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Problem \(1 \quad\) Bivariate normal distributionAssume that \(\boldsymbol{X}\) is a bivariate normal random variable with$$ \boldsymbol{\mu}=E \boldsymbol{X}=\left(\begin{array}{l} 0 \\ 2 \end{array}\right) \quad \text { and } \quad \Sigma=\operatorname{Cov} \boldsymbol{X}=\left(\begin{array}{ll} 3 & 1 \\ 1 & 3 \end{array}\right) $$Let$$ \boldsymbol{Y}=\left(\begin{array}{l} Y_{1} \\ Y_{2} \end{array}\right)=\left(\begin{array}{lr} 1 / \sqrt{2} & -1 / \sqrt{2} \\ 1 / \sqrt{2} & 1 / \sqrt{2} \end{array}\right) \boldsymbol{X} $$a) Find the mean vector and covariance matrix of \(Y\). What is the distribution of \(Y ?\) Are \(Y_{1}\) and...
Graybill, 1961]. Let x -(X1, X2) have a bivariate normal distribution with pdf where Q-2x-x1x2 + 4 _ 11x1-5T2 + 19, and k is a constant. Find a constant a such that P(3X1-X2 < a) 0.01.
8. An important distribution in the multivariate setting is the multivariate normal distribution. Let X be a random vector in Rk. That is Xk with X1, X2, ..., xk random variables. If X has a multivariate normal distribution, then its joint pdf is given by f(x) = {27}</2(det 2)1/2 exp {=} (x – u)?g="(x-1)} is the covariant matrix. Note with parameters u, a vector in R", and , a matrix in Rkxk that det is the determinant of matrix ....
. In probability theory, the Normal Distribution (sometimes called a Gaussian Distribution or Bell Curve) is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Describing the normal distribution using a mathematical function is called a probability distribution function (PDF) which is given here: H The mean of the distribution ơ-The standard deviation f(x)--e 2σ We can...
Let X have a normal distribution with mean μ and variance σ ^2 . The highest value of the pdf is equal to 0.1 and when the value of X is equal to 10, the pdf is equal to 0.05. What are the values of μ and σ?
Let fy(x, μ, σ) stand for the probability distribution function (PDF) for the normal distribution with parameters μ and σ. Let X be a random variable with a PDF defined as follows: where t is a fixed constant between O and 1. What is E[XI? None of these
Let X and Y have a bivariate normal distribution with parameters μX = 4, μY = 2, σX = 2, σY = 4, and ρ = 1/2. Find two different lines, a(x) and b(x), parallel to and equidistant from E(Y|x), such that P[a(x) < Y < b(x)|X = x] = 0.6827 for all real x.
On January 1, 2021, Julee Enterprises borrows $34,000 to purchase a new Toyota Highlander by agreeing to a 6%, 4-year note with the bank. Payments of $798.49 are due at the end of each month with the first installment due on January 31, 2021. Record the issuance of the note payable and the first two monthly payments. (If no entry is required for a particular transaction/event, select "No Journal Entry Required" in the first account field. Do not round intermediate...
(a) Show that (Xi, X2) has a bivariate normal distribution with means μ1 , μ2, variances 어 and 05, and correlation coefficient ρ if and only if every linear combination c Xc2X2 has a univariate normal distr bution with mean c1μι-c2μ2, and variance c?σ? + c3- +2c1c2ρσ12, where cı and c2 are real constants, not both equal to zero. (b) Let Yİ = Xi/ởi, i = 1,2. Show that Var(Y-Yo) = 2(1-2).
Suppose that X and Y form a bivariate normal distribution. You are given that E[X] = E[Y] = 0, with o x = 3, 0y = 2. Further, the correlation between X and Y is 0.5. Find P(X<Y + 1).