Joint, marginal and conditional distributions

Idea: The joint table

A joint distribution gives a chance to every pair of outcomes at once.

• Buy signal $X$ and up close $Y$, each $0$ or $1$
• Cells: $0.30$, $0.10$, $0.20$, $0.40$
• First check: $0.30+0.10+0.20+0.40=1$

Formula: Marginal

$P(X=x)=\sum _{y}P(X=x,Y=y)$

Add a variable away: sum the joint over every value of the other.

Formula: Conditional

$P(Y=y\mid X=x)=\frac{P(X=x,Y=y)}{P(X=x)}$

Keep one slice of the table and rescale it by its total: joint over marginal.

Example: Up close after a buy signal

$P(X=1,Y=1)=0.30$ and $P(X=1,Y=0)=0.10$. What is $P(Y=1\mid X=1)$?

1. Slice $X=1$: $0.30+0.10=0.40$
2. Rescale: $\frac{0.30}{0.40}=\frac{3}{4}$

Common mistake: Reading a joint cell as a conditional

Calling $0.30$ the chance of an up close after a buy signal. That cell is measured over all mornings.

Instead Condition: divide by $P(X=1)=0.40$ and get $\frac{3}{4}$.

Idea: Is the signal informative?

Independent means every conditional equals the marginal.

• $P(Y=1\mid X=1)=\frac{3}{4}$ against $P(Y=1)=\frac{1}{2}$
• Conditionals average back: $0.40\times \frac{3}{4}+0.60\times \frac{1}{3}=\frac{1}{2}$
• Marginals never give the joint back

Quick check: Covariance of the pair

$P(X=1,Y=1)=0.30$, $P(X=1)=0.40$, $P(Y=1)=0.50$. What is $\mathrm{Cov}(X,Y)$?

Idea: Densities work the same way

For continuous variables, sums become integrals and a chance is a volume.

• $f(x,y)=x+y$ on the unit square
• Marginal: $f_X(x)=x+\frac{1}{2}$
• Not independent: at the corner the product is $\frac{1}{4}$, the density $0$

Idea: Say it out loud

Name what you compute before you touch a number.

• Joint: measured over every case
• Marginal: add a variable away
• Conditional: shrink to one slice, then rescale