Probability and Statistics Quiz
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Questions and Answers

What is the purpose of a probability matrix in probability theory?

  • To compute the marginal distribution of a discrete bivariate random variable
  • To define the independence of two random variables
  • To express a probability mass function (PMF) (correct)
  • To calculate the covariance of two random variables

    • What is the covariance of two random variables a measure of?

  • The correlation between the two variables
  • The independence of the two variables
  • The extent of linear relationship between the two variables (correct)
  • The variance of the sum of the two variables

    • What is the formula for computing the variance of a weighted sum of two random variables?

  • V(aX + bY) = aV(X) + bV(Y) - 2abCov(X, Y)
  • V(aX + bY) = a^2V(X) + b^2V(Y) + 2abCov(X, Y) (correct)
  • V(aX + bY) = aV(X) + bV(Y) + 2abCov(X, Y)
  • V(aX + bY) = a^2V(X) + b^2V(Y) - 2abCov(X, Y)

    • What is the conditional expectation of a component of a bivariate random variable?

    <p>The expected value of the component given the value of the other component</p> Signup and view all the answers

    What is the key feature of an iid sequence of random variables?

    <p>The variables are independent and have the same distribution</p> Signup and view all the answers

    What is the effect of applying linear transformations on the covariance and correlation between two random variables?

    <p>The covariance and correlation are scaled by a constant factor</p> Signup and view all the answers

    What is the purpose of computing the marginal distribution of a discrete bivariate random variable?

    <p>To compute the conditional distribution of one variable given the other</p> Signup and view all the answers

    What is the relationship between the covariance and correlation of two random variables?

    <p>The correlation is a standardized measure of the covariance</p> Signup and view all the answers

    How does the iid property help in computing the mean and variance of a sum of iid random variables?

    <p>It simplifies the computation by allowing the sum of the means and variances</p> Signup and view all the answers

    What happens to the covariance of two random variables when they are independent?

    <p>The covariance becomes zero</p> Signup and view all the answers

    What is the formula to calculate the conditional probability P(A|B) given two events A and B?

    <p>P(A ∩ B) / P(B)</p> Signup and view all the answers

    What is the purpose of calculating the conditional distribution of a random variable?

    <p>To describe the probability of an outcome of a random variable conditioned on another variable</p> Signup and view all the answers

    What is the relationship between the joint probability and the marginal probabilities of two random variables?

    <p>The joint probability is the product of the marginal probabilities if the variables are independent</p> Signup and view all the answers

    What is the formula for calculating the conditional distribution of a bivariate random variable?

    <p>f(X1|X2) = f(X1,X2) / f(X2)</p> Signup and view all the answers

    What can be inferred if the joint probability of two events is equal to the product of their marginal probabilities?

    <p>The events are independent</p> Signup and view all the answers

    What is the purpose of calculating the marginal distribution of a discrete bivariate random variable?

    <p>To describe the probability distribution of a single random variable</p> Signup and view all the answers

    What is the advantage of using conditional distributions in bivariate probability?

    <p>It enables us to describe the probability of an outcome of a random variable conditioned on another variable</p> Signup and view all the answers

    What is the relationship between the joint probability and the marginal probabilities of two independent random variables?

    <p>The joint probability is equal to the product of the marginal probabilities</p> Signup and view all the answers

    What is the purpose of conditional distributions in finance?

    <p>To manage risk by computing the conditional distribution of interest rates</p> Signup and view all the answers

    What is a characteristic of independent and identically distributed (iid) random variables?

    <p>Each random variable has the same probability distribution and is mutually independent</p> Signup and view all the answers

    What is an application of iid random variables?

    <p>Time series analysis</p> Signup and view all the answers

    What is the formula for the conditional distribution of X1 given X2?

    <p>f(X1│X2)(x1│X2 = x2) = fX1,X2(x1, x2) / fX2(x2)</p> Signup and view all the answers

    What is the key feature of a fair coin?

    <p>The probability of head vs. tail is 50:50</p> Signup and view all the answers

    What is the distribution of iid variables generated by a normal distribution?

    <p>Each variable has the same probability distribution</p> Signup and view all the answers

    What is the purpose of conditional distributions?

    <p>To manage risk by computing the conditional distribution of interest rates</p> Signup and view all the answers

    What is a characteristic of iid random variables in time series analysis?

    <p>Each random variable has the same probability distribution and is mutually independent</p> Signup and view all the answers

    What is the purpose of calculating the conditional distribution of bond returns?

    <p>To determine the probability of a bond return given a positive analyst rating</p> Signup and view all the answers

    What is the formula for calculating the conditional probability of a bond return given a positive analyst rating?

    <p>f(X1│X2)(x1│X2 = 1) = fX1,X2(x1, X2 = 1) / fX2(x2 = 1)</p> Signup and view all the answers

    What can be inferred from the fact that the conditional PMF obeys the laws of probability?

    <p>The conditional probability sums to 1</p> Signup and view all the answers

    What is the marginal probability of a positive analyst rating?

