Bivariate Probability Distribution

1
Chapter 5

• Bivariate Probability Distribution

2
Section 5.2

• Bivariate Probability Distributions

3
Bivariate Probability Distributions

• Joint Discrete Random Variables
• Joint cumulative distribution function (cdf)
  F(y1,y2)
• Joint probability mass function (pmf) p(y1,y2)
• Joint Continuous Random Variables
• Joint cumulative distribution function (cdf)
  F(y1,y2)
• Joint probability density function (pdf) p(y1,y2)

4
Joint probability mass function (pmf)

• Definition 5.1 Let Y1 and Y2 be discrete random
  variables. The joint (or bivariate) probability
  mass distribution for Y1 and Y2 is given by
• p(y1,y2)P(Y1y1, Y2y2)
• where -??y1 ??, -??y2 ??
• The function p(y1,y2) is called the joint
  probability mass function (pmf) of Y1 and Y2 .

5
Properties of pmf

• Theorem 5.1 If Y1 and Y2 be discrete random
  variables with joint probability mass
  distribution p(y1,y2), then
• (1) p(y1,y2) ?0 for all y1,y2
• (2) ?y1 ?y2 p(y1,y2) 1, where the sum is over
  all values (y1,y2) that are assigned nonzero
  probabilities.

6
Cumulative distribution function( cdf)

• Definition 5.2 For any random variables Y1 and
  Y2, the joint (bivariate) distribution function
  (or cumulative distribution function, cdf)
  F(y1,y2) is given by
• F(y1,y2)P(Y1?y1, Y2?y2)
• where -??y1 ??, -??y2 ??

7
Joint Probability Density Function (pdf)

• Definition 5.3 Let Y1 and Y2 be continuous
  random variables with joint distribution function
  F(y1,y2). If there exists a nonnegative function
  f(y1,y2) such that
• then Y1 and Y2 are said to be jointly continuous
  random variables. The function f(y1,y2) is called
  the joint probability density function (pdf).

8
Properties of cdf

• Theorem 5.2 If Y1 and Y2 are random variables
  with joint cumulative distribution function
  F(y1,y2), then
• (1) F(-?,-?)F(-?,y2)F(y1,-?)0
• (2) F(?, ?)1
• (3) If y1?y1 and y2?y2, then
• F(y1,y2)-F(y1,y2)-F(y1,y2)F(y1,y2) ?0

9
Properties of pdf

• Theorem 5.3 If Y1 and Y2 are jointly continuous
  random variables with a joint density function
  (pdf) given by f(y1,y2), then
• (1) f(y1,y2) ?0 for all y1, y2
• (2)

10
Section 5.3

• Marginal Probability Distribution
• Conditional Probability Distribution

11
Marginal Probability Functions

• Definition 5.4
• (1) Let Y1 and Y2 be jointly discrete random
  variables with joint probability mass function
  p(y1,y2). Then the marginal probability functions
  of Y1 and Y2, respectively, are given by
• (2) Let Y1 and Y2 be jointly continuous random
  variables with joint density function f(y1,y2).
  Then the marginal density function of Y1 and Y2,
  respectively, are given by

12
Conditional Discrete Probability Function

• Definition 5.5 If Y1 and Y2 are discrete random
  variables with joint probability mass
  distribution p(y1,y2) and marginal probability
  function p1(y1) and p2(y2), respectively, then
  the conditional discrete probability function of
  Y1 given Y2 is

13
Conditional Cumulative Distribution Function

• Definition 5.6 If Y1 and Y2 are jointly
  continuous random variables with joint
  probability density function f(y1,y2), then the
  conditional cumulative distribution function of
  Y1 given Y2y2 is

14
Conditional Probability Density Function

• Definition 5.7 If Y1 and Y2 are jointly
  continuous random variables with joint
  probability density function f(y1,y2) and
  marginal density function f1(y1) and
  f2(y2),respectively. For any y2 such that
  f2(y2)gt0, the conditional density function of Y1
  given Y2y2 is given by
• and, for any y1 such that f1(y1)gt0, the
  conditional density of Y2 given Y1y1 is given by
  

15
Section 5.4

• Independent Random Variables

16
Independent Random Variables

• Definition 5.8 Let Y1 and Y2 have marginal
  cumulative distribution function F1(y1)and
  F2(y2), respectively, and Y1 and Y2 have joint
  cumulative distribution function F(y1, y2). Then
  Y1 and Y2 are said to be independent if only if
• F(y1, y2) F1(y1) F2(y2)
• for every pair of real numbers (y1, y2).

