We are given the probability mass function (PMF) of two discrete random variables $X$ and $Y$ in a tabular form. Let’s calculate the required probabilities step by step.

1. Find $P(X = 0, Y = 2)$:

From the table, the probability $P(X = 0, Y = 2)$ corresponds to the entry in the row $X = 0$ and column $Y = 2$. From the table:

$P(X = 0, Y = 2) = 0.075$

---

2. Find the marginal PMF of $X$:

The marginal PMF of $X$, denoted as $P(X = x)$, is obtained by summing over all values of $Y$ for each fixed $X = x$:

$P(X = x) = \sum_{y} P(X = x, Y = y)$

Calculation:

• For $X = 0$: $P(X = 0) = P(X = 0, Y = 0) + P(X = 0, Y = 1) + P(X = 0, Y = 2) = 0.05 + 0.15 + 0.075 = 0.275$

• For $X = 1$: $P(X = 1) = P(X = 1, Y = 0) + P(X = 1, Y = 1) + P(X = 1, Y = 2) = 0.045 + 0.05 + 0.025 = 0.12$

• For $X = 2$: $P(X = 2) = P(X = 2, Y = 0) + P(X = 2, Y = 1) + P(X = 2, Y = 2) = 0.1 + 0.04 + 0.065 = 0.205$

• For $X = 3$: $P(X = 3) = P(X = 3, Y = 0) + P(X = 3, Y = 1) + P(X = 3, Y = 2) = 0.2 + 0.05 + 0.15 = 0.4$

Thus, the marginal PMF of $X$ is: $P(X = 0) = 0.275, \quad P(X = 1) = 0.12, \quad P(X = 2) = 0.205, \quad P(X = 3) = 0.4$

---

3. Find $P(X > Y)$:

To compute $P(X > Y)$, we sum the probabilities where $X > Y$. This means summing over all table entries where the value of $X$ is strictly greater than the corresponding value of $Y$. From the table:

When $X = 1$:

• $P(X = 1, Y = 0) = 0.045$

When $X = 2$:

• $P(X = 2, Y = 0) = 0.1$
• $P(X = 2, Y = 1) = 0.04$

When $X = 3$:

• $P(X = 3, Y = 0) = 0.2$
• $P(X = 3, Y = 1) = 0.05$
• $P(X = 3, Y = 2) = 0.15$

Now, sum these probabilities: $P(X > Y) = 0.045 + 0.1 + 0.04 + 0.2 + 0.05 + 0.15 = 0.585$

---

Final Results:

1. $P(X = 0, Y = 2) = 0.075$
2. Marginal PMF of $X$: $P(X = 0) = 0.275, \quad P(X = 1) = 0.12, \quad P(X = 2) = 0.205, \quad P(X = 3) = 0.4$
3. $P(X > Y) = 0.585$

---

Do you want me to verify the calculations or explain a specific part in more detail?

---

Related Questions:

1. How is the joint PMF of $X$ and $Y$ used to calculate marginal probabilities?
2. What does $P(X > Y)$ represent in terms of real-world probability?
3. How do we calculate $P(Y > X)$, and would it complement $P(X > Y)$?
4. Can we use marginal PMFs to check if $X$ and $Y$ are independent?
5. What if we were asked for $P(X \leq Y)$? How would we compute it?

Tip:

When working with probabilities, always check if the total probability of all events sums to 1 to verify the correctness of your PMF table.