
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
13
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
15
3D plotting for example problem
16
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
Example – Solution
17
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
18
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
Example – Solution
19
Find g(x) and h(y) for the joint density function ofthe previous example :
elsewhere,0
10,10),32(),( 5
2 yxyxyxf
Example
20
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
Example – Solution
21
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
22
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
23
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
Example – Solution
24
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
25
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
26
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
27
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
Example – Solution
X
28
We find the three probabilities f(0,1), g(0),and h(1) to be
Clearly, and therefore X and Y arenot statistically independent.