3.6 Conditional Distributions

Transcription

1 STAT 42 Lecture Notes Conditional Distributions Definition Suppose that X and Y have a discrete joint distribution with joint p.f. f and let f 2 denote the marginal p.f. of Y. For each y such that > 0, the conditional p.f. of X given Y is. () For a particular choice of Y, say y 0, g ( y 0 ) is called the the conditional p.f. of X given that Y y 0. It is straightforward to derive equation () from the definition of conditional probability since Pr(X, Y y), f 2 () Pr(Y y) and Pr(X Y y). To verify that g ( y 0 ) is truly a p.f., note that the sum of g ( y) over all values of in the support of X is since f 2(y). In addition, 0 and 0 0 / g ( y). Eample Returning to the Titanic eample, Table shows the joint p.f. of the class and survivorship random variables. It is presumed that the eperiment of interest is that of drawing an individual at random from among the 220 individuals. Table : Outcome probabilities for the Titanic passengers and crew. Class First Second Third Crew Pr(Outcome) Survived Died Pr(Class) Let X denote survivorship according to, if the individual survives, X 0, otherwise, and let Y denote class according to, if the individual is a first-class ticket holder, 2, if the individual is a second-class ticket holder, Y 3, if the individual is a third-class ticket holder, 4, if the individual is a crew member, 0, otherwise.

2 STAT 42 Lecture Notes 59 The conditional distributions of X given Y are shown in Table 2. For eample, g ( ) Pr(X Y ) The probability that an indvidual survived given that they.48 held a first-class ticket is much larger than the conditional probability of survival for the other ticket classes. Table 2: Conditional distribution of survivorship given ticket class for the Titanic passengers and crew. Class First (y ) Second (y 2) Third (y 3) Crew (y 4) Survived ( ) Died ( 2) Question : Determine P (Y X ) and P (Y 2 X ). Definition Suppose that X and Y have a continuous joint distribution with joint p.d.f. f, and let f 2 denote the marginal p.d.f. of Y. For each y such that > 0, the conditional p.d.f. of X given Y is defined to be for all R. It s straightforward to verify that g ( y) is a probability density function (nonnegative and the integral over R is ) (see Theorem 3.6..). In this contet, the random variable is X, not Y. In essence, the distribution of X has been modified (unless X and Y are independent) by the condition that Y has taken on a particular realization (value), namely, y. Eample Suppose that a manufacturing process takes places in stages. The random variable Y is the time elapsed during the first stage and X is time elapsed over the entire process. Suppose that the joint p.d.f. of X and Y is e I {(r,s) 0 s r} (, y). The marginal distribution of Y is and the conditional distribution of X given Y is y e d e y I {0 y< } (y), ey I {(r,s) s r} (, y).

3 STAT 42 Lecture Notes 60 For eample, the conditional probability that X < 5 given that Y 3 is Pr(X < 5 Y 3) 5 3 e 3 d e e Question 2 : Determine Pr(X < 5 Y ) and Pr(X < 5 Y 4). Eample Recall the earlier eample where y, 0 2 y <, 0 otherwise. The support of f is graphed below. The lower boundary of the support is the graph of the equation y 2 and the upper boundary is the graph of the line y. The boundaries are determined by from the constraints 0 2 y <. y The marginal p.d.f. of X is f () 2 2 y dy y (2 6 ), for, 0, otherwise ( 4 ), for, 0, otherwise The conditional p.d.f. g 2 (y ) /f () is defined everywhere on the (, ) interval ecept at since f (0) 0. Thus, g 2 (y ) y ( 4 ) 2y, for 0 < 4 2 y. 0, otherwise. Mied distributions Let X be discrete and Y continuous. Then, the conditional distribution of X given Y

