Let's break down and solve each part of this problem.

Problem 1: Joint Probability Density Function

Given: $f(x, y) = \frac{1}{5} (x + 2y) \quad \text{for } 0 < x < 1 \text{ and } 0 < y < 2$ $f(x, y) = 0 \quad \text{elsewhere}$

We are asked to find the expected value $E(xy)$.

To find $E(xy)$, we use: $E(xy) = \int_{0}^{1} \int_{0}^{2} x y f(x, y) \, dy \, dx$ Substitute $f(x, y) = \frac{1}{5}(x + 2y)$: $E(xy) = \int_{0}^{1} \int_{0}^{2} x y \cdot \frac{1}{5} (x + 2y) \, dy \, dx$

Steps:

1. Expand the integrand $x y \cdot \frac{1}{5}(x + 2y)$.
2. Integrate with respect to $y$ first, from $0$ to $2$.
3. Integrate the result with respect to $x$, from $0$ to $1$.

Problem 2: Joint Probability Mass Function

The table of the joint probability mass function $p(x, y)$ is given for $x \in \{0, 1\}$ and $y \in \{0, 1, 2, 3\}$.

Part (a): Find $P(x + y < 2)$

This probability is the sum of $p(x, y)$ values where $x + y < 2$. From the table:

• For $(x = 0, y = 0): x + y = 0$
• For $(x = 0, y = 1): x + y = 1$
• For $(x = 1, y = 0): x + y = 1$

So, $P(x + y < 2) = p(0, 0) + p(0, 1) + p(1, 0)$ Using the table values: $P(x + y < 2) = 0.1 + 0.25 + 0.07 = 0.42$

Part (b): Find the Marginal Probability Density of $x$

The marginal probability of $x$, $p(x)$, is found by summing the probabilities across all values of $y$ for each $x$:

• For $x = 0$: $p(x=0) = p(0, 0) + p(0, 1) + p(0, 2) + p(0, 3) = 0.1 + 0.25 + 0.2 + 0.1 = 0.65$
• For $x = 1$: $p(x=1) = p(1, 0) + p(1, 1) + p(1, 2) + p(1, 3) = 0.07 + 0.0 + 0.15 + 0.13 = 0.35$

So, the marginal probability density of $x$ is: $p(x=0) = 0.65, \quad p(x=1) = 0.35$

Would you like further details on the integration steps for $E(xy)$ or any specific part?

Related Questions

1. What is the marginal probability density of $y$ in Problem 2?
2. How do we interpret the expected value $E(xy)$ in a real-world context?
3. What conditions make a joint probability density function valid?
4. How would $E(x)$ and $E(y)$ be computed for the distribution in Problem 1?
5. In Problem 2, how would we calculate $P(x + y = 3)$?

Tip

For joint distributions, remember that marginal probabilities are calculated by summing or integrating over the unwanted variable.