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8 Joint Distributions
8. Joint Distributions
8.1 Bivariate distribution
Def Let X and Y be 2 discrete random variables defined on the same sample space. Let the sets of possible values of X and Y be A and B, respectively. The function
p(x, y)=P(X=x, Y=y)
is called the joint probability mass function of X and Y.
Bivariate distribution
Def Let X and Y have joint probability mass function p(x,y). Let the sets of possible values of X and Y be A and B, respectively. Then the functions
are called, respectively, the marginal probability mass functions of X and Y.
Bivariate distribution
Ex 8.1 A small college has 90 male and 30 female professors. An ad hoc committee of 5 is selected at random to write the vision and mission of the college. Let X and Y be the number of men and women on this committee, respectively.
(a) Find the joint probability mass function of X and Y.
(b) Find pX and pY, the marginal functions of X and Y.
Bivariate distribution
Sol:
Bivariate distribution
Ex 8.2 Roll a balanced die and let the outcome be X. Then toss a fair coin X times and let Y denote the number of tails. What is joint probability mass function?
and what are pX and pY?
Bivariate distribution
Sol:
y
pX(x)
0
1
2
3
4
5
6
x
1
1/12
1/12
0
0
0
0
0
1/6
2
1/24
2/24
1/24
0
0
0
0
1/6
3
1/48
3/48
3/48
1/48
0
0
0
1/6
4
1/96
4/96
6/96
4/96
1/96
0
0
1/6
5
1/192
5/192
10/192
10/192
5/192
1/192
0
1/6
6
1/384
6/384
15/384
20/384
15/384
6/384
1/384
1/6
pY(y)
63/384
120/384
99/384
64/384
29/384
8/384
1/384
Bivariate distribution
Thm 8.1 Let f(x,y) be the probability mass function of discrete r. v. X and Y. If h is a function of two variables from R2 to R, then Z=h(X,Y) is a random variable with the expected value given by
Corollary of Thm 8.1
E(X+Y)= E(X)+E(Y)
Proof:
Bivariate distribution
Def Two r. v.’s X and Y, defined on the same sample space, have a continuous joint distribution if there exists a n
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