• Given two random variables X and Y, both with nonzero variance, define the function h(t) = E[((X — ux)+t(Y – My))]. (a) Show that h(t) ≥ 0. (b) Show that h(t) = Var(X) + 2 Cov(X, Y)t + Var(Y)t² (c) h is a nonnegative parabola, so it has at most one root. Thus its discriminant is at most 0. Use this to deduce that the correlation coefficient p satisfies -1 ≤p ≤ 1.
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Random variables and variance
![• Given two random variables X and Y, both with nonzero variance, define the function h(t) = E[((X —
ux)+t(Y – ly))].
(a) Show that h(t) ≥ 0.
(b) Show that h(t) = Var(X) + 2 Cov(X, Y)t + Var(Y)t²
(c) h is a nonnegative parabola, so it has at most one root. Thus its discriminant is at most 0. Use
this to deduce that the correlation coefficient p satisfies −1 ≤ p≤1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2F1034038a-de23-46e6-ad47-931a7a172a7a%2Fwys2y6q_processed.png&w=3840&q=75)
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.Find two nonnegative number x and y for which x+3y=30, such that x2y is maximized.A (t) is a random process having mean = 2 and auto correlation function Rxx (7) = 4 [e- 0.2 ld+ 1. Let Y and Z be the random variables obtained by sampling X (t) at t = 2 and t = 4 respectively. Find the variance of the random variable W = Y -Z.
- X (t) is a random process having mean = 2 and auto correlation function. Rxx (7) = 4 [e-0.2 ltl + 1. Let Y and Z be the random variables obtained by sampling X (t) att = 2 and t = 4 respectively. Find the variance of the random variable W = Y -Z.Suppose X is a random variable, whose pdf is defined as follows: 2x = (²x) (u(x) - u(x − 3)) where u(x) is the unit step function. Determine the conditional pdf fx(x 1Let Y be a continuous random variable with √ ½ (2 f(y) = = Find the mean (u) and variance (o²) of Y. (2−y), 0≤ y ≤ 2 elsewhere.Let fix) = 2x, 0Assume that X and Y are independent random variables where X has a pdf given by fx(x) = 2aI(0,1)(x) and Y has a pdf given by fy(y) = 2(1– y)I(0,1)(y). Find the distribution of X + Y.Prove that the variance of b, V(b) = o²(X'X)−¹A firm's revenue R is stochastically related to the effort exerted by its employee. Effort is a continuous variable. The employee can choose any level of effort e E [0, ). The choice of effort affects revenue so that: E(R|e) = e and Var(R|e) = 1 %3D where E(R|e) and V ar(R|e) denote the expected value and variance, respectively, of rev- enue when the employee exerts effort level e. The employer cannot observe the level of effort exerted by the employee. The employer wants to design a wage contract w based on the revenue and considers only contracts of the form: w-α+ βR and so the employee is guaranteed a payment a and then a bonus payment ßR which de- pends on revenue. The employee is a risk-averse expected utility maximiser. A contract w gives expected utility: Eu(w\e) = E(w\e)-eV ar(w]e) – c(e) where E(wle) and Var(wle) denote the expected value and variance of the contract, re- spectively, conditional on effort e, p is a parameter of risk aversion, and c(e) denotes the disutility of…Let Y be a continuous random variable. Let c be a constant. PROVE Var (Y) = E (Y2) - E (Y)2An investor has found that company1 have an expected return on E(X) = 4% and variance for the return equal V(X) = 0.49. Company 2 has E(Y) = 6% and variance V(Y) = 0.64. The correlation between the companies return is ρ(X,Y) = 0.3. The investor wants to invest p (0<p<1) in company1 and (1-p) in company2. The combined investment have a return: R = pX + (1-p)Y. Let p=0.4 such that R= 0.4X +0.6Y. Find the Expectation and variance of R. how are these results in comparison with X and Y separatley?SEE MORE QUESTIONSRecommended textbooks for youTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning
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