Topic B5 — Discrete Distributions

Table of contents
  1. Topic B5 — Discrete Distributions
    1. Discrete vs Continuous RV
    2. Probability Mass Function vs. Probability Density Function
    3. Expected value of a discrete RV
    4. Variance of a discrete RV
    5. Properties of the expected value and the variance of two RV
    6. Expected value
    7. Var/covar
    8. Risk neutral vs. risk averse vs. risk lovers
    9. Portfolio risk and return
    10. Bernoulli trial
    11. Binomial distribution interpretation
  2. Properties of the binomial distribution

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)} $

Properties of the expected value and the variance of two RV

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 $E(X) <0$

Portfolio risk and return

Bernoulli trial

Binomial distribution interpretation

Properties of the binomial distribution