Question

4. Suppose that the joint p.d.f. of two random variables X and Y is as follows: $f(x, y) = \begin{cases} c(x + y^2) & \text{for } 0 \le x \le 1 \text{ and } 0 \le y \le 1, \ 0 & \text{otherwise.} \end{cases}$ Determine (a) the conditional p.d.f. of X for every given value of Y, and (b) Pr(X < \frac{1}{2}|Y = \frac{1}{2}).

          4. Suppose that the joint p.d.f. of two random variables X and Y is as follows:
$f(x, y) = \begin{cases} c(x + y^2) & \text{for } 0 \le x \le 1 \text{ and } 0 \le y \le 1, \
0 & \text{otherwise.} \end{cases}$
Determine (a) the conditional p.d.f. of X for every given value of Y, and (b) Pr(X < \frac{1}{2}|Y = \frac{1}{2}).

4. Suppose that the joint p.d.f. of two random variables X and Y is as follows:
f(x, y) = c(x + y^2) for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1, 0 otherwise.
Determine (a) the conditional p.d.f. of X for every given value of Y, and (b) Pr(X < (1)/(2)|Y = (1)/(2)).

Added by Alfredo R.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Donna Densmore

Step 1

d.f. of Y. To find the marginal p.d.f. of Y, we need to integrate the joint p.d.f. over the range of X. Since the joint p.d.f. is only non-zero for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1, we can write the integral as follows: f_Y(y) = ∫[0,1] f(x,y) dx = ∫[0,1] c(x+y) dx = Show more…

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Suppose that the joint p.d.f. of two random variables X and Y is as follows: f(x,y) = c(x+y) for 0 â‰¤ x â‰¤ 1 and 0 â‰¤ y â‰¤ 1, 0 otherwise. Determine the conditional p.d.f. of X for every given value of Y, and find Pr(X|Y).