If (X,Y) are discrete random variables with values x,y=0,1 and the joint PMF is defined by f(0,0)=1/4, f(0,1)=1/8, f(1,0)=1/2, f(1,1)=1/8, what are the values of the conditional PMF, f(y|x=1), of Y given X=1?

To find the conditional probability mass function (PMF) \( f(y \mid x = 1) \), we use the formula for conditional probability:

\[
f(y \mid x = 1) = \frac{f(x = 1, y)}{P(X = 1)}
\]

### Step 1: Find \( P(X = 1) \)
To find \( P(X = 1) \), we sum the joint probabilities for all values of \( y \) when \( X = 1 \):

\[
P(X = 1) = f(1, 0) + f(1, 1)
\]

From the given joint PMF:

- \( f(1, 0) = \frac{1}{2} \)
- \( f(1, 1) = \frac{1}{8} \)

Thus,

\[
P(X = 1) = \frac{1}{2} + \frac{1}{8} = \frac{4}{8} + \frac{1}{8} = \frac{5}{8}
\]

### Step 2: Compute the conditional PMF \( f(y \mid x = 1) \)
Now, we calculate \( f(y \mid x = 1) \) for each possible value of \( y \) (which can be 0 or 1):

- For \( y = 0 \):

\[
f(0 \mid x = 1) = \frac{f(1, 0)}{P(X = 1)} = \frac{\frac{1}{2}}{\frac{5}{8}} = \frac{1}{2} \times \frac{8}{5} = \frac{4}{5}
\]

- For \( y = 1 \):

\[
f(1 \mid x = 1) = \frac{f(1, 1)}{P(X = 1)} = \frac{\frac{1}{8}}{\frac{5}{8}} = \frac{1}{8} \times \frac{8}{5} = \frac{1}{5}
\]

### Final Conditional PMF
Thus, the conditional PMF \( f(y \mid x = 1) \) is:

\[
f(y \mid x = 1) =
\begin{cases}
\frac{4}{5} & \text{if } y = 0 \\
\frac{1}{5} & \text{if } y = 1
\end{cases}
\]

Let me know if you'd like further details or have any questions! Here's a list of related questions for further exploration:

1. How do you derive the marginal PMF from a joint PMF?
2. What is the definition of conditional independence in probability?
3. How do you compute the expected value of a discrete random variable?
4. What is the relationship between joint and conditional distributions in probability theory?
5. Can you explain Bayes' theorem and its applications in probability?

**Tip**: Always remember that the sum of probabilities in a conditional distribution must equal 1. For example, \( f(0 \mid x = 1) + f(1 \mid x = 1) = 1 \), which you can use as a quick check.

This page explains how to compute the conditional PMF f(y | x = 1) of a discrete random variable Y given X = 1. It includes a step-by-step breakdown of the joint PMF and conditional probability formulas. The problem involves the joint PMF values f(0,0)=1/4, f(0,1)=1/8, f(1,0)=1/2, and f(1,1)=1/8. Suitable for high school students studying probability theory.

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