Discrete Random Variables

This article is part of the EconHelp Tutoring Wiki

A discrete random variable maps outcomes to values of a countable set (e.g., the integers), with each value in the range having probability greater than or equal to zero.

Discrete Probability

$\begin{array}{rl}& P(x)\in [0,1]\\ & \sum _{x}P(x)=1\\ & F(a)=\sum _{x\le a}P(x)=P(X\le a)\\ & F(a)\in [0,1]\\ & F(a)\le F(b)\mathrm{\forall }a<b\\ & P(X>a)=1-P(X\le a)=1-F(a)\end{array}$

Mean and Variance

Expected Value

$E(X)=\sum _{x}xP(x)={\mu }_x$

Variance

$\begin{array}{rl}Var(X)& =E[(X-{\mu }_x{)}^2]\\ & =\sum _x(x-{\mu }_x{)}^{2}P(x)\\ & =E(X^2)-{\mu }_x^2\end{array}$

Binomial Distribution

$X\backsim B(p)$, where p is the probability of successes in n independent trials.

$\begin{array}{rl}& {\mu }_{X_B}=E(X_B)=p\\ & Var(X_B)=p(1-p)\\ & P(x)=p^x(1-p{)}^{n-x}{C}_x^n\end{array}$

Jointly distributed variables

$P_{X,Y}(x,y)=P(X=x,Y=y)$

$\sum _X\sum _Y{P}_{X,Y}(x,y)=1$

$\begin{array}{rl}{P}_X(x)& =\sum _Y{P}_{X,Y}(x,y)\\ P_Y(y)& =\sum _X{P}_{X,Y}(x,y)\end{array}$

Conditional Probabilities

$\begin{array}{rl}E(Y|X=x)& =\sum _{y}y{P}_{Y|X}(y|x)={\mu }_{Y|X}\\ Var(Y|X& =\sum _y(y-{\mu }_{Y|X}{)}^2{P}_{Y|X}(y|x)\\ & =E(Y^2|X=x)-{\mu }_{(}^{2}Y|X)\end{array}$

Covariance and Correlation

$\begin{array}{rl}Cov(X,Y)& =E[(X-{\mu }_X)(Y-{\mu }_Y)]\\ & =\sum _x\sum _y(x-{\mu }_x)(y-{\mu }_y)P_{X,Y}(x,y)\\ & =E(XY)-{\mu }_X{\mu }_Y\end{array}$

$\rho =\frac{Cov(X,Y)}{{\sigma }_X{\sigma }_Y},-1\le \rho \le a$

Other useful results

$\begin{array}{rl}Var(X+Y)& =Var(X)+Var(Y)+2Cov(X,Y)\\ Var(X-Y)& =Var(X)+Var(Y)-2Cov(X,Y)\\ Var(aX+bY)& =a^{2}Var(X)+b^{2}Var(Y)+2abCov(X,Y)\end{array}$