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 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.

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

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

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)

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.

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).

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.

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

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 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))

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

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]

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)

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)

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 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

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

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

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