Agenda

<aside> 📌 Reminders

• Term test 2 November 11, 2024 </aside>

Joint Density Function

<aside> 💡 Joint density function $f_{X,Y}(x, y)$: describes the probability distribution of two continuous random variables $X$ and $Y$. It satisfies:

• $f_{X,Y}(x, y) \geq 0$ for all $x, y$
• $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f_{X,Y}(x, y) \, dx \, dy = 1$ </aside>

Marginal Density Function

<aside> 💡 Marginal density function: of $X$ (or $Y$) is obtained by integrating the joint density over the other variable: $f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x, y) \, dy$ or $f_Y(y) = \int_{-\infty}^{\infty} f_{X,Y}(x, y) \, dx$

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Example

Given $f_{X,Y}(x, y) = C \cdot e^{-(x + y)}$, the marginal density of $X$ is:

$$ f_X(x) = \int_0^{\infty} C \cdot e^{-(x + y)} \, dy = C \cdot e^{-x} \int_0^{\infty} e^{-y} \, dy  $$

Conditional Density Function

<aside> 💡 Conditional density function $f_{X|Y}(x|y)$ describes the distribution of $X$ given $Y = y$ and is defined as: $f_{X|Y}(x|y) = \frac{f_{X,Y}(x, y)}{f_Y(y)}$

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Properties

• Integrates to 1 over the support of $X$:

$$ \int_{-\infty}^{\infty} f_{X|Y}(x|y) \, dx = 1 $$

Fubini's Theorem

<aside> 💡 Fubini's Theorem: allows us to compute double integrals as iterated integrals (under certain conditions **)

$\int_{\mathbb{R}^2} f(x, y) \, dA = \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty} f(x, y) \, dy \right) dx = \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty} f(x, y) \, dx \right) dy$

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**Fubini’s theorem can be applied only when $f(x, y)$ is Lebesegue integrable over the rectangle of intergration

Practice