Multivariate Random Variables
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Multivariate Random Variables

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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?

  • 0.657 (correct)
  • 0.450
  • 0.752
  • 0.415
  • 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?

  • Probability of realization based on two variables
  • Probability of occurrence of a specific event
  • Probability of joint outcomes (correct)
  • Probability of realization based on one variable
  • The probability matrix is a tabular representation of the ____.

    <p>PMF</p> Signup and view all the answers

    If two events A and B are independent, the joint distribution of their components equals the product of their marginal distributions.

    <p>False</p> Signup and view all the answers

    What is the formula for the variance of the sum of two random variables?

    <p>Var(X1 + X2) = Var(X1) + Var(X2) + 2Cov(X1, X2)</p> Signup and view all the answers

    In the context of random variables, what does a conditional expectation represent?

    <p>A conditional expectation is the mean value of a random variable after certain prior conditions have occurred.</p> Signup and view all the answers

    What is the formula for calculating conditional variance in the context of random variables?

    <p>Var(X1 | X2 = x2) = E(X1^2 | X2 = x2) - [E(X1 | X2 = x2)]^2</p> Signup and view all the answers

    What does the conditional distribution describe?

    <p>The probability of an outcome of a random variable conditioned on the other random variable taking a particular value.</p> Signup and view all the answers

    How is the conditional distribution defined in bivariate distributions?

    <p>The conditional distribution of X1 given X2 is defined as the joint distribution of X1 and X2 divided by the marginal distribution of X2.</p> Signup and view all the answers

    Which laws of probability does the conditional probability mass function (PMF) obey?

    <p>Both a and b</p> Signup and view all the answers

    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 ___.

    <p>{+1, 0}</p> Signup and view all the answers

    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! (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

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    Understanding probability matrices, marginal and conditional distributions, expectation, and covariance in bivariate discrete random variables.

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