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In: Statistics and Probability

Let X and Y be two continuous random variables with the joint probability density function of...

Let X and Y be two continuous random variables with the joint probability density function of for 0 < x < 2, 0 < y < 2, x + y < 1,where c is a constant. (In all the following answers, you do NOT need to find what the value of c is; just treat it as a number.)

(a) Write out the marginal distribution of Y.

(b) P(Y < 1/3) = ?

(c) P(X < 1.5, Y < 0.5)= ? (This question may not be as easy as it looks; be careful.)

(d) Write out the conditional distribution of Xgiven Y = 1/2; that is, write out .

(e) E[X|Y = 1/2] = ?

(f) It is given thatE[X] = E[Y] = c/60.Calculate the covariance ofXandY. (In case you may need,(1 - x)^3 = 1 - 3x + 3 x^2 - x^3.)

(g)Are X and Y independent? (Yes or no, you have to provide mathematical justifications; otherwise, no credit will be given.)

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orchestra answered 2 years ago
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