Joint, Marginal and Conditional Distributions

Class: STAT 131 Subject: probability Date: 2025-02-24 Teacher: **Prof. Marcela

Discrete

Joint

Joint CDF

• Joint CDF. The joint CDF of r.v.s $X$ and $Y$ is the function $F_{X,Y}$ given by: $F_{X,Y}(x,y)=P(X\le x,Y\le y)$

Joint PMF

• Joint PMF of a discrete r.v.s X and Y is the function px,y given by $px,y(x,y)=P(X=x,Y=y)$
• If X and Y are independent, then pxy(x, y) = px(x)py(y)

Marginal

Marginal PMF

• for discrete r.v.s X and Y, the marginal PMF of X is: $P(X=x)=\sum P(X=x,Y=y)$

Conditional PMF

• for conditional r.v.s X and Y, the conditional PMF of Y is given as X = x is: $P(Y=y|X=x)=\frac{P(X=x|Y=y)}{P(X=x)}$

Example

a) $P(X=0,Y=0)+P(X=0,Y=1)=1/6+1/4=5/12$ b) $P(Y=0)=1/6+1/8=7/24$ $P(Y=1)=1/4+1/6=5/12$ $P(Y=2)=1/8+1/6=7/24$ If they don’t sum up to 1, then you made a mistake somewhere!

c) $P(Y=1|X=0)=\frac{P(Y=1,X=0)}{P(X=0)}$ $P(Y=1,X=0)=1/4$ $P(X=0)=13/24$ = $\frac{1/4}{13/24}=6/13$ d) Just show that for some $P(X=x,Y=y)\ne P_X(x)\ast P_Y(y)$ $P(X=0,Y=0)\ne P_X(0)\ast P_Y(0)$ because $1/6\ne 13/24\ast 7/24$

Continuos

Joint

Joint PDF

• if X and Y are continuous, their joint PDF is the derivative of the joint CDF with respect to x and y:

Example

$P(X<3,1<Y,4)={\int }_1^4{\int }_{-\infty }^3{f}_{X,Y}(x,y)dxdy$

Marginal

Marginal PDF

• for continuous r.v.s X and Y with joint PDF fx,y, the marginal PDF of X is:

Conditional

Conditional PDF

• for continuous r.v.s X and Y with joint PDF fx,y, the marginal PDF of Y given X=x is:

Example

b) $c/10=1,c=10$

c)

d)

e) $=\frac{P(Y\le x/4,Y\le x/2)}{P(Y\le x/2)}=4P(Y\le x/4)$ we ignore $Y\le 2$ because