1. [Random Vectors] Let X and Z are independent and identically distributed discrete random variables with support Sx = {−1,1} and probability mass function (PMF) is given by 0.5, x = 1 TX (x): = 0.5, x = -1 There given other random variable, Y, that is defined as Y = (a) Characterize support and PMF for Y. 1, if X = 1 Z, if X = −1 (b) Find variance-covariance matrix for random vector (X,Y). Are X and Y uncorrelated? (c) Find the optimal (in the mean squared error sense) predictor of Y given X, E(Y|X). (d) Find the optimal (in the mean squared error sense) linear predictor of Y given X. This implies finding the coefficients a and b from g(X) = a+b. X, such that E(Y - 9(X))² is minimized.

Q: what is the probability of the event (X < }, Y < })?

A:

Q: Let X_1 ∼ N (1, 1) and X_2 ∼ N (2, 4) be two normally distributed and statisti- cally independent…

A: Let's analyze each statement one by one:(a) P(X1 &gt; 1) = P(X2 (b) Cov(X1, X2) = 0: Since X1 and X2…

Q: 1 V(X) for i) X ~ U{1,n}, ii) X~ Geometric (p), iii) X P

A: We have given that X be a random variable which follows 1) Uniform distribution, i.e. X∼U{1,n}2)…

Q: Consider a random variable K with parameter p, whose probability mass function (PMF) is given by…

A: Solution: From the given information, the probability mass function is

Q: i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and…

A: It is given that  X follows N(θ, 1), then E(X) = θ and V(X) = 1, And the moment genarating function…

Q: 3- , thei a) E{f,(y\x)} =- exp(- o/27(2-r²) 20ʻ(2-r²)' 1 b) E{f, (x)f;(y)}=- 2.noV4-r

A:

Q: then the value of variance of X is

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Q1. Suppose that X and Y are jointly continuous random variable (у — х for 0<x < 1 and 1 < y < 2 - x…

A: We have given joint pdf of X and Y we have to find answers of related questions.at bartleby we…

Q: QUESTION 2 Consider the following information about 3 random variables, X Y and Z. X|Y- Gamma(2Y2,).…

A: Given: X/Y ~ Gamma( 2Y2, Y) Y~ N(2, 5) From given: E(X/Y)=2Y2Y Var(X/Y)=2Y2Y2 We know that for…

Q: Ex//let xa random variable with pm f p X=xN N , x=1,2,.., N and let Y=a+bx where a, bER find Ey and…

A:

Q: Suppose the random variable has possible values {1, 2, 3, 4, 5, 6} and that the probability mass…

A: Hi! Thank you for the question. As you have posted a question with multiple sub-parts, according to…

Q: Random variables X1, X2, ..., Xn are independent and identically distributed; 0 is a parameter of…

A: (a) A pivotal function is a function under which the function's probability distribution does not…

Q: Obtain variance covariance matrix of , Var-Cov( ) and the standard errors.

A: Regression is used when you want to predict a continuous dependent variable from a number of…

Q: 4. For two independent random variables such that X sin N(0, 4), Y ~ U[0, 4], Var(2X +3Y) = Var(2X -…

A:

Q: The distribution of random variable X is Y(X=1)=0^2, P(X=2)=20(1-0), P(X=3)=(1-0)^2 a) Give the…

A: Given that the distribution of random variable X is PX=1=θ2 PX=2=2θ1-θ PX=3=1-θ2

Q: iii) Show using the moment generating function technique that Y is a standard normal random…

A: It is given that  X follows N(θ, 1), then E(X) = θ and V(X) = 1, And the moment genarating function…

Q: 2. Consider two sets of multivariate random variables: x = and y = (y₁, y2, ....ya)', where E[x]…

A:

Q: Let X,Y be two random variables and take values in {0,1} and whose joint distribution is given by: (…

A: First, we need to calculate the expected values of X, Y, X^2, and Y^2. The expected value of a…

Q: Let X,Y ~ U(0, 1) be independent random variables uniformly distributed over (0, 1) and Z = X+ (a)…

A: Introduction:- We would like to estimate the value of an unobserved random variable X, given that we…

Q: If X and Y are independent U (–1, 1) random variables then E (min (X,Y))

A:

Q: X1, X2, , nare n independent zero mean Gaussian random variables M] / nis with equal variance o².…

A:

Q: Let X be à discrete random variable with pmf fax² p(¹) = | e z=0,1,2,3,4 otherwise à) If p(0) = .14,…

A: Here , we have given a pmf which is describes in the question.

Q: The joint cdf of a random variable X and Y is given by Fxy(x, y) = -e¯ª)(l-e-by) _x≥0, y ≥0, a, b>0…

A:

Q: and let Y; = Xí, i = 1, ..., n. %3D Find the distribution of W 2(Y1+Y2+Y3) 3(Y4+Y5)

A:

Q: 3- Show that, if the random variables X and Y are N(0 6,0",r) , thên 1 a) E{ fy(y|x)} = =exp{- 20…

A:

Q: ad e; are independent random variables. that it makes sense to do a least squa east squares estimate…

A: (a) The assumptions as, 1. The expected value of the mean of the error terms εi will be zero when…

Q: 2. Let X and Y be two random variables. Suppose XY ~ Binomial (n, Y) where n is fixed and Y stands…

A:

Q: Suppose that X,Y are both standard normal random variables and Cov(X,Y) = 1 (a) Compute E (X² – Y²)|…

A:

Q: 2. If X is a random variable taking values 0,1,2,.... and P(X) = ab*, where a and b are positive…

A: i) First part:ii) second part:

Q: Consider a binary choice model in random utility framework: U1 = Bo + B1x1+ + Bræk + u = x'B+ u and…

A: Given:

Q: i) Let X,X2, be two identical independent distributed (iid)random variables ,each X- X2 having chi –…

A: Hello! As you have posted 2 different questions, we are answering the first question. In case you…

Q: Let Y~Po(1) be a poisson distributed variable with parameter 2 = 1. Suppose that, designated on Y, X…

A:

Q: Suppose X₁, X2, · ·, Xn are independent random variables and have CDF F(x). Find the CDF of Y =…

A: To find the cumulative distribution function (CDF) of Y=min{X1,X2,…,Xn}, we need to consider the…

Q: X and Y are discrete random variables. If var(X)= var(Y) = o², cov(X,Y)= find var(2X-3Y)

A:

Q: Let X_1 and X_2 be independent random variables for which P(X_i = 1) = 2/5 and P(X_i = 2) = 3/5.…

A: