If X is a random variable for which E (X) 10 and Var (X) = 25, for what positive value of 'a' and b' does Y= a X-b have expectation 0 and variance 1. %3D
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- Imagine a family member looks over your shoulder as you look at the variance equation Σki=1(xi − μ)2P(X = xi) and asks why the P(X = xi) term is there. What would you say?3 Suppose the following model yi=β1+ui Where ui is assumed to be normally distributed with mean zero and variance σ^2. Calculate analytically, following OLS technique, βHAT1and Var(βHAT1)For theoretically modelling the economic development of national economy scenarios the following 2 models for GDP increment are analysed: a. Yt = Yt-1 + at b. Yt = 1.097 Yt-1 - 0,97 Yt-2 + at, where Y - GDP increment a - White noise with zero mean and constant variance o2 =100 t- Time (quarters starting with Q1, 1993) Check by an algebraic criterion which one is stationary.
- Suppose Y is a random variable with E(Y) = 13 and Var(Y) = 6. Solve for the following: (show complete solution) a. E(3Y+ 10) = ? b. E(Y^2) = ? c. Var (10) = ? d. Var (5) = ?If a discrete random variable X has the following probability distribution: X -2, - 1, 0, 1, 2 P(X) 0.2, 0.3, 0.15, 0.2, 0.15 Use this to find the following: (a) The mean of X and E[X^2]. (b) The probability distribution for Y = 2X^2 + 2 (i.e, all values of Y and P(Y )). (c) Using part (b) (i.e, the probability distribution forY ), find E[Y ]. (d) Using part (a), verify your answer in part (c) for E[Y ]. **Note: Please do not just copy from Chegg!1) Suppose the following model y; = B1 + u; Where is assumed to be normally distributed with mean zero and variance ơ² . Calculate analytically, following OLS technique, ß, and Var(ß,).
- 4An electrochemical engineer has manufactured a new type of fuel cell (a type of battery)which has to undergo testing to prove its duration: the time it takes to go from fullycharged to completely uncharged, under a fixed nominal load. From the computational simulation models she has, the variance of the duration is σ2 = 4 h2 (hours squared)but she wants to estimate the mean duration time μ. To achieve this she is determinedto do the tests multiple times in independent but identical conditions. Can you findwhat is the smallest number of these tests that she has to do in order for her estimatedmean duration to be within ±0.2 h tolerance of the true mean with 95% certaintyA researcher that wanted to estimate the expectation AY of a random variable Y got three independent observations, Y, Y, Y The researcher knows the value o, of the variance of Y and is considering the following estimators: Pi = 4) Yi + (+) ¥ Py = () Yn + (;) ¥½ + () Y½ in (}) Yi + (}) ¥z + (() %D Which of the following is correct? ONone of the above Ois an unbiased estimator of l and it has the smallest variance of the three estimators. Ois an unbiased estimator of µ and it has the smallest variance of the three estimators. O and i, are both unbiased estimators of fl and Var (ſîz) < Var (îì3). is an unbiased estimator of µ and it has the smallest variance of the three estimators.
- Heteroskedasticity arises because of non-constant variance of the error terms. We said proportional heteroskedasticity exists when the error variance takes the following structure: Var(et)=σt^2=σ^2 xt. But as we know, that is only one of many forms of heteroskedasticity. To get rid of that specific form of heteroskedasticity using Generalized Least Squares, we employed a specific correction – we divided by the square root of our independent variable x. And the reason why that specific correction worked, and yielded a variance of our GLS estimates that was sigma-squared, was because of the following math: (Picture 1) Where var(et)=σ^2 according to our LS assumptions. In other words, dividing everything by the square root of x made this correction work to give us sigma squared at the end of the expression. But if we have a different form of heteroskedasticity (i.e. a difference variance structure), we have to do a different correction to get rid of it. (a) what correction would you use…'Melanoma' is a form of skin cancer and each year 17% of the patients who suffer from the disease die. A random sample of 10,000 melanoma patients is formed at the start of a year. Let Y denote the number of patients in this sample who will die during the year? (a) What is the expected value of Y? That is, compute E(Y). Answer: [Select] (b) What is the variance of Y? That is, compute var(Y). Answer: [Select] (c) What is the probability that Y exceeds 1,800? That is, compute P(Y> 1,800). Answer: [Select]Let Y be a random variable with the mgfM(t) = .35e^(−4t) + .1 + .25e^(2t) + .3e^(4t).(a) What is the pmf of Y?(b) Calculate the variance of Y directly from the pmf you determined in part (a).(c) Calculate E[Y^4] directly from the mgf.