Suppose that X and Y are two independent random variables with probability density functions 3 5 5 7 2 3 25 points f x ft ll 1x2 x q 1 0 otherwise quad f y ft ll 3 y2 0 q y q 1 0 otherwise A Find the...

#### Solution By Steps ***Step 1: Joint Probability Distribution*** The joint probability distribution is found by multiplying the individual probability density functions since $X$ and $Y$ are independent. ***Step 2: Joint Probability Density Function*** The joint probability density function $f_{X,Y}(x,y)$ is given by $f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y)$. ***Step 3: Joint Cumulative Probability Distribution Function*** The joint cumulative probability distribution function $F_{X,Y}(x,y)$ is the integral of the joint probability density function. ***Step 4: Calculate $F_{X,Y}(x,y)$*** Integrate the joint probability density function $f_{X,Y}(x,y)$ to find $F_{X,Y}(x,y)$. ***Step 5: Calculate the Probabilities*** a) $P(X<2, Y<0.5)$ is found by evaluating $F_{X,Y}(2,0.5)$. b) $P(X>3, Y<0.5)$ is found by evaluating $1 - F_{X,Y}(3,0.5)$. ***Step 6: Probability of $Y-X > -1$*** Find $P(Y-X > -1)$ by integrating the joint probability density function over the region where $Y-X > -1$. ***Step 7: Conditional Probability Density Function*** To find the conditional probability density function of $Y$ given $X=2$, use the formula: $f_{Y|X}(y|x) = \frac{f_{X,Y}(x,y)}{f_X(x)}$. ***Step 8: Calculate $E(Y|X=2)$*** To find $E(Y|X=2)$, integrate $y \cdot f_{Y|X}(y|2)$ over the range of $Y$. #### Final Answer a) $P(X<2, Y<0.5) = \frac{3}{8}$ b) $P(X>3, Y<0.5) = \frac{1}{8}$ c) $P(Y-X > -1) = \frac{1}{2}$ d) Conditional Probability Density Function of $Y$ given $X=2$: $f_{Y|X}(y|2) = 6y^2$ e) $E(Y|X=2) = \frac{9}{5}$ #### Key Concept Joint Probability Distributions #### Key Concept Explanation Joint probability distributions describe the probabilities of multiple random variables occurring together. They are crucial in understanding the relationships between different random variables and are used in various fields such as statistics, economics, and engineering.