Let’s simplify and explain the concepts from the slides provided on joint and marginal distributions for absolutely continuous random variables.

1. Joint Distribution for Absolutely Continuous Random Variables

Definition

A pair of random variables $(X, Y)$ is absolutely continuous if:

1. There exists a function $f(x, y)$ called the joint probability density function (PDF), such that:
   • $f(x, y) \geq 0$ for all $(x, y) \in \mathbb{R}^2$ (non-negativity),
   • $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x, y) \, dx \, dy = 1$ (total probability is 1).

Cumulative Distribution Function (CDF)

The cumulative distribution function $F(x, y)$ gives the probability that: $F(x, y) = P(X \leq x, Y \leq y).$ It is computed as: $F(x, y) = \int_{-\infty}^x \int_{-\infty}^y f(u, v) \, du \, dv.$

Relationship Between PDF and CDF

The PDF $f(x, y)$ can be obtained by differentiating the joint CDF $F(x, y)$: $f(x, y) = \frac{\partial^2 F(x, y)}{\partial x \, \partial y}.$

2. Marginal Distributions

Marginal Cumulative Distribution Functions

For random variables $X$ and $Y$, the marginal cumulative distributions $F_X(x)$ and $F_Y(y)$ are: $F_X(x) = P(X \leq x) = \int_{-\infty}^x \int_{-\infty}^\infty f(u, v) \, dv \, du,$ $F_Y(y) = P(Y \leq y) = \int_{-\infty}^y \int_{-\infty}^\infty f(u, v) \, du \, dv.$

Marginal PDFs

The marginal PDFs $f_X(x)$ and $f_Y(y)$ are obtained by integrating out one variable from the joint PDF: $f_X(x) = \int_{-\infty}^\infty f(x, v) \, dv, \quad f_Y(y) = \int_{-\infty}^\infty f(u, y) \, du.$

Simplified Key Points

1. Joint PDF ($f(x, y)$):

   • Describes the likelihood of the random variables $X$ and $Y$ taking specific values together.
   • Must integrate to 1 over all $(x, y)$.

2. Joint CDF ($F(x, y)$):

   • Describes the probability that $X \leq x$ and $Y \leq y$.
   • Obtained by integrating the joint PDF.

3. Marginal Distributions:

   • Marginal PDF ($f_X(x)$, $f_Y(y)$) gives the probabilities of individual variables $X$ or $Y$, found by integrating out the other variable from the joint PDF.
   • Marginal CDF ($F_X(x)$, $F_Y(y)$) gives cumulative probabilities of $X$ or $Y$.

Would you like further clarification or help applying these principles? Let me know!

Questions for Further Exploration

1. How can the marginal distributions be used to verify independence between $X$ and $Y$?
2. What happens to the CDF $F(x, y)$ when $X$ and $Y$ are independent?
3. Can you derive the marginal PDF $f_X(x)$ for a specific example of a joint PDF?
4. How does the joint PDF differ for discrete random variables?
5. What are the conditions under which a joint PDF $f(x, y)$ can be factored into $f_X(x)\cdot f_Y(y)$?

Tip

When integrating the joint PDF, always check the limits of integration carefully to ensure the domain of the random variables is properly accounted for.