Fundamentals of Probability and Statistics for Engineers

62 Fundamentals of Probability and Statistics for Engineerswhich is expected. It gives the relationship between the joint jpmf and theconditional mass function. As we will see in Example 3.9, it is sometimes moreconvenient to derive joint mass functions by using Equation (3.35), as conditionalmass functions are more readily available.If random variables X and Y are independent, then the definition of independence,Equation (2.16), impliesp XY …xjy† ˆp X …x†;…3:36†and Equation (3.35) becomesp XY …x; y† ˆp X …x†p Y …y†:…3:37†Thus, when, and only when, random variables X and Y are independent, theirjpmf is the product of the marginal mass functions.Let X be a continuous random variable. A consistent definition of theconditional density function of X given Y ˆ y, f XY xjy), is the derivative ofits corresponding conditional distribution function. Hence,where F XY (xjy) is defined in Equation (3.33). To see what this definition leadsto, let us considerIn terms of jpdf f XY (x, y), it is given byBy setting x 1 ˆ 1, x 2 ˆ x, y 1 ˆ y,and y 2 ˆ y ‡ y, and by taking the limity ! 0, Equation (3.40) reduces toprovided that f Y (y) 6ˆ 0.f XY …xjy† ˆdF XY…xjy†; …3:38†dxP…x 1 < X x 2 jy 1 < Y y †ˆP…x 1 < X x 2 \ y 1 < Y y 2 †2 : …3:39†P…y 1 < Y y 2 †P…x 1 < X x 2 jy 1 < Y y 2 †ˆˆZ y2Z x2y 1 x 1Z y2Z x2y 1F XY …xjy† ˆx 1f XY …x;y†dxdyZ y2Z 1f XY …x;y†dxdyy 1 1Z y2f XY …x;y†dxdy f Y …y†dy: …3:40†y 1Z x1f XY …u; y† duf Y …y†; …3:41†TLFeBOOK