2. Let the bivariate continuous random vector (X,Y) have the following joint probability density function: c 0

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2. Let the bivariate continuous random vector (X,Y) have the following joint probability density
function:
{;
c 0<y<1- x? and – 1< x <1
f (x, y) =
0 otherwise
a. Graph the support on the two dimensional X-Y plane, and find the value of c. Hint: Integrate
with respect to y first, dydx.
Transcribed Image Text:2. Let the bivariate continuous random vector (X,Y) have the following joint probability density function: {; c 0<y<1- x? and – 1< x <1 f (x, y) = 0 otherwise a. Graph the support on the two dimensional X-Y plane, and find the value of c. Hint: Integrate with respect to y first, dydx.
b. Find the marginal probability density function of X, fr(x).
c. Find the marginal probability density function of Y, fy(y).
d. Find the conditional probability density function of X give Y=y, fæy(x|y).
e. Are X and Y independent? Why or why not.
f. Find P(Y < X + 1). Do not work out the entire integral, just set up the double integral that
needs to be solved and put in the correct limits. Hint: Sketch out the region using the support from
part a.
Transcribed Image Text:b. Find the marginal probability density function of X, fr(x). c. Find the marginal probability density function of Y, fy(y). d. Find the conditional probability density function of X give Y=y, fæy(x|y). e. Are X and Y independent? Why or why not. f. Find P(Y < X + 1). Do not work out the entire integral, just set up the double integral that needs to be solved and put in the correct limits. Hint: Sketch out the region using the support from part a.
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