    <p>40%</p> Signup and view all the answers

    What is the conditional probability of a bond return of -10% given a positive analyst rating?

    <p>12.5%</p> Signup and view all the answers

    What is the purpose of the joint probability matrix in this example?

    <p>To calculate the conditional probability of a bond return given an analyst rating</p> Signup and view all the answers

    What is the formula for calculating the conditional probability of a bond return given an analyst rating?

    <p>f(X1│X2)(x1│X2 = x2) = P(X1 = x1 |X2 = x2) / fX2(x2 = x2)</p> Signup and view all the answers

    What is the conditional probability of a bond return of 10% given a positive analyst rating?

    <p>75%</p> Signup and view all the answers

    What does the PMF of a bivariate random variable represent?

    <p>The probability that the components of X take specific values</p> Signup and view all the answers

    What is the trinomial distribution a generalization of?

    <p>Binomial distribution</p> Signup and view all the answers

    What is the purpose of the CDF of a bivariate discrete random variable?

    <p>To find the total probability that each component is less than or equal to a given value</p> Signup and view all the answers

    What is the property of the PMF that states it is always non-negative?

    <p>Property 1: fX1,X2(x1, x2) ≥ 0</p> Signup and view all the answers

    What is the formula for the PMF of the trinomial distribution?

    <p>fX1,X2(x1, x2) = px1 px2 (1 − p1 − p2)n−x1−x2</p> Signup and view all the answers

    What is the total number of trials represented by the parameter 'n' in the trinomial distribution?

    <p>The total number of trials</p> Signup and view all the answers

    What is the third component of the trinomial distribution?

    <p>n − X1 − X2</p> Signup and view all the answers

    What is the probability of observing n − X1 − X2 in the trinomial distribution?

    <p>1 − p1 − p2</p> Signup and view all the answers

    What is the conditional distribution of a random variable given by?

    <p>f(X1│X2) = f(X1,X2) / f(X2)</p> Signup and view all the answers

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

    <p>Var(aX + bY) = a^2 Var(X) + b^2 Var(Y)</p> Signup and view all the answers

    What is the relationship between the covariance and correlation of two random variables?

    <p>Covariance is proportional to correlation</p> Signup and view all the answers

    What happens to the covariance of two random variables when they are independent?

    <p>The covariance becomes zero</p> Signup and view all the answers

    What is the formula for calculating the conditional distribution of a bivariate random variable?

    <p>f(X1│X2) = f(X1,X2) / f(X2)</p> Signup and view all the answers

    What is the purpose of calculating the conditional expectation of a component of a bivariate random variable?

    <p>To find the mean of the variable</p> Signup and view all the answers

    What is the relationship between the joint probability and the marginal probabilities of two independent random variables?

    <p>The joint probability is equal to the product of the marginal probabilities</p> Signup and view all the answers

    What can be inferred if the joint probability of two events is equal to the product of their marginal probabilities?

    <p>The events are independent</p> Signup and view all the answers

    Study Notes

    Conditional Variance and Standard Deviation

    • The conditional variance of X1 given X2 is defined as: Var(X1 | X2 = x2) = E(X1^2 | X2 = x2) - [E(X1 | X2 = x2)]^2
    • The conditional standard deviation is the square root of the conditional variance: σ(X1 | X2 = x2) = √Var(X1 | X2 = x2)

    Bivariate Random Variables

    • Multivariate random variables are used to model the dependence between two or more random variables
    • The concepts of expectations, moments, and distributions are similar to those of univariate random variables
    • Multivariate random variables are vectors of random variables, e.g., X = (X1, X2) with realizations x1 and x2

    Joint Probability Mass Function (PMF)

    • The joint PMF of a bivariate random variable is a function that gives the probability that the components of X take the values X1 = x1 and X2 = x2
    • The joint PMF has the following properties:
      • fX1,X2(x1, x2) ≥ 0
      • ∑x1 ∑x2 fX1,X2(x1, x2) = 1

    Conditional Distribution

    • The conditional distribution describes the probability of an outcome of a random variable conditioned on the other random variable taking a particular value
    • The conditional distribution is defined as: f(X1|X2)(x1|X2 = x2) = fX1,X2(x1, x2) / fX2(x2)
    • The conditional distribution is used in finance, such as risk management, to compute the conditional distribution of interest rates given a huge loss

    Independence and Conditional Distribution

    • If the distributions of the components of the bivariate distribution are independent, then: f(X1,X2)(x1, x2) = fX1(x1) fX2(x2)
    • If the distributions are independent, then the conditional distribution is equal to the marginal distribution: fX1(x1) = f(X1|X2)(x1|X2 = x2)

    Multivariate Discrete Random Variables

    • Multivariate discrete random variables involve defining several random variables simultaneously on a sample space
    • The PMF or PDF for a multivariate random variable gives the probability that the random variables each take a certain value

    Independent, Identically Distributed (IID) Random Variables

    • A collection of random variables is IID if each random variable has the same probability distribution as the others and all are mutually independent
    • IID variables are commonly used in time series analysis
    • The mean and variance of IID variables can be calculated using the normal distribution

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