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8. Joint Distributions

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**1. **2012/4/2 1 8. Joint Distributions

**2. **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.

**3. **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.

**4. **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.

**5. **Bivariate distribution
Sol:

**6. **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?

**7. **Bivariate distribution Sol:

**8. **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

**9. **Corollary of Thm 8.1
E(X+Y)= E(X)+E(Y)
Proof:

**10. **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 nonnegative function of 2 variables, f(x,y) on RXR, such that for any region S in the xy-plane that can be formed from rectangles by a countable number of set operations,
The function f(x,y) is called the joint (probability) density function of X and Y.

**11. **Bivariate distribution
Def Let X and Y have joint density function f(x,y), then the functions
are called, respectively, the marginal density functions of X and Y.

**12. **Bivariate distribution
Def Let X and Y be two r. v.’s (discrete, continuous, or mixed). The joint distribution function of X and Y, is defined by

**13. **Bivariate distribution

**14. **Bivariate distribution
Ex 8.4 The joint density function of random variables X and Y is given by
(a) Determine the value of .
(b) Find the marginal density function of X and Y.

**15. **Bivariate distribution

**16. **Bivariate distribution
Ex 8.5
Determine if F is the joint distribution function of 2 random variables X and Y.

**17. **Bivariate distribution

**18. **Bivariate distribution
Ex 8.6 A circle of radius 1 is inscribed in a square with sides of length 2. A point is selected at random from the square we mean that the point is selected in a way that all the subsets of equal areas of the square are equally likely to contain the point.

**19. **Bivariate distribution

**20. **Bivariate distribution
Def Let S be a subset of the plane with area A(S). A point is said to be randomly selected from S if for any subset R of S with area A(R), the probability that R contains the point is A(R)/A(S).
Buffon’s needle problem(p.336)
(an interesting geometric probability problem)

**21. **Bivariate distribution
Ex 8.7 A man invites his fiancee to a fine hotel for a Sunday brunch. They decide to meet in the lobby of the hotel between 11:30 A.M. and 12 noon. If they arrive at random times during this period, what is the probability that they will meet within 10 minutes?

**22. **Bivariate distribution
Sol:

**23. **Bivariate distribution
Thm 8.2 Let f(x,y) be the density function of 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

**24. **Bivariate distribution
Corollary For r. v.’s X and Y,
E(X+Y) = EX + EY
Proof:

**25. **Bivariate distribution
Ex 8.9 Let X and Y have joint density function
Find E(X2+Y2)

**26. **Bivariate distribution
Sol:

**27. **8.2 Independent random variables
Def Two random variables are called independent if, for A, B, in R, the events
are independent, that is, if
Using the axioms of probability, we can prove that X and Y are independent iff for any real numbers a and b,

**28. ** Independent random variables
Thm 8.3 Let X and Y be two random variables defined on the same sample space. If F is the joint distribution function of X and Y, then X and Y are independent iff for all real numbers t and u,
F(t, u)=FX(t)FY(u).

**29. ** Independent random variables
Thm 8.4 Let X and Y be two discrete random variables defined on the same sample space. If p(x, y) is the joint probability function of X and Y, then X and Y are independent iff for all real numbers x and y,
p(x, y)=pX(x)pY(y).

**30. ** Independent random variables
Thm 8.5 Let X and Y be independent random variables and g: R?R and h: R?R be real-valued functions; then g(X) and h(Y) are also independent random variables.
Thm 8.6 Let X and Y be independent random variables and g: R?R and h: R?R be real-valued functions; then
E[g(X)h(Y)]=E[g(X)]E[h(Y)]

**31. ** Independent random variables
By Thm 8.6, if X and Y are independent, then EXY=EXEY but the converse is not necessarily true.
Ex 8.11 Let X be a r. v. with p(-1)=p(0)=p(1) =1/3. Letting Y=X2, we have
EX=(-1+0+1)/3=0,
EY=E(X2)=((-1)2+02+(1)2)/3=2/3,
EXY=E(X3)=((-1)3+03+(1)3)/3=0.
Thus EXY=EXEY while, clearly, X and Y are dependent.

**32. ** Independent random variables
Thm 8.7 Let X and Y be two continuous random variables defined on the same sample space. If f(x, y) is the joint density function of X and Y, then X and Y are independent iff for all real numbers x and y,
f(x, y)=fX(x)fY(y).

**33. ** Independent random variables
Ex 8.12 Store A and B, which belong to the same owner, are located in 2 different towns. If the density function of the weekly profit of each store, in thousands of dollars, is given by
and the profit of one store is independent of the other, what is the probability that next week one store makes at least $500 more than the other store?

**34. ** Independent random variables
Sol:

**35. ** Independent random variables

**36. **Ex 8-14 Buffon’s Needle Problem – A plane is ruled with parallel lines a distance d apart. A needle of length l, l<d, is tossed at random onto the plane. What is the probability that the needle intersects one of the parallel lines?
Sol:
X: the distance from the center of the needle to the closest line = U(0, d/2)
?: the angle between the needle and the line = U(0,?)

**38. ** Independent random variables
Ex 8.15 Prove that 2 random variables X and Y with the following joint density function are not independent.

**39. ** Independent random variables
Sol:

**40. **8.3 Conditional distributions
Def Let X and Y be two discrete random variables. The conditional probability function of X given that Y=y is defined as follows:

**41. ** Conditional distributions

**42. ** Conditional distributions
Def The conditional expectation of the random variable X given that Y=y is
(discrete)
(continuous)

**43. ** Conditional distributions Ex 8.18 Calculate the expected number of aces in a randomly selected poker hand that is found to have exactly 2 jacks.

**44. ** Conditional distributions

**45. ** Conditional distributions
Def Let X and Y be two continuous random variables with joint density
f(x, y). The conditional density function of X given that Y=y is defined as follows:

**46. ** Conditional distributions

**47. ** Conditional distributions Ex 8.21 Let X and Y be continuous random variables with joint density function

**48. ** Conditional distributions

**49. ** Conditional distributions Ex 8.22 First, a point Y is selected at random from the interval (0, 1). The another point X is chosen at random from the interval (0,Y). Find the density function of X.

**50. ** Conditional distributions

**51. ** Conditional distributions Ex 8.24 Let X and Y be continuous random variables with joint density function

**52. ** Conditional distributions

**53. ** Chapter 9 Multivariate Distributions
Skip
8.4 Transformations of 2 r. v.’s
9.1 Multivariate distributions
9.2 Order statistics
9.3 Multinomial distributions