1 (Expected Shortfall) Suppose the price of an asset Pt follows a normal random walk, i.e., Pt = Po+r₁ + ... + rt with r₁, r2,... being IID N(μ, o²). Po+r1+. ⚫ Suppose the VaR of rt is VaRq(rt) at level q, find the VaR of the price in T days, i.e., VaRq(Pt – Pt–T). - • If ESq(rt) = A, find ES₁(Pt – Pt–T).
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- Derive the variance of the Student's t distribution using the definition Var (x) = ELX2] - EX)² .(1 point) Let Y be a Binomial variable counting the number of successes in 70 independent trials where the probability of success at each trial is 0.57; that is, Y - Bin(70, 0.57). Part a) Use the pbinom function in R to find P(35 0.25.The number of clients a store has during a week is Poisson distributed with - 25. The amount of money spent by each client follows a random = el,000t+250t2. Assuming the mean A variable with moment generating function o(t) independence between the number of clients, and the amount of money spent by each client. Find the mean and variance of the amount of money received by the store every week.
- Suppose that X and Y are random variables where the conditional expected value of Y given X is given by E (Y|X) = 3X + 6 If X has the expected value 0 and variance Var(X)=9 determine the minimal possible value of the variance Var(Y) of Y./Let ZZ be a discrete random variable taking one of the four distributions covered in Chapter 10. Suppose you know that Var(Z)=(k+1)E(Z)Var(Z)=(k+1)E(Z) where kk is the last non-zero digit of your student ID number. Determine the distribution of ZZ and find its parameter(s), explaining your argument carefully.The Pareto distribution is frequently used as a model in the study of incomes and has the cumulative distribution function F(x;8,,0,)=1-(6, /x), x2 6,, where 6, > 0 and 0, > 0. If X1,X2, .. , X, is a random sample from this distribution, find the maximum likelihood estimators of 0, and 0,.
- Find the z-score z_0.025The average wait time to get seated at a popular restaurant in the city on a Friday night is 9 minutes. Is the mean wait time different for men who wear a tie? Wait times for 13 randomly selected men who were wearing a tie are shown below. Assume that the distribution of the population is normal. 7, 8, 9, 10, 10, 8, 10, 9, 10, 8, 11, 11, 9 What can be concluded at the the αα = 0.01 level of significance level of significance? For this study, we should use The null and alternative hypotheses would be: H0:H0: H1:H1: The test statistic = (please show your answer to 3 decimal places.) The p-value = (Please show your answer to 4 decimal places.) The p-value is αα Based on this, we should the null hypothesis. Thus, the final conclusion is that ... The data suggest the population mean is not significantly different from 9 at αα = 0.01, so there is statistically insignificant evidence to conclude that the population mean wait time for…let Y be a single observation from a normal distribution with mean θ and variance θ^2, θ>0.
- 4. The battery life of a fully charge iPhone 13 Pro on a standby follows a normally distributed with mean 7 hour and variance o² fl = 1 hours. = (a) Write out the integral form of the probability that a fully-charged iPhone 13 Pro last between 7 and 9 hours on standby. then use the pnorm function in R to find this probability. (b) What is the probability that a fully-charged iPhone 13 Pro last less than 5 hours? (c) What is the probability that a fully-charged iPhone 13 Pro last more than 9 hours? (d) What is the minimum time a fully-charged iPhone 13 Pro need to last in order to be in the top 10% best quality?For each random variable defined, describe the set of possible values for the variable, and state whether the variable is discrete or continuous. (a) U = number of times a surfer has to paddle in front of a wave before catching one (b) X = length of a randomly selected angelfishLet Y1, . . . , YN be a random sample from the Normal distribution Yi ∼ N(ln β, s2) where s^2is known.Find the maximum likelihood estimator of b from first principles.Find the Score function, the estimating equation and the information matrix using GLM