# Topic B5 — Discrete Distributions

## Discrete vs Continuous RV

Discrete vs Continuous

## Probability Mass Function vs. Probability Density Function

• Relative freq
• Mass for discrete
• Density for continuous
• Discrete properties
• Prob of each value is btw 0 and 1
• Sum of prob = 1
• Cumul distrib fnct (CDF)
• “Proba that X is at least val x” P(X <= x)

## Expected value of a discrete RV

Expected value converges to the pop mean

$E(X) = \mu = \sum x_iP(X=x_i)$ where $X = x_i$ is the probability of hitting $x_i$

## Variance of a discrete RV

$Var(X) = \sum (x_i-\mu)^2P(X=x_i)$ just the variance multiplied by the probability.

SD $\sqrt{Var(X)}$

## Expected value

• $E(X+Y) = E((X)+E(Y)) = E(X) + E(Y)$
• Constant $\alpha$ : $E(\alpha) = \alpha, E(\alpha X) = \alpha E(\alpha)$

## Var/covar

$Covar(X+Y) = E(XY) - E(X)E(Y)$

$E[(X-\mu_x)(Y-\mu_y)]$

or $E(XY) - E(X)E(Y)$ where $E(XY)$ is the joint probability

Prop

• $Var(X+Y) = Var(X) + Var(Y) - 2Cov(X,Y)$
• $Var(\alpha) = 0$ and $Var(\alpha X) = \alpha^2 Var(X)$

## Risk neutral vs. risk averse vs. risk lovers

• Averse: might decline a risky prospect even if $E(X) >0$
• Neutral: always accepts if $E(X) >0$
• Loving: might decline a risky prospect even if $E(X) <0$