Question

The joint p.d.f of \(X\) and \(Y\) is given by

$$ f(x, y)=\left\{\begin{array}{ll} c(1-y), & 0 \leq x \leq y \leq 1 \\ 0 & \text { otherwise. } \end{array}\right. $$

Determine the value of \(c\). Find the marginal density of \(X\) and the marginal density of \(Y\) Find the conditional density of \(X\) given \(Y\). Are \(X\) and \(Y\) independent? Why? Find \(E(X-2 Y)\).

Answer 1

TOPIC:Joint pdf,marginal pdf,conditional pdf,independence and Expectation of random variables.

- The joint bold of x and y is Hx, a) = { c (1-9); OS X = 4 21. o otherwise. since, have, sex, I) is a joint pdf; we must - SS tem, da by S'{S: C(4) dy } de 12 → So [- ] dx= * S [(-1) - (x-3)) dx =1 * S. (1-z* 3) de al 9 [- + 12.

So, the joint pdf of (x,x) reduces to =) dx,y) = 6(1-7) ; Jon, osasysli) ; otherwise, The marginal pdf of x is - de 39 - Hexe 7) dy +60+3) 8y * 6 [(-4)-(2-3) = 3 (x²= 2x +D -3 (2-7², when, of asli (dx (x) = { 3 (x-1)² ; oenssi j otherwise.

» Again, the marginal pdf of Y is - → 0, (+) = Home) de. 1861) dx. = 6(1-1). [2] 64 (1-2); do osys11 So, we have, y dy(a) = $ 64(1-4) ; 03761 jotherwise . Ang ( The conditional pdf of x given y is - OxJy dx = (x) = f(x, y) d () 64(1-7) (2) - 3470527 1 0 < x (iv) observe that *tx (). Jy (4) = [3 (x-2²]. [64 (1-2)]. = 18 8 (1-4) (x-1) ². & HX, Y);&x, J. So, we have, JexeY) & dx (2). dy lf) ; x (x,y) Hence, X and Y are not independent. Ans. Now, we have, E(X) = {xdx (x) dx . 3. f x (x-1) de 23.2x2 +a) di 3 بابا [ +]. Again, we have *E(Y) = 1 y. dy gdy 65 841-4) dy. = 6 f (y? _ 43) dy. -6 ( 26.(-4)- 2 Hence, E (x-27). = E(+1-2.E(Y). - - 2x Linearity Expectation.

(iv) observe that *tx (). Jy (4) = [3 (x-2²]. [64 (1-2)]. = 18 8 (1-4) (x-1) ². & HX, Y);&x, J. So, we have, JexeY) & dx (2). dy lf) ; x (x,y) Hence, X and Y are not independent. Ans. Now, we have, E(X) = {xdx (x) dx . 3. f x (x-1) de 23.2x2 +a) di

3 بابا [ +]. Again, we have *E(Y) = 1 y. dy gdy 65 841-4) dy. = 6 f (y? _ 43) dy. -6 ( 26.(-4)- 2 Hence, E (x-27). = E(+1-2.E(Y). - - 2x Linearity Expectation.

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