Download Chapter 5 Joint Probability Distributions

Chapter 5
Joint Continuous Probability
Distributions
Doubling our pleasure with two
random variables
Chapter 5C
Today in Prob/Stat
5-2 Two Continuous Random Variables
5-2.1 Joint Probability Distribution
Definition
Computing Joint Probabilities
b d
Pr a  X 1  b, c  X 2  d     f xy ( x, y ) dy dx
a c
That’s a mighty fine
formula you got there.
Example: Computing Joint
Probabilities
In a healthy individual age 20 to 29 yrs, the calcium level in the
blood, X, is usually between 8.5 and 10.5 mg/dl and the
cholesterol level is usually between 120 and 140 mg/dl. Assume for
a healthy individual the random variables (X,Y) is uniformly
distributed with a PDF given by:
f xy ( x, y )  c, 8.5  x  10.5, 120  y  240
10.5 240
10.5
8.5 120
8.5
  c dy dx  1 or c   240  120  dx  1
120c 10.5  8.5   1
240c  1 or c  1/ 240  .0041667
find c!
Example: Computing Joint Probabilities

Thinking outside the cuboid
A cuboid is a solid figure
bounded by six rectangular
faces: a rectangular box
Now find:
Pr 9  X  10 and 125  Y  140
10 140
10
1
1
15
 
dy dx 
 .0625
140  125 dx 

240
240 9
240
9 125
c
1
240
9 10
125
x
140
y
Volume = (1) (140-125) / 240 = .0625
Example 5-12
Example 5-12
Figure 5-8
The joint probability density function of X and Y is nonzero
over the shaded region.
Example 5-12
Figure 5-9
Region of integration for the probability that X < 1000 and Y
< 2000 is darkly shaded.
5-2 Two Continuous Random Variables
5-2.2 Marginal Probability Distributions
Definition
Returning to our blood example


Let X = a continuous RV, the calcium level in the blood,
Y = a continuous RV, the cholesterol level in the blood
1
f xy ( x, y ) 
, 8.5  x  10.5, 120  y  240
240
240
1
240  120 1
f x ( x)  
dy 
 , 8.5  x  10.5
240
240
2
120
10.5
f y ( y) 

8.5
1
2
1
dx 

, 120  y  240
240
240 120
Pr 150  Y  200 
1
200  150
dy

 .41667
150 120
120
200
Example 5-13
Example 5-13
5-2 Two Continuous Random Variables
5-2.3 Conditional Probability Distributions
Definition
5-2.3 Conditional Probability Distributions
Example 5-14
Example 5-14
Figure 5-11 The
conditional probability
density function for Y, given
that x = 1500, is nonzero
over the solid line.
Conditional Mean and Variance
5-2 Two Continuous Random Variables
5-2.4 Independence
Definition
Is our blood example
independent?
1
 1  1 
f xy ( x, y ) 
  
  f x ( x) f y ( y )
240  2  120 
A Dependent Example


Let X = a continuous RV, the fraction of student applicants
(deposits) that visit the University campus
Let Y = a continuous RV, the fraction of student applicants
(deposits) that do not attend the University.
12
f xy ( x, y )  x  2  x  y  , 0  x  1, 0  y  1
5
1
1
12
12 
xy 2 
2
0 0 5 x  2  x  y  dy dx  5 0 2 xy  x y  2  dx
0
1 1
1
1
12 
x
12  2 x3 x 2   12  12  4  3 
2
   2 x  x   dx   x      
 1
5 0
2
5 
3 4  0  5  12

students matriculating
Marginal Distribution of X
1
1
12
12 x
f x ( x)   x  2  x  y  dy 
 2  x  y  dy

5
5 0
0
1
12 x 
y  12 x

2 y  xy   
1.5  x 

5 
2 0
5
2
12 x 2  3
12  3 2

3
E[ X ]  

x
dx

x

x



 dx

5 2
5 02


0
1
1
1
12  x x   12  1  3
          .6
5  2 4  0  5  4  5
3
4
Marginal Distribution of Y
1
1
12
12
f y ( y )   x  2  x  y  dx    2 x  x 2  xy  dx
5
5 0
0
1
12  2 x x y  12  2 y 
 x  
   

