# 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.

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