Chapter 4 Multiple Random Variables

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1 Review for the previous lecture Definition: n-dimensional random vector, joint pmf (pdf), marginal pmf (pdf) Theorem: How to calculate marginal pmf (pdf) given joint pmf (pdf) Example: How to calculate marginal pmf (pdf), joint pmf (pdf) Chapter 4 Multiple Random Variables Chapter 4 Conditional Distributions and Independence Definition 41: Let (, ) be a discrete bivariate random vector with joint pmf f ( xy, ) and marginal pmfs f ( y ) For any such x such that P ( = x) = f ( x) > 0, the conditional pmf of given that = x is the function of y denoted by f ( y x ) : f ( y x) = P( = y = x) = f( x, y)/ f ( x) For any such y such that P ( = y) = f ( y) > 0, the conditional pmf of given that = y is the function of x denoted by f ( x y ) : f ( x y) = P( = x = y) = f( x, y)/ f ( y) Note: f ( y x ) is a valid pmf since f ( y x) 0, because f ( xy, ) 0and f ( x) 0 being joint and marginal pmfs, and since f( y x) = f( x, y)/ f ( x) = f ( x)/ f ( x) = 1 y y Example 4: Define the joint pmf of (, ) by f (0,10) = f (0,0) = /18, f (1,10) = f (3,10) = 3/18, f (1, 0) = 4 /18, f (,30) = 4/18, that is 1

2 yx, f ( y ) 10 /18 3/18 3/18 8/18 0 /18 4/18 6/ /18 4/18 f ( x ) 4/18 7/18 4/18 3/18 1 (a) Obtain the conditional distribution of given (b) Find P ( > 10 = 1) = x Definition 43: Let (, ) be a continuous bivariate random vector with joint pdf f ( xy, ) and marginal pdfs f ( y ) For any such x such that f ( x ) > 0, the conditional pdf of given that = x is the function of y denoted by f ( y x ) : f ( y x) = f( x, y)/ f ( x) For any y such that f ( y ) > 0, the conditional pdf of given that = y is the function of x denoted by f ( x y ) : f ( x y) = f( x, y)/ f ( y) Example 44: Let the continuous random vector (, ) have joint pdf (a) Find the marginal pdf of (b) For any x such that f ( x ) > 0, find f ( y x ) x (a) (0, ) f ( x) = e I ( x) f ( xy, ) = e I ( xy, ) y, {( uv, ):0 < u< v< }

3 (b) f ( x, y) f y x = = e I y x>, ( y x) ( ) ( x, ) ( ), 0 f ( x) Calculating Expected Values Using Conditional pmfs or pdfs: Let (, ) be a discrete (continuous) bivariate random vector with joint pmf (pdf) f ( xy, ) and marginal pmfs (pdfs) f ( y ), and g ( ) is a function of, then the conditional expected value of g( ) given that = x is denoted by E( g ( ) x ) and is given by E( g ( ) x) = g( y) f( y x) ( E( g ( ) x) = g( y) f( y xdy ) ) y Example 44: (continued) (c) Find E( = x) (d) Find Var( = x) ( ) ( ) y x 1 x (c) E = x = ye dy= + x (d) ( ) y x z E = x = y e dy= ( z+ x) e dz = + x+ x, therefore, ( ) x 0 Var x E x E ( = ) = ( = ) ( ( )) = 1 Definition 45: Let (, ) be a bivariate random vector with joint pdf or pmf f ( xy, ) and marginal pdfs or pmfs f ( y ) Then and are called independent random variables, if for every x R and y R, f ( xy, ) = f ( xf ) ( y) Consequently, if and are independent, f ( y x) = f ( y) and f ( x y) = f ( x) 3

4 Technical Note: If f ( xy, ) is the joint pdf for the continuous random vector (, ) where f ( xy, ) f( xf ) ( y) on a set Asuch that f ( x, y ) dxdy= 0, then and are still called independent random A variables Example 46: Consider the discrete bivariate random vector (, ), with joint pmf given by f (10,1) = f(0,1) = f(0,) = 1/10, f (10,) = f (10,3) = 1/5, and f (0,3) = 3/10 Find the marginals of and Are and independent? Question: Can we check for independence without knowing the marginals? Lemma 47: Let (, ) be a bivariate random vector with joint pdf or pmf f ( x, y ) Then and are independent random variables if and only if there exist functions g( x ) and h( y) such that, for every x R and y R, f ( xy, ) = gxhy ( ) ( ) In this case f ( x) = cg( x) and f ( y) = dh( y), where c and d are some constants that would make f ( y ) valid pdfs or pmfs Example 48: Consider the joint pdf independent random variables f( x, y) y ( x/) = x y e, 0 x > and y > 0 By Lemma 47, and are Notes: 4

5 1 Consider the set {( x, y) : x A and y B}, where A= { x: f ( x) > 0} and B= { y: f ( x) > 0}, then this set is called a cross-product denoted by A B If f ( xy, ) is a joint pdf or pmf such that the set {( x, y) : f( x, y ) > 0} is not a cross-product, then the random variables and with joint pdf or pmf f ( xy, ) are not independent 3 If it is known that and are independent random variables with marginal pdfs (or pmfs) f ( x ) and f ( y ), then the joint pdf (or pmf) of and is given by f ( xy, ) = f( xf ) ( y) (See Example 49 for discrete case) Theorem 410: Let and be independent random variables a For any A R and B R, P ( A, B) = P ( AP ) ( B) b Let g( x ) be a function only of x and h( y ) be a function only of y Then E( g( ) h( )) = E( g( )) E( h( )) Proof: Example 411: Let and be independent exponential(1) random variables a Find the joint pdf of and b Find P ( 4, < 3) c Find E( ) Theorem 41: Let and be independent random variables with moment generating functions M () t and M () t Then the moment generating function of the random variable Z = + is given by M Z() t = M () t M() t 5

6 Proof: Example 413: Let ~ N ( µ, σ ) and ~ n( γ, τ ) Find the pdf of Z = + (This is a very important result!!) 6

Chapter 4 Multiple Random Variables

Chapter 4 Multiple Random Variables Chapter 41 Joint and Marginal Distributions Definition 411: An n -dimensional random vector is a function from a sample space S into Euclidean space n R, n -dimensional