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PROBLEM 1. [5 points] Value-at-Risk (VaR). The random variable Y measures the change in market va...
PROBLEM 1. [5 points] Value-at-Risk (VaR). The random variable Y measures the change in market value...
PROBLEM 1. [5 points] Value-at-Risk (VaR). The random variable Y measures the change in market value of a portfolio during a given time period. The variable Y is assumed to be normal with N(μ, σ, (a) Calculate the VaR (Value-at-Risk) with confidence level 1-a, 0 < α < 1, (b) In particular, calculate VaR with confidence level 80% if Y-N(0.1, 0.22)
Problem 2: 20 points 10 5 + 5) A continuous random variable (Y) has a density,...
Problem 2: 20 points 10 5 + 5) A continuous random variable (Y) has a density, fY (3e-3V for y>0 and f () 0, elsewhere. Given Y y, a discrete random variable, N, is Poisson distributed with the rate equal to y TA 1. Derive the marginal distribution of N 2. Determine the marginal expectation of N, EIN 3. Determine the marginal variance of N, Var[N]
Problem 1. (a) Let X be a Binomial random variable such that E(X) 4 and Var(x)...
Problem 1. (a) Let X be a Binomial random variable such that E(X) 4 and Var(x) 2. Find the parameters of X (b) Let X be a standard normal random variable. Write down one function f(t) so that the random variable Y-f(X) is normal with mean a and variance b.
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y)...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
QUESTION 2 1 points Save a Tina Ming is a senior portfolio manager at Flusk Pension...
QUESTION 2 1 points Save a Tina Ming is a senior portfolio manager at Flusk Pension Fund (Flusk). Flusk's portfoliois composed of fixed Income instruments structured to match Flusk's liabilities. Mingworks with Shrikant McKee, Flusk's risk analyst.Ming and McKee discuss the latest risk report. McKee calculated value at risk (VaR)for the entire portfolio using the historical method and assuming a lookback period offive years and 250 trading days per year. McKee presents VaR measures in Exhibit 1. Exhibit 1: Flusk...
Problem 3 [5 points) (a) [2 points] Let X be an exponential random variable with parameter...
Problem 3 [5 points) (a) [2 points] Let X be an exponential random variable with parameter 1 =1. find the conditional probability P{X>3|X>1). (b) [3 points] Given unit Gaussian CDF (x). For Gaussian random variable Y - N(u,02), write down its Probability Density Function (PDF) [1 point], and express P{Y>u+30} in terms of (x) [2 points)
Problem 4 Suppose X ~N(0, 1) (1) Explain the density of X in terms of diffusion process. (2) Calculate E(X), E(X2), and Var(X). (3) Let Y = μ +ơX. Calculate E(Y) and Var(Y). Find the density of Y....
Problem 4 Suppose X ~N(0, 1) (1) Explain the density of X in terms of diffusion process. (2) Calculate E(X), E(X2), and Var(X). (3) Let Y = μ +ơX. Calculate E(Y) and Var(Y). Find the density of Y.
Problem 4 Suppose X ~N(0, 1) (1) Explain the density of X in terms of diffusion process. (2) Calculate E(X), E(X2), and Var(X). (3) Let Y = μ +ơX. Calculate E(Y) and Var(Y). Find the density of Y.
Problem 1 (16 points). Suppose that Y is normally distributed random variable with u-10 and σ-2...
Problem 1 (16 points). Suppose that Y is normally distributed random variable with u-10 and σ-2 and X is another normally distributed random variable with μ-: 5 and σ-5. Y and X are independent. Calculate the following probabilities according to a normal distribution table (e.g., a normal table found from the Internet) (1) (4 points) Pr(Y> 12) (2) (4 points) Pr(2 < X <4) (3) (4 points) Pr(Y> 12 and 2< X <4) and Pr(Y> 12 or 2< X <4)...
Problem 10: 10 points Assume that a random variable (L) follows the exponential distribution with intensity...
Problem 10: 10 points Assume that a random variable (L) follows the exponential distribution with intensity λ-1. Given L-u, a random variable Y has the Poisson distribution with parameter - u. 1. Derive the marginal distribution of Y and evaluate probabilities, PY=n] , for n = 0,1,2, 2. Find the expectation of Y, that is E Y 3. Find the variance of Y, that is Var Y
3. (10 points) The random variable Y has a normal probability distribution with the density funct...
