Question

1. Let (X,Y ) be a pair of random variables with joint pdf given by f(x,y) = 1(0 < x < 1,0 < y < 1).

(a) Find P(X + Y ≤ 1).

(b) Find P(|X −Y|≤ 1/2).

(c) Find the joint cdf F(x,y) of (X,Y ) for all (x,y) ∈R×R.

(d) Find the marginal pdf fX of X. (e) Find the marginal pdf fY of Y .

(f) Find the conditional pdf f(x|y) of X|Y = y for 0 < y < 1.

Answer #1

1) The given joint PDF is .

a)Consider the sketch below. The support is shown shaded.

The probability is the area of the region below the red line.

b)Consider the sketch below. The probability is the area of the region shown shaded.

The area is (total area - area of two white triangles = 1 - 2(0.5)(0.5)(0.5))

c) The joint CDF is

d) The marginal PDF of X is

e) The marginal PDF of Y is

f) The conditional PDF is

4. Let X and Y be random variables having joint probability
density function (pdf) f(x, y) = 4/7 (xy − y), 4 < x < 5 and
0 < y < 1
(a) Find the marginal density fY (y).
(b) Show that the marginal density, fY (y), integrates to 1
(i.e., it is a density.)
(c) Find fX|Y (x|y), the conditional density of X given Y =
y.
(d) Show that fX|Y (x|y) is actually a pdf (i.e., it integrates...

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(c) Find the pdf of Y by taking the derivative of FY (y) with
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a. Find the conditional density F xly (xly)
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fX,Y(x, y) = nx^ne^(?xy) , 0 < x < 1, y > 0,
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Let X and Y be continuous random variables with joint density
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may be more than one)
E[X^2Y^3]=(∫0 TO 1 x^2 dx)(∫0 TO 1 y^3dy)
E[X^2Y^3]=∫0 TO 1∫0 TO 1x^2y^3 f(x,y) dy dx
E[Y^3]=∫0 TO 1 y^3 fX(x) dx
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