The density function of Xis (2(x-1), f(x)= (a) Find the mean E(4X+3); (b) Find the variance σ3x+2. 0, 1 < x < 2, elsewhere.
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- Yearly returns on two assets (in percent) are denoted by rx and ry for asset X and Y respectively. Their joint distribution is multivariate normal with with mean (6, 2) and a covariance matrix 0.4 -0.2 -0.2 0.3 • Find the best linear approximation of the relationship between returns for a given return of asset X.Suppose X has PDF fX (x) = 4x(1 − x^2) for 0 < x < 1 (0 everywhere else). Find all mediansand modes for X.(a) Find all medians of X.(b) Find all modes of X.Find the method of moments estimator for the parameters of the double exponential distribution DE(0,n) f(x) Le-lz-nl/, - 0 0. 20
- (b) (i) Briefly explain how moment generating function of a random variable, y may be used to Generate its moments. (ii) Let the random variable, y have probability density function S(t) = kye-1/2y y20 , elsewhere Where k> 0 is a constant. Find the moment generating function of y and use it to evaluate the mean and variance of y leaving your results in terms of k.Let X₁,..., Xn be an iid random sample from a Uniform(0, 0), where 0 > 0. Let X(n) = max{X₁, (a) Find the mean and variance of X(n). (b) Find the limiting distribution of Zn = n (0 – X (n)). (c) Find the limiting distribution of log(Zn). Xn}Consider the data (1,3), (2,5), (3,9.67) (a) Derive the normal equations for the best fit of the form f(x) = - - + Dx? (b) Find C, D using the normal equations derived above and given data. (c) Find a suitable Linearization for f (x) =- + Dx² (Don't find C, D)
- Let X1, X2, ..., X50 be a random sample of size 50 from a distribution with density for 0Pv. Don't provide handwriting solution(b) Let X represent the lifetime in months of a certain brand of light bulb. The pdf of X is given by f(x)=K(6-x), 0Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON