Let's solve the given problem step by step.

1. Evaluate the marginal distribution of $X$ and $Y$:

The table shows the joint probability distribution of two random variables $X$ and $Y$. To determine the marginal distributions, we sum the probabilities along the rows for $Y$ and the columns for $X$.

a) Marginal distribution of $X$:

For $X$, we sum the probabilities column-wise.

$P(X = 1) = 0.05 + 0.05 + 0 = 0.1$ $P(X = 2) = 0.05 + 0.1 + 0.2 = 0.35$ $P(X = 3) = 0.1 + 0.35 + 0.1 = 0.55$

Thus, the marginal distribution of $X$ is: $P(X = 1) = 0.1, \quad P(X = 2) = 0.35, \quad P(X = 3) = 0.55$

b) Marginal distribution of $Y$:

For $Y$, we sum the probabilities row-wise.

$P(Y = 1) = 0.05 + 0.05 + 0.1 = 0.2$ $P(Y = 2) = 0.05 + 0.1 + 0.35 = 0.5$ $P(Y = 3) = 0 + 0.2 + 0.1 = 0.3$

Thus, the marginal distribution of $Y$ is: $P(Y = 1) = 0.2, \quad P(Y = 2) = 0.5, \quad P(Y = 3) = 0.3$

2. Find $P(Y = 3 \mid X = 2)$:

To compute the conditional probability $P(Y = 3 \mid X = 2)$, we use the formula: $P(Y = 3 \mid X = 2) = \frac{P(X = 2, Y = 3)}{P(X = 2)}.$

From the table:

• $P(X = 2, Y = 3) = 0.2$
• $P(X = 2) = 0.35$ (from the marginal distribution).

Substitute these values: $P(Y = 3 \mid X = 2) = \frac{0.2}{0.35} = \frac{20}{35} = \frac{4}{7}.$

Final Answers:

1. Marginal distribution of $X$:

   • $P(X = 1) = 0.1$
   • $P(X = 2) = 0.35$
   • $P(X = 3) = 0.55$

2. Marginal distribution of $Y$:

   • $P(Y = 1) = 0.2$
   • $P(Y = 2) = 0.5$
   • $P(Y = 3) = 0.3$

3. $P(Y = 3 \mid X = 2) = \frac{4}{7}$.

Do you want me to explain any step in more detail or clarify any concepts?

Here are 5 related questions you can consider:

1. How do you calculate conditional probabilities in joint probability distributions?
2. What is the difference between marginal probability and joint probability?
3. How can we verify that a probability distribution sums to 1?
4. How would you find $P(X = 2 \mid Y = 3)$ for this table?
5. What is the concept of independence between $X$ and $Y$ in probability?

Tip:

Always ensure that the sum of all joint probabilities equals 1 to confirm that the given distribution is valid.