3. (10 points) The random variable Y has a normal probability distribution with the density function (a) Verify,Ef(y) dy=1; (b) Show that E(Y) = μ; (c) Let F(u) be the distribution function of Y. Prove that e2 1 dr
3. (10 points) The random variable Y has a normal probability distribution with the density function (a) Verify,Ef(y) dy=1; (b) Show that E(Y) = μ; (c) Let F(u) be the distribution function of Y. Prove that e2 1 dr
PROBLEM 1. [5 points] Value-at-Risk (VaR). The random variable Y measures the change in market value of a portfolio during a given time period. The variable Y is assumed to be normal with N(μ, σ, (a) Calculate the VaR (Value-at-Risk) with confidence level 1-a, 0 < α < 1, (b) In particular, calculate VaR with confidence level 80% if Y-N(0.1, 0.22)
Problem 2: 20 points 10 5 + 5) A continuous random variable (Y) has a density, fY (3e-3V for y>0 and f () 0, elsewhere. Given Y y, a discrete random variable, N, is Poisson distributed with the rate equal to y TA 1. Derive the marginal distribution of N 2. Determine the marginal expectation of N, EIN 3. Determine the marginal variance of N, Var[N]
Problem 1. (a) Let X be a Binomial random variable such that E(X) 4 and Var(x) 2. Find the parameters of X (b) Let X be a standard normal random variable. Write down one function f(t) so that the random variable Y-f(X) is normal with mean a and variance b.
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
QUESTION 2 1 points Save a Tina Ming is a senior portfolio manager at Flusk Pension Fund (Flusk). Flusk's portfoliois composed of fixed Income instruments structured to match Flusk's liabilities. Mingworks with Shrikant McKee, Flusk's risk analyst.Ming and McKee discuss the latest risk report. McKee calculated value at risk (VaR)for the entire portfolio using the historical method and assuming a lookback period offive years and 250 trading days per year. McKee presents VaR measures in Exhibit 1. Exhibit 1: Flusk...
Problem 3 [5 points) (a) [2 points] Let X be an exponential random variable with parameter 1 =1. find the conditional probability P{X>3|X>1). (b) [3 points] Given unit Gaussian CDF (x). For Gaussian random variable Y - N(u,02), write down its Probability Density Function (PDF) [1 point], and express P{Y>u+30} in terms of (x) [2 points)
Problem 4 Suppose X ~N(0, 1) (1) Explain the density of X in terms of diffusion process. (2) Calculate E(X), E(X2), and Var(X). (3) Let Y = μ +ơX. Calculate E(Y) and Var(Y). Find the density of Y.
Problem 4 Suppose X ~N(0, 1) (1) Explain the density of X in terms of diffusion process. (2) Calculate E(X), E(X2), and Var(X). (3) Let Y = μ +ơX. Calculate E(Y) and Var(Y). Find the density of Y.
Problem 1 (16 points). Suppose that Y is normally distributed random variable with u-10 and σ-2 and X is another normally distributed random variable with μ-: 5 and σ-5. Y and X are independent. Calculate the following probabilities according to a normal distribution table (e.g., a normal table found from the Internet) (1) (4 points) Pr(Y> 12) (2) (4 points) Pr(2 < X <4) (3) (4 points) Pr(Y> 12 and 2< X <4) and Pr(Y> 12 or 2< X <4)...
Problem 10: 10 points Assume that a random variable (L) follows the exponential distribution with intensity λ-1. Given L-u, a random variable Y has the Poisson distribution with parameter - u. 1. Derive the marginal distribution of Y and evaluate probabilities, PY=n] , for n = 0,1,2, 2. Find the expectation of Y, that is E Y 3. Find the variance of Y, that is Var Y
3. (10 points) The random variable Y has a normal probability distribution with the density function (a) Verify,Ef(y) dy=1; (b) Show that E(Y) = μ; (c) Let F(u) be the distribution function of Y. Prove that e2 1 dr
3. (10 points) The random variable Y has a normal probability distribution with the density function (a) Verify,Ef(y) dy=1; (b) Show that E(Y) = μ; (c) Let F(u) be the distribution function of Y. Prove that e2 1 dr
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