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Q: Assistance in understanding and solving this example from Probability and Statistical (Conditional Distributions) with the...

Q: Assistance in understanding and solving this example from Probability and Statistical (Conditional Distributions) with the steps of the solution to better understand, thanks.

**Please give the step by steps with details to completely see how the solution came about.

1) Suppose X and Y both take values in [0,1] with joint probability density f(x,y) = 4xy.

a) Find fx(x) and fy(y), the marginal probability density functions.

b) Are the two random variables independent? Why or why not?

c) Compute the means and variances of X and Y.

d) Find P(X \leq Y).

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Answer #1

1)

a)

f(x) = int f(x,y)dy = int 4xy dy = 2x

similarly

-jj | r» (z.vydr _ tryiz-Zy

b)

Yes , two variables are independent as

f(x,y) = f(x) f(y)

c)

E(X) = int x f(x) dx =

Z.2xd.r=- (Decimal : 0.66667 )

Var(X) = E(X^2) - (E(X))^2

E(X2)=/ z? . 2rdr_ (Decimal : 0.5)

hence

Var(X) = 1/2 - (2/3)^2

= 0.05555

E(Y) = E(X) = 2/3

Var(Y) = Var(X) = 0.05555    {X and Y are symmetric}

d)

PlXSM.OC 4xydydx=- (Decimal : 0.5)

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