Chapter 5 Summary: Discrete Probability Distributions

Random Variables

• A random variable assigns a number to each outcome of an experiment.

• The value of a random variable is a numerical event.

  Discrete vs. Continuous Variables

• Discrete variables:

• Come from a counting process (e.g., number of classes).

• Continuous variables:

• Come from a measurement (e.g., salary, weight).

  Discrete Variables

• Assume a countable number of values.

• Examples:

• Rolling a die twice: X = number of times 4 occurs (0, 1, or 2).

• Tossing a coin 5 times: X = number of heads (0, 1, 2, 3, 4, or 5).

  Probability Distribution for a Discrete Variable

• A mutually exclusive list of all possible numerical outcomes for that variable and a probability of occurrence associated with each outcome.

  Conditions for Probability Distributions

• $0 ≤ P(x) ≤ 1$ for all $x$

• $Σ_{all x} P(x) = 1$

  Expected Value of Discrete Variables

• Measuring Center

• Expected Value (or mean) of a discrete variable (Weighted Average):

• $μ = E(X) = Σ<em>{i=1}^{N} x</em>i P(X = x_i)$

  Rules of Expectations

• $E(c) = c$

• $E(X + c) = E(X) + c$

• $E(c \cdot X) = c \cdot E[X]$

• $E[aX + b] = a \cdot E[X] + b$

  Variance and Standard Deviation of Discrete Variables

• Measuring Dispersion

• $σ^2 = Σ<em>{i=1}^{N} [x</em>i - E(X)]^2 P(X = x_i)$

• $σ = \sqrt{Σ<em>{i=1}^{N} [x</em>i - E(X)]^2 P(X = x_i)}$

  Calculation Formula for Variance

• $σ^2 = E[X^2] - [E(X)]^2 = Σ<em>{i=1}^{N} x</em>i^2 P(X = x_i) - μ^2$

  Rules of the Variance

• $Var(c) = 0$

• $Var(X + c) = Var(X)$

• $Var(cX) = c^2Var(X)$

• $Var(aX + b) = a^2 \cdot Var(X)$

  Probability Distributions

• Discrete:

• Binomial

• Poisson

• Hypergeometric

• Continuous:

• Normal

• Uniform

  Binomial Probability Distribution

• Fixed number of observations, $n$.

• Each observation is classified into one of two mutually exclusive categories.

• Probability of event of interest, $π$, is constant.

• Observations are independent.

  Binomial Distribution Formula

• $P(X=x | n,π) = \frac{n!}{x!(n-x)!} π^x (1-π)^{n-x}$

• where:

• $x$ = number of “events of interest” in sample

• $n$ = sample size (number of trials or observations)

• $\pi$ = probability of “event of interest”

  Binomial Distribution Characteristics

• Mean: $μ = E(X) = nπ$

• Variance: $σ^2 = nπ(1-π)$

• Standard Deviation: $σ = \sqrt{nπ(1-π)}$

  Cumulative Probabilities

• $P(X ≤ k)$

• $P(X = k) = P(X ≤ k) – P(X ≤ [k–1])$

• $P(X ≥ k) = 1 – P(X ≤ [k–1])$

  Poisson Distribution

• Interested in the number of times an event occurs in a given area of opportunity.

• Area of opportunity: continuous unit or interval of time, volume, or area.

  Poisson Distribution Formula

• $P(X = x | λ) = \frac{e^{-λ} λ^x}{x!}$

• where:

• x = number of events in an area of opportunity

• λ = expected number of events

• e = base of the natural logarithm system (2.71828…)

  Poisson Distribution Characteristics

• Mean: $μ = λ$

• Variance: $σ^2 = λ$

• Standard Deviation: $σ = \sqrt{λ}$

  Hypergeometric Distribution

• Selecting from a finite population without replacement.

• "n" trials in a sample taken from a finite population of size N.

• Sample taken without replacement.

• Outcomes of trials are dependent.

  Hypergeometric Distribution Formula

• $P(X = x | n, N, E) = \frac{\binom{E}{x} \binom{N-E}{n-x}}{\binom{N}{n}}$

• Where

• N = population size

• E = number of items of interest in the population

• N – E = number of events not of interest in the population

• n = sample size

• x = number of items of interest in the sample

• n – x = number of events not of interest in the sample

  Properties of the Hypergeometric Distribution

• Mean: $μ = E(X) = n\frac{E}{N}$

• Standard Deviation: $σ = \sqrt{n(\frac{E}{N})(1-\frac{E}{N})(\frac{N-n}{N-1})}$

• $\frac{N-n}{N-1}$ is the “Finite Population Correction Factor”

  Discrete Bivariate Distributions

• Probabilities of combinations of TWO variables.

• Also called joint probabilities.

• $P(x, y) = P(X = x \text{ and } Y = y)$

  Requirements for a Bivariate Distribution

• $0 ≤ P(x, y) ≤ 1$

• $Σ<em>{all x} Σ</em>{all y} P(x, y) = 1$

  Marginal Probabilities

• Summing across rows and down columns to determine the probabilities of $X$ and $Y$ individually.

  Covariance

• Measures the strength of the linear relationship between two discrete variables.

• Positive covariance: positive relationship.

• Negative covariance: negative relationship.

  Covariance Formula

• $σ<em>{XY} = Σ</em>{i=1}^{N} [x<em>i - E(X)][y</em>i - E(Y)] P(X=x<em>i, Y=y</em>i)$

• $COV(X, Y) = σ<em>{xy} = Σ</em>{all x} Σ<em>{all y} x</em>i y<em>j P(x</em>i, y<em>j) - μ</em>X μ_Y$

• $COV(X, Y) = σ_{xy} = E(XY) - E(X)E(Y)$

  Coefficient of Correlation

• $ρ = \frac{COV(X, Y)}{σ<em>X σ</em>Y}$

  Sum of Two Random Variables

• A bivariate distribution allows us to develop the probability distribution of any combination of the two variables, of particular interest is the SUM of two variables.

  Independence

• If two random variables are independent, the covariance is equal to zero: $COV(X, Y) = 0$

• Then the coefficient of correlation, ρ, is also equal to zero: $ρ = \frac{COV(X, Y)}{σ<em>X σ</em>Y} = 0$

  Rules For The Sum Of Two Random Variables

• $E(X + Y) = E(X) + E(Y)$

• $V(X + Y) = V(X) + V(Y) + 2COV(X, Y)$

• If X and Y are independent, $COV(X, Y) = 0$ and thus: $V(X + Y) = V(X) + V(Y)$

• NOTE:

• $E(aX + bY) = aE(X) + bE(Y)$

• $V(aX + bY) = a^2V(X) + b^2V(Y) + 2abCOV(X, Y)$

  Applications in Finance

• Computing the Mean for Investment Returns

• Computing the Standard Deviation for Investment Returns

• Computing the Covariance for Investment Returns

• Interpreting the Results for Investment Returns

  Portfolio Risk and Return

• Portfolio expected return (weighted average return): $E(P) = wE(X) + (1 − w)E(Y)$

• Portfolio risk (weighted variability): $σ<em>p = \sqrt{w^2 σ</em>X^2 + (1 − w)^2 σ<em>Y^2 + 2w(1 − w) σ</em>{xy}}$

• where

• $w$ = portion of portfolio value in asset X

• $(1 − w)$ = portion of portfolio value in asset Y