(Solved): e joint pdf of \( (X, Y) \) is \( f_{(X, Y)}(x, y)=(6-x-y) / 8 \quad 0 \leq x \leq 2, \quad 2 \leq ...

e joint pdf of \( (X, Y) \) is \( f_{(X, Y)}(x, y)=(6-x-y) / 8 \quad 0 \leq x \leq 2, \quad 2 \leq y \leq 4 \) the marginal pdf of \( X \) is \( f_{X}(x)=(6-2 x) / 8 \quad 0 \leq x \leq 2 \). a. The marginal pdf of \( Y \) is \( f_{Y}(y)=(5-y) / 4 \quad 2 \leq y \leq 4 \). Find the (unconditional) probability that \( \mathrm{Y} \) is between 3 and 7 or \( \mathrm{P}(3<\mathrm{Y}<7) \). b. Find the conditional pdf of \( \mathrm{Y} \) given \( \mathrm{X}=\mathrm{x} \) : \[ f_{Y \mid X}(y \mid x)=\left\{\begin{array}{cc} \bar{n} & \ldots \ldots \leq y \leq \ldots . . \\ 0 & \text { otherwise } \end{array} ; \text { where } \ldots \ldots \leq x \leq \ldots \ldots\right. \] c. Use the conditional pdf of \( Y \) given \( X=x \) to find the conditional probability that \( Y \) is between 3 and 7 given that \( \mathrm{X}=\mathrm{x} \) or \( \mathrm{P}(3<\mathrm{Y}<7 \mid \mathrm{X}=\mathrm{x}) \). Note: your answer will be in terms of \( x \). d. Evaluate part c) for the following 3 cases: i. when \( X=0.5 \) resulted Compare these answers to the ii. when \( X=1 \) resulted answer in part a) in order to see iii. when \( X=2 \) resulted how observing \( X=x \) changes the probability \( 3<\mathrm{Y}<7 \).