Multivariate Random Variables

Multivariate Random Variables

• Multivariate random variables accommodate the dependence between two or more random variables.
• The concepts of multivariate random variables (e.g. expectations and moments) are analogous to those of univariate random variables.

Multivariate Discrete Random Variables

• Multivariate discrete random variables involve defining several random variables simultaneously on a sample space.
• A bivariate random variable X can be a vector with two components X1 and X2 with corresponding realizations x1 and x2.

Probability Mass Function (PMF)

• The PMF of a bivariate random variable gives the probability that the components of X take specific values.
• The PMF, fX1, X2(x1, x2), has the following properties:
  • fX1, X2(x1, x2) ≥ 0
  • ∑x1 ∑ x2 fX1, X2(x1, x2) = 1

Example: Trinomial Distribution

• The trinomial distribution is a distribution of n independent trials with three possible outcomes.
• The PMF of the trinomial distribution is given by: n! / (x1! x2! (n-x1-x2)!) * px1 * px2 * (1-p1-p2)n-x1-x2

Cumulative Distribution Function (CDF)

• The CDF of a bivariate discrete random variable returns the total probability that each component is less than or equal to a given value.
• The CDF is given by: FX1, X2(x1, x2) = P(X1 ≤ x1, X2 ≤ x2) = ∑ t1 ≤ x1 ∑ t2 ≤ x2 fX1, X2(t1, t2)

Probability Matrices

• A probability matrix is a tabular representation of the PMF.
• Each cell in the matrix represents the probability of a joint outcome.

Marginal Distributions

• The marginal distribution gives the distribution of a single variable in a joint distribution.
• The marginal PMF of X1 is computed by summing up the probabilities for X1 across all values in the support of X2.

Independence of Random Variables

• If two events A and B are independent, then P(A ∩ B) = P(A)P(B).
• This principle applies to bivariate random variables as well.
• If the distributions of the components of the bivariate distribution are independent, then fX1, X2(x1, x2) = fX1(x1)fX2(x2).

Conditional Distributions

• The conditional distribution of X1 given X2 is defined as fX1|X2(x1 | x2) = fX1, X2(x1, x2) / fX2(x2).
• The conditional PMF must sum across all outcomes in the set that is conditioned on.

Expectations

• The expectation of a function of a bivariate random variable is defined as: E(g(X1, X2)) = ∑ x1 ∑ x2 g(x1, x2)fX1, X2(x1, x2).
• The expectation of a nonlinear function g(x1, x2) is not equal to g(E(X1), E(X2)).### Calculating the Expectation
• The formula to calculate the expectation of a function g(x1, x2) given a probability mass function fX1, X2(x1, x2) is: E(g(X1, X2)) = ∑ ∑ g(x1, x2)fX1, X2(x1, x2)
• In the given example, g(x1, x2) = x1x2 and the probability mass function is given
• The expectation is calculated as: E(g(X1, X2)) = 2.80

Moments

• The first moment is defined as the expectation: E(X) = [E(X1), E(X2)] = [μ1, μ2]
• The second moment involves the covariance between the components of the bivariate distribution X1 and X2
• The formula for the second moment is: Var(X1 + X2) = Var(X1) + Var(X2) + 2Cov(X1, X2)
• Covariance is defined as: Cov(X1, X2) = E[(X1 - E[X1])(X2 - E[X2])]
• If X1 and X2 are independent, then Cov(X1, X2) = 0

Correlation

• Correlation is defined as: Corr(X1, X2) = Cov(X1, X2) / (σ1 * σ2)
• Correlation measures the strength of the linear relationship between two random variables
• Correlation is always between -1 and 1
• If X2 = α + βX1, then Corr(X1, X2) = β / √(β²Var(X1))

Portfolio Variance and Hedging

• The variance of a portfolio of securities can be calculated using the formula: σ²A+B = σA² + σB² + 2ρABσAσB
• The minimum variance achievable is given by: min[σ²P] = σA²(1 - ρ²AB)
• The hedge ratio can be calculated as: h* = -ρAB σA / σB

Covariance Matrix

• The covariance matrix is a 2x2 matrix that displays the covariance between the components of X
• The covariance matrix is given by: Cov(X) = [σ12, σ12; σ12, σ22]

Variance of Sums of Random Variables

• The variance of the sum of two random variables is given by: Var(X1 + X2) = Var(X1) + Var(X2) + 2Cov(X1, X2)
• If the random variables are independent, then Cov(X1, X2) = 0 and Var(X1 + X2) = Var(X1) + Var(X2)
• For weighted random variables, the variance is given by: Var(aX1 + bX2) = a²Var(X1) + b²Var(X2) + 2abCov(X1, X2)

Conditional Expectation

• A conditional expectation is the mean calculated after a set of prior conditions has happened
• The conditional expectation uses the same expression as any other expectation and is a weighted average where the probabilities are determined by a conditional PMF
• For a discrete random variable, the conditional expectation is given by: E(X1|X2 = x2) = ∑ x1f(X1|X2 = x2)

Conditional Variance

• The conditional variance of X1 conditional on X2 is given by: Var(X1|X2 = x2) = E(X12|X2 = x2) - [E(X1|X2 = x2)]²

Continuous Random Variables

• Continuous random variables use PDFs instead of PMFs
• The joint PDF is always nonnegative, and the double integration yields a value of 1
• The joint PDF is given by: FX1,X2(x1, x2) = ∫∫ fX1,X2(x1, x2)dx1 dx2

Joint Cumulative Distribution Function (CDF)

• The joint CDF is given by: FX1,X2(x1, x2) = ∫∫ fX1,X2(t1, t2)dt1 dt2

Marginal Distributions

• The marginal distribution is given by: fX1(x1) = ∫ fX1,X2(x1, x2)dx2

• Similarly, fX2(x2) = ∫ fX1,X2(x1, x2)dx1### IID Random Variables

• IID random variables are typically defined as x iii d ∼ N(μ, σ²)

• The expected mean of IID random variables is given by E(∑ Xi) = ∑ E(Xi) = ∑ μ = nμ

• The variance of IID random variables is given by Var(∑ Xi) = ∑ σ² = nσ²

Importance of Independence

• The independence property is important because it affects the variance of the sum of multiple random variables
• The variance of the sum of IID random variables is the sum of their individual variances
• The variance of a multiple of a single random variable is not equal to the variance of the sum of multiple random variables

Variance of IID Random Variables

• If X1 and X2 are IID with variance σ², then Var(X1 + X2) = 2σ²
• Var(2X1) = 4Var(X1) = 4σ²

Joint Density Function and Marginal PMF

• The joint density function of X and Y is given by f(x, y) = 6[1 - (x + y)], x > 0, y > 0, x + y < 1
• The marginal PMF of Y is given by fY(y) = 3 - 6y + 3y², 0 < y < 1
• The probability of Y taking on a value less than 0.3 is given by P(Y < 0.3) = 0.657