Week 5.3: Bivariate Normal Distribution

Prerequisites: Joint Continuous (Week 5.1: Joint Continuous Distributions), Normal (Week 4.2: Common Continuous Distributions) Cross-links: BSMA3012 (Linear Stat Models) — multivariate normal Core question: What is the most important joint continuous distribution?

1. Intuition: The "Correlated Normal"

• Height and weight of individuals
• Exam scores in two subjects
• Stock returns on consecutive days

2. Definition

3. Key Properties

1. Marginals are normal: $X \sim \mathcal{N}(\mu_X, \sigma_X^2)$, $Y \sim \mathcal{N}(\mu_Y, \sigma_Y^2)$.
2. $\rho$ is the correlation coefficient between $X$ and $Y$.
3. If $\rho = 0$, $X$ and $Y$ are independent (unique to normal!).
4. Conditional distributions are normal:

4. Visualising the Bivariate Normal

5. Practice Questions

Q1 (Easy)

Full Solution

$\mathbb{E}[Y \mid X=70] = 60 + 0.6 \cdot \frac{12}{10}(70-50) = 60 + 0.6 \cdot 1.2 \cdot 20 = 60 + 14.4 = 74.4$.

Q2 (Medium)

Full Solution

$\operatorname{Var}(Y \mid X=70) = \sigma_Y^2(1-\rho^2) = 144(1-0.36) = 144 \cdot 0.64 = 92.16$.

Q3 (Hard)

Full Solution

If $\rho=0$, then $Q(x,y) = \frac{(x-\mu_X)^2}{\sigma_X^2} + \frac{(y-\mu_Y)^2}{\sigma_Y^2}$, and $\sqrt{1-\rho^2}=1$.

$f_{XY}(x,y) = \frac{1}{2\pi\sigma_X\sigma_Y} \exp\left(-\frac{(x-\mu_X)^2}{2\sigma_X^2} - \frac{(y-\mu_Y)^2}{2\sigma_Y^2}\right) = f_X(x) \cdot f_Y(y)$.

Thus $X$ and $Y$ are independent.

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