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Week 5.3: Bivariate Normal Distribution

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Week 5.3: Bivariate Normal Distribution

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Now · 1. Intuition: The "Correlated Normal"

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"

The bivariate normal is the natural joint distribution for two normally-distributed variables that may be correlated. Examples:

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

2. Definition

(X,Y) \sim \mathcal{N}(\mu_X, \mu_Y, \sigma_X^2, \sigma_Y^2, \rho)

has PDF:

f_{XY}(x,y) = \frac{1}{2\pi\sigma_X\sigma_Y\sqrt{1-\rho^2}} \exp\left( -\frac{1}{2(1-\rho^2)} Q(x,y) \right),

where

Q(x,y) = \frac{(x-\mu_X)^2}{\sigma_X^2} - 2\rho\frac{(x-\mu_X)(y-\mu_Y)}{\sigma_X\sigma_Y} + \frac{(y-\mu_Y)^2}{\sigma_Y^2}.

3. Key Properties

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

Y \mid X = x \sim \mathcal{N}\left( \mu_Y + \rho\frac{\sigma_Y}{\sigma_X}(x-\mu_X),\; \sigma_Y^2(1-\rho^2) \right).

The conditional mean is linear in

x

— this is the foundation of linear regression.

4. Visualising the Bivariate Normal

(Diagram)

5. Practice Questions

Q1 (Easy)

(X,Y)

bivariate normal with

\mu_X=50

\mu_Y=60

\sigma_X=10

\sigma_Y=12

\rho=0.6

. Find

\mathbb{E}[Y \mid X=70]

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)

For the same distribution, find

\operatorname{Var}(Y \mid X=70)

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)

X

and

Y

are bivariate normal with

\rho=0

, show they are independent.

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.

Next topic: Week 6.1: Moment Generating Functions — Moment Generating Functions. Join Discord PreviousWeek 5.2: Law of Large Numbers & Central Limit TheoremNextWeek 6.1: Moment Generating Functions