5 
3
2 0 5  3 2 
3
2
1
12 y  2 y 
12  2 y y 2 
E[Y ]  
  dy
   dy   
5 3 2
5 0 3
2 
0
1
1
y   12  1  2
 12   y
            .4
6  0  5  6  5
 5  3
2
3
The Variances
1
12  3 3
12  3x
x 
21
4
E[ X ]    x  x  dx  
  
5 02
5  8
5  0 50

21
2
 x   .62  .06;   .245
50
1
4
5
2
1
12  2 y
y 
y   12  2 1  7
 12   2 y
E[Y ]   
  dy    
       
5 0 3
2 
8  0  5  9 8  30
 5  9
7
2
 y   .42  .0733;  y  .271
30
1
2
2
3
3
4
The Covariance and Correlation
1
1

12
12 2 
2
E[ XY ]     xy  x  2  x  y  dy dx   x    2 y  xy  y  dy  dx
5
5 0
0 0
0

1 1
1
1
12 2  2
y
y 
12
12 
x x 
 x 1
x  y  x   dx   x 2 1    dx    x 2    dx
5 
2
3 0
5 0  2 3
5 0
2 3 0
0
1

2
3
1
1
3
4
x3   12  1 1 1  7
 12   x x
             
 5   3 8 9  0  5  3 8 9  30
7
Cov( XY )   xy 
 (.4)(.6)  .006667
30
 xy
.006667
 xy 

 .10
 x y .245 .271
1
3
2
A Conditional Distribution
12
f xy ( x, y ) 5 x  2  x  y  2  x  y
fY | x ( y ) 


3
12  3
f x ( x)

x
x  x
2
5 2


Given that 50 percent of the students that submit applications
visit the campus, what is the expected number that will not
attend?
3
y
3
fY | x .5 ( y )  2
 y
3 1 2

2 2
1
 3 y 2 y3  9  4
3

E (Y | x  .5)   y   y  dy  
  
 .4167
3 0
12
2

 4
0
1
E (Y | x  .8)  .3814
Example 5-18
5-4 Bivariate Normal Distribution
Definition
Figure 5-17. Examples of bivariate normal distributions.
Example 5-30
Figure 5-18
Marginal Distributions of Bivariate
Normal Random Variables
Figure 5-19
Marginal probability
density functions of a
bivariate normal
distributions.
5-4 Bivariate Normal Distribution
Example 5-31
A Reproductive Continuous
Distribution

If X1, X2, …,Xp are independent normal RV with E{xi] = i
and V[Xi] = i2, then
p
Y   ci X i is normal
i 1
p
p
i 1
i 1
with E[Y ]   ci i and V [Y ]   ci2 i2
Gosh, everything
is so normal.
The Last Example


Let Xi = a normal RV, the daily demand for item i stocked by the
Mall-Mart Discount Store. Mall-Mart has a daily cost
requirement of $1100. Will they have a cash flow problem?
Then Y 
p
c X
i 1
Item
1
2
3
4
5
6
7
i
i
is the daily sales revenue
Mean
Variance Selling Price
8.5
1.2
$22.50
3.4
2.3
$18.75
2.1
0.87
$5.67
7.8
2.4
$24.50
11.2
3.2
$15.99
8.4
5.9
$36.80
15.1
6.7
$12.95
Expected
revenue Variance
191.25
607.50
63.75
808.59
11.907
27.97
191.1 1,440.60
179.088
818.18
309.12 7,990.02
195.545 1,123.61
1141.76 12816.46
Std dev.
$113.21
Pr{Y  1100}
 Y   y 1100  1141.76 
 Pr 



113.21


y
 Pr{z  .37}  .3561
Other Reproductive Continuous
Distributions



The sum of independent exponential RVs having the
same mean is Erlang (gamma)
The sum of independent gamma RVs having the
same scale parameter is gamma
The sum of independent Weibull RVs having the
same shape parameter is Weibull
The overachieving student will
want to research the
derivations of these results.
So Ends Chapter 5
Another great adventure has just ended.
Can you read me the story
again about the marginal and
conditional distributions. I
liked that one the best!
… and thus our story ends with
the X random variable having a
strong positive correlation with
the Y random variable.