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Problem 4 (35 points) An asset price is modeled by using a sequence of independent and...
74. Let X1, X2, ... be a sequence of independent identically distributed contin- uous random variables....
74. Let X1, X2, ... be a sequence of independent identically distributed contin- uous random variables. We say that a record occurs at time n if X > max(X1,..., Xn-1). That is, X, is a record if it is larger than each of X1, ... , Xn-1. Show (i) P{a record occurs at time n}=1/n; (ii) E[number of records by time n] = {}_1/i; (iii) Var(number of records by time n) = 2/_ (i - 1)/;2; (iv) Let N =...
4.) Let X1, X2 and X3 be independent uniform random variables on [0,1]. Write Y =...
4.) Let X1, X2 and X3 be independent uniform random variables on [0,1]. Write Y = X1 + X, and Z X2 + X3 a.) Compute E[X, X,X3]. (5 points) b.) Compute Var(x1). (5 points) c.) Compute and draw a graph of the density function fy (15 points)
3. (a) (5 points) Let Xi,... be a sequence of independent identically distributed random variables e...
3. (a) (5 points) Let Xi,... be a sequence of independent identically distributed random variables e of tnduqendent idente onm the interval (o, 1] and let Compute the (almost surely) limit of Yn (b) (5 points) Let X1, X2,... be independent randon variables such that Xn is a discrete random variable uniform on the set {1, 2, . . . , n + 1]. Let Yn = min(X1,X2, . . . , Xn} be the smallest value among Xj,Xn. Show...
Problem 5. Indicator variables S points possible (graded) Consider a sequence of n 1 independent tosses...
Problem 5. Indicator variables S points possible (graded) Consider a sequence of n 1 independent tosses of a biased coin, at times k = 0,1,2,...,n On each toss, the probability of Heads is p, and the probability of Tails is 1 -p {1,2,.., at time for E resulted in Tails and the toss at time - 1 resulted in A reward of one unit is given if the toss at time Heads. Otherwise, no reward is given at time Let...
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 of a portfolio during a given time period. The variable Y is assumed to be normal with N(μ,02). (a) Calculate the VaR (Value-at-Risk) with confidence level 1-α, 0 < α < 1, (b) In particular, calculate VaR with confidence level 75% if Y N(0.2, 0.32)
PROBLEM 1. [5 points] Value-at-Risk (VaR). The random variable Y measures the change in market value of a portfolio...
5. (15 points) The shopping times of n = 64 randomly selected customers at a local...
5. (15 points) The shopping times of n = 64 randomly selected customers at a local supermarket were recorded. The average and variance of the 64 shopping times were 33 minutes and 356 minutes, respectively. Estimate u, the true average shopping time per customer, with a confidence coefficient of 1-a = 0.90. 6. (10 points) Let X1, X2, ..., Xn denote n independent and identically distributed Bernoulli random vari- ables s.t. P(X; = 1) = p and P(Xi = 0)...
4) Repair time of a car can be modeled by linear regression using months since the type of repair (X1), drive type (X2), servicing frequency (X3), and driving frequency (X4) as predictors. The model o...
4) Repair time of a car can be modeled by linear regression using months since the type of repair (X1), drive type (X2), servicing frequency (X3), and driving frequency (X4) as predictors. The model obtained was Y = 2.5 +1.7X1 – 3.5X2 + 6.2X3 – 4.7X4. The model was built based on 40 cars and the standard errors for X1, X2, X3, and X4 were 3.4, 2.6, 2.8, and 3.9 respectively. If the F-stat is 5.8, a) Find R2. b)...
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 5 (10 points). Suppose that the independent Bernoulli trials each with success probability p, are...
Problem 5 (10 points). Suppose that the independent Bernoulli trials each with success probability p, are performed independently until the first success occurs, Let Y be the number of trials that are failure. (1) Find the possible values of Y and the probability mass function of Y. (2) Use the relationship between Y and the random variable with a geometric distribution with parameter p to find E(Y) and Var(Y).
[PLEASE USE HINT] Problem 10: 10 points Suppose that (Xi, X2,... are independent identically distributed binary...
[PLEASE USE HINT]
Problem 10: 10 points Suppose that (Xi, X2,... are independent identically distributed binary variables taking (0 1) values with probability PX here 0<q Introduce the new variable, M such that the event {M = k} occurs when three consecutive successes appear at the first time. In other words, event [M = kj, where k-3, occurs if and only if and there is no previous occurrence of three consecutive successes. Use conditioning techniques to derive the expected value,...
74. Let X1, X2, ... be a sequence of independent identically distributed contin- uous random variables. We say that a record occurs at time n if X > max(X1,..., Xn-1). That is, X, is a record if it is larger than each of X1, ... , Xn-1. Show (i) P{a record occurs at time n}=1/n; (ii) E[number of records by time n] = {}_1/i; (iii) Var(number of records by time n) = 2/_ (i - 1)/;2; (iv) Let N =...
4.) Let X1, X2 and X3 be independent uniform random variables on [0,1]. Write Y = X1 + X, and Z X2 + X3 a.) Compute E[X, X,X3]. (5 points) b.) Compute Var(x1). (5 points) c.) Compute and draw a graph of the density function fy (15 points)
3. (a) (5 points) Let Xi,... be a sequence of independent identically distributed random variables e of tnduqendent idente onm the interval (o, 1] and let Compute the (almost surely) limit of Yn (b) (5 points) Let X1, X2,... be independent randon variables such that Xn is a discrete random variable uniform on the set {1, 2, . . . , n + 1]. Let Yn = min(X1,X2, . . . , Xn} be the smallest value among Xj,Xn. Show...
Problem 5. Indicator variables S points possible (graded) Consider a sequence of n 1 independent tosses of a biased coin, at times k = 0,1,2,...,n On each toss, the probability of Heads is p, and the probability of Tails is 1 -p {1,2,.., at time for E resulted in Tails and the toss at time - 1 resulted in A reward of one unit is given if the toss at time Heads. Otherwise, no reward is given at time Let...
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,02). (a) Calculate the VaR (Value-at-Risk) with confidence level 1-α, 0 < α < 1, (b) In particular, calculate VaR with confidence level 75% if Y N(0.2, 0.32)
PROBLEM 1. [5 points] Value-at-Risk (VaR). The random variable Y measures the change in market value of a portfolio...
5. (15 points) The shopping times of n = 64 randomly selected customers at a local supermarket were recorded. The average and variance of the 64 shopping times were 33 minutes and 356 minutes, respectively. Estimate u, the true average shopping time per customer, with a confidence coefficient of 1-a = 0.90. 6. (10 points) Let X1, X2, ..., Xn denote n independent and identically distributed Bernoulli random vari- ables s.t. P(X; = 1) = p and P(Xi = 0)...
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 5 (10 points). Suppose that the independent Bernoulli trials each with success probability p, are performed independently until the first success occurs, Let Y be the number of trials that are failure. (1) Find the possible values of Y and the probability mass function of Y. (2) Use the relationship between Y and the random variable with a geometric distribution with parameter p to find E(Y) and Var(Y).
[PLEASE USE HINT]
Problem 10: 10 points Suppose that (Xi, X2,... are independent identically distributed binary variables taking (0 1) values with probability PX here 0<q Introduce the new variable, M such that the event {M = k} occurs when three consecutive successes appear at the first time. In other words, event [M = kj, where k-3, occurs if and only if and there is no previous occurrence of three consecutive successes. Use conditioning techniques to derive the expected value,...
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