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:

\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.