4 STAT 42 Lecture Notes 6 is defined in the same manner as when X and Y are both discrete (or both continuous), that is for all R, provided that > 0, where is the marginal p.d.f. of Y. Similarly, the conditional distribution of Y given X is g 2 (y ) f () for all y R, provided that f () > 0, where f () is the marginal p.d.f. of Y. Theorem (below) generalizes the conditional probability formula Pr(A B) Pr(A B) Pr(B). Theorem Suppose that X and Y are random variables and that the marginal and conditional distributions are denoted as usual. Then, g ( y) for all wherever > 0. Eample Suppose that p is the probability that a defective part is produced by a machine and that p is the realization of a random variable distributed on the unit interval. Thus, p is a realization of P Unif(0, ). If n parts are collected when the machine is operating with a defective rate of p, then, the number that are defective, given that P p, has a binomial distribution: g ( p) ( ) n p ( p) n, {0,,..., n}. Since P Unif(0, ), f 2 (p) I [0,] (p), and f(, p) g ( p)f 2 (p) ( ) n p ( p) n I [0,] (p), {0,,..., n}. This eample illustrates how the joint distribution can be determined when the situation is such that the conditional and marginal distributions are easily deduced or specified, as is the case with problem 2 on page 52. Eample: waiting times Let X be a random variable representing the time until an event occurs, say time until recovery of an individual infected with a disease. A common model for the distribution of X is the gamma p.d.f. ye y I {r 0} ().

5 STAT 42 Lecture Notes 62 This distribution uses y as a parameter that determines the epected, or mean time until recovery (more precisely, the epected time to recovery is /y). Since individuals vary across a population with respect to the epected time to recovery (because of variation in resistance to the disease), a distribution is assumed for Y, say e y I {r 0} (y). The joint distribution of X and Y is the product ye (+)y I {r 0} ()I {r 0} (y). Theorem Law of Total Probability for Random Variables The marginal p.f. or p.d.f. of the random variable X can be computed from the marginal distribution of the discrete random variable Y and the conditional p.f./p.d.f. g ( y) by computing f () y g ( y). If Y is continuous, then the marginal p.f./p.d.f. is f () g ( y) dy. Theorem If is the marginal p.f. or p.d.f. of a random variable Y and g ( y) is the conditional p.f. or p.d.f. of a random variable X given Y y, then the conditional p.f. or p.d.f. of Y given X is where f () is obtained by computing g 2 (y ) g ( y) f () f () y g ( y) or f () g ( y) dy. Eample: Choosing points from uniform distributions Suppose that X Unif(0, ) and that after a realization has been observed, a second point is randomly selected from the interval [, ]. The outcome of the draw from [, ] is represented by the random variable Y, and the objective is to determine the marginal distribution of Y. First, note that. The statement X Unif(0, ) implies that f () I (0,) ().

6 STAT 42 Lecture Notes The statement that a second point is randomly selected from the interval [, ] implies that the second point is a realization of Y Unif(, ), and hence To obtain the marginal p.d.f. of Y, we use g 2 (y ) I [,](y). First, the joint p.d.f. is Then, the marginal is d. g 2 (y )f () I [,](y)i (0,) () I {(r,s) 0<r s<}(, y). I (0,) (y) I {(r,s) 0<r s<}(, y) d y 0 d I (0,) (y) log( ) y 0 log( y)i (0,) (y). It s possible now to compute the conditional distribution of X given that Y y: I {(r,s) 0<r s<}(, y) log( y)i (0,) (y) ( ) log( y) I {(r,s) 0<r s<}(,y). (2) The indicator function I (0,) (y) in formula (2) is omitted because I (0,) (y) for all points in the set S {(, y) 0 < y < }; hence, I {(r,s) 0<r s<} (, y) I (0,) (s). Theorem Independent random variables Random variables X and Y are independent if and only if for every y > 0, f (). In other words, the conditional p.f./p.d.f. of X given Y y is the same as the marginal p.f./p.d.f. if and only if X and Y are independent. The proof is straightforward since if f (),

7 STAT 42 Lecture Notes 64 then the joint p.f./p.d.f. is f (). But f () if and only if X and Y are independent (Corollary 3.5.., p. 36). On the other hand, if X and Y are independent, then f (), and f () f ().