We start out with a couple of defintions and examples.
Definition: Let X and Y have joint pdf f(x,y). The conditional pdf
of Y given X = x (resp. of X given Y = y) is defined by
h(y|x) = f (x, y) resp. g(x|y) = f (x, y) f1(x) f2(y)
If A is a subset of the real line, then
P(Y ∈A|X =x)= h(y|x)dy resp. P(X ∈A|Y =y)= g(x|y)dx . AA
Example 1 (seen in class) Consider the joint pdf of the random variables X and Y : 2, if 0 < x ≤ y < 1
Recall, the marginal pdf of X if f1(x) = 2(1−x), x ∈ (0,1) and the marginal pdf of Y is f2(y) = 2y, y ∈ (0,1). Both X and Y have the interval (0,1) as their support (i.e., their pdf’s are equal to 0 outside (0, 1)).
f (x, y) = 0, otherwise
Given y ∈ (0, 1), the conditional density of X , given Y = y
is
g(x|y)= f(x,y) = 2 = 1. x∈(0,1).
f2(y) 2y y It should be clear that g(x|y) = 0 if x ̸∈ (0, 1). Therefore,
1/y, if 0 < x < y, y ∈ (0, 1)
g(x|y) = 0, otherwise (1)
Keep in mind that in equation (1), y is held constant while x varies, subject to the stated constraint. It is easy to see that given Y = y (y ∈ (0, 1)), the conditional distribution of X is the uniform distribution over the interval (0, y).
As a numerical application, let’s compute the conditional probability of the event X ≤ 1/4) given Y = 1/2. By definition,
1/4 1/4 P((X ≤ 1/4|Y = 1/2) = g(x|1/2)dx =
00
2dx = 1/2.
Your turn: We reverse the roles of X and Y . (use the back page
if you run out of space)
(i) What is the conditional pdf, h(y|x), of Y given X = x, x ∈ (0,
1). Present your result in the form of
equation (1) above and name the distribution (it’s recognaizable). (ii) Compute P (Y > 5/9|X = 2/3).
Example 2 (also seen in class) Consider the joint pdf of the random variables X and Y : 2e−x−y, if 0 < x ≤ y < ∞
Recall, the marginal pdf of X if f1(x) = 2e−2x, x ∈ (0, ∞) and the marginal pdf of Y is f2(y) = 2e−y(1−e−y), y ∈ (0,∞). Both X and Y have the interval (0,∞) as their support (i.e., their pdf’s are equal to 0 outside (0, ∞)).
Given x ∈ (0, ∞), the conditional density of Y , given X = x is f (x, y) 2e−x−y
f (x, y) = 0, otherwise
h(y|x) = f1(x) = 2e−2x = ex−y. y > x. It should be clear that h(y|x) = 0 if y ∈ (0, x]. Therefore,
ex−y, if y > x, x > 0
h(y|x) = 0, otherwise (2)
Keep in mind that in equation (2,) x is held constant while y varies, subject to the stated constraint (h(y|x) is not a recognizable pdfl–it’s sometimes called the shifted exponential distribution).
As a numerical application, let’s compute the conditional probability of the event Y ≤ 3) given Y = 1. By definition,
3 3
P((Y ≤3|X=1)= h(y|1)dy= e1−ydy=1−e−2.
11
Your turn: We reverse the roles of X and Y . (use the back page
if you run out of space)
(i) What is the conditional pdf, g(x|y), of X given Y = y, y ∈ (0,
∞). Present your result in the form of
equation (2) (the distribution is not recognaizable). (ii) Compute P (X > 1|Y = 2).
We start out with a couple of defintions and examples. Definition: Let X and Y have...
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2. Let the random variables X and Y have the joint PDF given below: 2e -y 0 xyo0 fxy (x, y) otherwise 0 (a) Find P(X Y < 2) (b) Find the marginal PDFs of X and Y (c) Find the conditional PDF of Y X x (d) Find P(Y< 3|X = 1)
2. Let the random variables X and Y have the joint PDF given below: S 2e-2-Y 0 < x < y < fxy(x,y) = { 0 otherwise (a) Find P(X+Y < 2). (b) Find the marginal PDFs of X and Y. (c) Find the conditional PDF of Y|X = r. (d) Find P(Y <3|X = 1).
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Suppose X and Y have the joint pdf f (x, y) = 3y, 0 < y < 1, y − 1 < x < 1 − y 0 otherwise a) Give an expression for P (X > Y ). b) Find the marginal pdfs for Y . c) Find the conditional pdf of X given Y = y, where 0 < y < 1. d) Give an expression for E[XY ]. e) Are X and Y independent?
5. Let X have a uniform distribution on the interval (0,1). Given X = x, let Y have a uniform distribution on (0, 2). (a) The conditional pdf of Y, given that X = x, is fyıx(ylx) = 1 for 0 < y < x, since Y|X ~U(0, X). Show that the mean of this (conditional) distribution is E(Y|X) = , and hence, show that Ex{E(Y|X)} = i. (Hint: what is the mean of ?) (b) Noting that fr\x(y|x) =...
Let X and Y be continuous rvs with a joint pdf of the form: ?k(x+y), if(x,y)∈?0≤y≤x≤1? f(x,y) = 0, otherwise (a) Find k. (b) Find the joint CDF F (x, y). 0, otherwise (c) Find the conditional pdfs f(x|y) and f(y|x) (d) Find P[2Y > X] (e) Find P[Y + 2X > 1]
PROBLEM 1 Let the joint pdf of (X,Y) be f(x, y)= xe", 0<y<<< a. Compute P(X>Y). b. What is the conditional distribution of X given Y=y? Are X and Y independent? c. Find E(X|Y = y). d. Calculate cov(X,Y).
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