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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! (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
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The marginal distribution is given by: fX1(x1) = ∫ fX1,X2(x1, x2)dx2
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Similarly, fX2(x2) = ∫ fX1,X2(x1, x2)dx1### IID Random Variables
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IID random variables are typically defined as x iii d ∼ N(μ, σ²)
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The expected mean of IID random variables is given by E(∑ Xi) = ∑ E(Xi) = ∑ μ = nμ
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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.