17
Properties of Independent Random Variables

• Theorem 5.4 If Y1 and Y2 are discrete random
  variables with joint probability mass
  distribution p(y1,y2) and marginal probability
  function p1(y1) and p2(y2), respectively, then Y1
  and Y2 are the independent if only if
• p(y1,y2) p1(y1) p2(y2)
• for all pairs of real numbers (y1,y2).
• If Y1 and Y2 are continuous random variables with
  joint probability density function f(y1,y2) and
  marginal densities functions f1(y1) and
  f2(y2),respectively, then Y1 and Y2 are
  independent if only if
• f(y1,y2) f1(y1) f2(y2)
• for all pairs of real numbers (y1,y2).

18
Theorem for Independence

• If Y1 and Y2 have joint probability density
  function f(y1,y2) and
• f(y1,y2)gt0, if a y1 b and cy2d
• f(y1,y2)0, elsewhere
• Then Y1 and Y2 are independent if only if
• f(y1,y2) g(y1) h(y2)
• where g(y1) is a nonnegative function of y1
  alone and h(y2) is nonnegative function of y2
  alone.

19
Section 5.5

• Expected Value of Function of Random Variables

20
Expected Value of Function of Random Variables

• Definition 5.9
• (1) Let g(Y1,Y2 ) be a function of the discrete
  random variables, Y1,Y2 with joint probability
  mass function p(y1,y2). Then the expected value
  of g(Y1,Y2 ) is
• (2) If Y1 and Y2 are continuous random variables
  with joint density function f(y1,y2). Then the
  expected value of g(Y1,Y2 ) is

21
Section 5.6

• Properties of Expected Value

22
Properties of Expected Value

• Theorem 5.6 Let c be a constant, then Ecc
• Theorem 5.7 Let g(Y1,Y2 ) be a function of the
  random variables Y1 and Y2, and let c be a
  constant. Then
• Theorem 5.8 Let g1(Y1,Y2 ), g2(Y1,Y2 ),,
  gk(Y1,Y2 ) be functions of random variables Y1
  and Y2, and let c1, c2,, c1 be constants. Then

23
Expectation of Independent Random Variables

• Let Y1 and Y2 be independent random variables and
  g(Y1) and h(Y2) be functions of only Y1 and Y2,
  respectively. Then
• provided that the expectations exists.

24
Section 5.7

• Covariance of Two Random Variables

25
Covariance and Correlation Coefficient

• Definition 5.10 If Y1 and Y2 are random
  variables with means and standard deviations
  ?1,?1 and ?2, ?2, respectively, the covariance of
  Y1 and Y2 is given by
• Cov(Y1,Y2)E(Y1- ?1)(Y2- ?2)
• Correlation Coefficient
• Values of ? and Interpretation -1 ? 1

26
Calculation of Covariance

• Theorem 5.10 Let Y1 and Y2 are random variables
  with means ?1 and ?2 , respectively, then
• Cov(Y1,Y2)E(Y1- ?1)(Y2- ?2)
• EY1Y2 - EY1EY2

27
Covariance of Independent Random Variables

• Theorem 5.11 Let Y1 and Y2 are independent
  random variables , then
• Cov(Y1,Y2)0
• If the covariance of two random variables is
  zero, the variables need not be independent.

28
Homework

• p253 5.75, 5.77, 5.78, 5.81

29
Section 5.8

• Expected Value and Variance of Linear Functions
  of Random Variables

30
Expected Value and Variance of Linear Functions
of Random Variables

• Let Y1,Y2 , Yn and X1, X2 ,Xm be random
  variables with EYi?i and EXj?j. Define
• for constants a1, a2 , an and b1, b2 , bm.
  Then the following hold
• (a)
• (b)
• (c)

31
Expected Value and Variance of Linear Functions
of Two Random Variables

• Let Y1,Y2 be random variables with EY1?1 and
  EY2?2. Define
• for constants a1, a2. Then the following hold
• (a)
• (b)
• Special case a1a21, then

32
Homework

• p261 5.87,5.89,5.92.