INTRODUCTION TO STATISTICS AND ECONOMETRICS

JOINT DISTRIBUTION OF DISCRETE AND CONTINUOUS RANDOM VARIABLES

In Section 3.2 we studied the joint distribution of discrete random vari­ables and in Section 3.4 we studied the joint distribution of continuous random variables. In some applications we need to understand the char­acteristics of the joint distribution of a discrete and a continuous random variable.

Let X be a continuous random variable with density /(x) and let Y be a discrete random variable taking values yit і — 1, 2, . . . , n, with prob­abilities Р(уг) . If we assume that X and Y are related to each other, the best way to characterize the relationship seems to be to specify either the conditional density/(x | уг) or the conditional probability P(yl x). In this section we ask two questions: (1) How are the four quantities/(x), Р(уг), f(x yi), and Р(Уіх) related to one another? (2) Is there a bivariate
function ф(х, уsuch that P{a < X < b, Y Є S) = fba ЕгЄ/ ф(х, уг)dx, where I is a subset of integers (1,2,..., n) and S = {$ | і Є /}?

Note that, like any other conditional density defined in Sections 3.3 and 3.4, /(x I yi) must satisfy

(3.7.1) I /(x I yi)dx = P(a < X < bY = yd J a

for any a ^ b.

Since the conditional probability P(y, |x) involves the conditioning event that happens with zero probability, we need to define it as the limit of P(yt x<X^x + e)ase goes to zero. Thus we have

(3.7.2) P(yi I x) = lim P(Y = y; | x s X < x + e)

e->0

by the mean value theorem,

which provides the answer to the first question. Next consider

(3.7.3) [ X /(* I ydPiyddx

a ІЄІ

= Y<p(a - x - bY = УдР(У = уд by (3.7.1)

ІЄІ

= ^ P{a ^ X < b, Y = y{) by Theorem 2.4.1

ІЄ1

= P(a<X <6, T Є S)

where 5 = [уі і Є I). Thus /(x у-)Р(у{) plays the role of the bivariate function ф(х, yi) defined in the second question. Hence, by (3.7.2), so does Р(уі I x)f(x).

EXERCISES

1. (Section 3.2.3)

Let X, Y, and Z be binary random variables each taking two values, 1 or 0. Specify a joint distribution in such a way that the three variables are pairwise independent but not mutually independent.

2. (Section 3.4.3)

Given the density f(x, y) = 2(x + y),0<x<l,0<y<x, calcluate

(a) P(X < 0.5, Y < 0.5).

(b) P(X < 0.5).

(c) P(Y < 0.5).

3. (Section 3.4.3)

Let X be the midterm score of a student and Y be his final exam score. The score is scaled to range between 0 and 1, and grade A is given to a score between 0.8 and 1. Suppose the density of X is given by

f(x) = 1, 0 < x < 1

and the conditional density of Y given X is given by

f(y I x) = 2xy + 2(1 — x) (1 — y), 0 < x < 1, 0 < у < 1.

What is the probability that he will get an A on the final if he got an A on the midterm?

4. (Section 3.4.3)

Let the joint density of (X, Y) be given by

f(x, y)= 2, 0 < x < 1, 0 < у < 1 — x.

(a) Calculate marginal density f{x).

(b) Calculate P(0 < Y < %| X = 0.5).

5. (Section 3.6)

Given f(x) = exp(—x), x > 0, find the density of the variable

(a) Y = 2X + 1.

(b) Y = X2.

(c) Y = 1/X.

(d) Y = log X. (The symbol log refers to natural logarithm through­out.)

6. (Section 3.6)

Let X have density f(x) = 0.5 for —1 < x < 1. Define Y by Y = X + 1 if 0 < X < 1 = —2X if-1 < X < 0.

Obtain the density of Y.

7. (Section 3.6)

Assuming f(x, y) = 1, 0 < x < 1, 0 < у < 1, obtain the density of Z = X-Y.

8. (Section 3.6)

Suppose that U and V are independent with density f(t) = exp( — t), t > 0. Find the conditional density of X given Y if X = U and Y = U + V.