Chapter Summary

Chapter 7 Summary

Key Points

• 1.

  The joint CDF $F_{X,Y}(x,y) = \mathbb{P}(X \le x, Y \le y)$ is the fundamental object that encodes all joint distributional information. It determines the marginals, but the marginals do not determine it.

• 2.

  Joint PMF/PDF: For discrete RVs, $P_{X,Y}(x_i, y_j) = \mathbb{P}(X = x_i, Y = y_j)$; for continuous RVs, $f_{X,Y}(x,y) = \partial^2 F_{X,Y} / \partial x\,\partial y$. Marginals are obtained by summing or integrating out the other variable.

• 3.

  Conditional distributions: $f(y \mid x) = f_{X,Y}(x,y)/f_{X}(x)$. The conditional expectation $\mathbb{E}[X \mid Y]$ satisfies the tower property $\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X \mid Y]]$ and the law of total variance.

• 4.

  Independence: $X$ and $Y$ are independent iff $f_{X,Y} = f_{X} \cdot f_{Y}$. Independence implies uncorrelatedness, but uncorrelated does not imply independent (except for jointly Gaussian RVs).

• 5.

  Jacobian method: For an invertible transformation $(U,V) = g(X,Y)$, $f_{U,V}(u,v) = f_{X,Y}(g^{-1}(u,v)) \cdot |J_{g^{-1}}|$.

• 6.

  Convolution: The PDF of $Z = X + Y$ for independent $X, Y$ is $f_{Z} = f_{X} * f_{Y}$. Gaussians are closed under convolution.

• 7.

  Order statistics: $F_{X_{(n)}}(x) = [F_{X}(x)]^n$ for the maximum; $F_{X_{(1)}}(x) = 1 - [1 - F_{X}(x)]^n$ for the minimum. The minimum of i.i.d. exponentials with rate $\lambda$ is exponential with rate $n\lambda$.

• 8.

  Covariance and correlation: $\text{Cov}(X,Y) = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]$; $\rho_{X,Y} \in [-1,1]$. The variance of a sum decomposes as $\text{Var}(\sum X_i) = \sum \text{Var}(X_i) + 2\sum_{i<j} \text{Cov}(X_i, X_j)$.