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Questions and Answers
What is the probability that the portion of a claim representing damage to the rest of the property is less than 0.3?
What is the function that gives the probability that the components of a bivariate random variable equal certain values?
Probability Mass Function (PMF)
What does the PMF explain?
The probability matrix is a tabular representation of the ____.
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If two events A and B are independent, the joint distribution of their components equals the product of their marginal distributions.
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What is the formula for the variance of the sum of two random variables?
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In the context of random variables, what does a conditional expectation represent?
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What is the formula for calculating conditional variance in the context of random variables?
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What does the conditional distribution describe?
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How is the conditional distribution defined in bivariate distributions?
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Which laws of probability does the conditional probability mass function (PMF) obey?
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The conditional distribution can be computed for one variable while conditioning on more than one variable by summing across all outcomes conditioned on a set S, represented as ___.
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Study Notes
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! (nx1x2)!) * px1 * px2 * (1p1p2)nx1x2
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 fX1X2(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(X1X2 = x2) = ∑ x1f(X1X2 = x2)
Conditional Variance
 The conditional variance of X1 conditional on X2 is given by: Var(X1X2 = x2) = E(X12X2 = x2)  [E(X1X2 = 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
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Description
Understanding probability matrices, marginal and conditional distributions, expectation, and covariance in bivariate discrete random variables.