To calculate $ F_X(1) $, we need to find the cumulative distribution function (CDF) of the random variable $ X $ at $ x = 1 $.

### Step 1: Define the CDF $ F_X(x) $

The cumulative distribution function $ F_X(x) $ is defined as:
\[
F_X(x) = P(X \leq x) = \int_{-\infty}^{x} f_X(t) \, dt,
\]
where $ f_X(x) $ is the marginal probability density function of $ X $. To find $ f_X(x) $, we integrate the joint PDF $ f_{XY}(x, y) $ over all values of $ y $.

### Step 2: Find the Marginal PDF $ f_X(x) $

The marginal PDF $ f_X(x) $ is obtained by integrating $ f_{XY}(x, y) $ over the range of $ y $:
\[
f_X(x) = \int_{-\infty}^{\infty} f_{XY}(x, y) \, dy.
\]
Given the joint PDF:
\[
f_{XY}(x, y) = 
\begin{cases} 
\frac{1}{8}xy & \text{for } 0 < x < 2 \text{ and } 1 < y < 3, \\
0 & \text{otherwise},
\end{cases}
\]
we integrate over the interval $ 1 < y < 3 $:
\[
f_X(x) = \int_1^3 \frac{1}{8}xy \, dy = \frac{x}{8} \int_1^3 y \, dy.
\]
Evaluate the integral:
\[
\frac{x}{8} \left[\frac{y^2}{2}\right]_1^3 = \frac{x}{8} \left(\frac{9}{2} - \frac{1}{2}\right) = \frac{x}{8} \cdot 4 = \frac{x}{2}.
\]
Thus, the marginal PDF $ f_X(x) $ is:
\[
f_X(x) = 
\begin{cases} 
\frac{x}{2} & \text{for } 0 < x < 2, \\
0 & \text{otherwise}.
\end{cases}
\]

### Step 3: Compute $ F_X(1) $

Now, compute the CDF $ F_X(1) $ by integrating $ f_X(x) $ from $ -\infty $ to 1:
\[
F_X(1) = \int_{-\infty}^{1} f_X(x) \, dx = \int_0^1 \frac{x}{2} \, dx,
\]
since $ f_X(x) = 0 $ for $ x \leq 0 $.

Evaluate the integral:
\[
\int_0^1 \frac{x}{2} \, dx = \frac{1}{2} \int_0^1 x \, dx = \frac{1}{2} \left[\frac{x^2}{2}\right]_0^1 = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}.
\]

### Final Answer

Therefore, $ F_X(1) = \frac{1}{4} $.

Would you like further explanations or help with another problem?

**Relative Questions:**
1. What does the cumulative distribution function $ F_X(x) $ represent?
2. How do you find the marginal CDF $ F_Y(y) $ from a joint PDF?
3. What is the relationship between the marginal and joint PDFs?
4. How would $ F_X(1) $ change if the range of $ y $ was different?
5. Can you explain why the integration limits for $ y $ were $ 1 $ to $ 3 $ in this context?

**Tip:** Understanding the connection between joint and marginal distributions is crucial in probability, as it helps in analyzing the behavior of individual random variables within a multivariate distribution.

Learn how to calculate $F_X(1)$ using the cumulative distribution function (CDF) for a given joint probability distribution. Includes step-by-step integration and marginal PDF derivation. Suitable for advanced undergraduate students studying probability theory.

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