Let X1, X2, and X3 be independent normal random variables with mean µ1, µ2, µ3 and variance σ1^2 , σ2^2 , and σ3^2 . What is the distribution of X1 − X2 + 2X3 − 10?
Let X1, X2, and X3 be independent normal random variables with mean µ1, µ2, µ3 and...
Let X1, X2, X3 be independent random variables with E(X1) = 1, E(X2) = 2 and E(X3) = 3. Let Y = 3X1 − 2X2 + X3. Find E(Y ), Var(Y ) in the following examples. X1, X2, X3 are Poisson. [Recall that the variance of Poisson(λ) is λ.] X1, X2, X3 are normal, with respective variances σ12 = 1, σ2 = 3, σ32 = 5. Find P(0 ≤ Y ≤ 5). [Recall that any linear combination of independent normal...
Let x1, x2, x3, x4
be independent standard normal random variables. Show that
,
,
are independent and each follows a
distribution
(x1 - r2)
Considering two Gaussian distributions N1~(μ1,σ1^2) and N2~(μ2,σ2^2), we pick two random variables x1 and x2 in order to compute the sum x3=x1+x2. We want to prove that: a) x3 follows a gaussian distribution b) estimate mean value μ3 and variance σ3^2 c) repeat the above steps for multivariate Gaussian distributions N1~(μ1,Σ1) and N2~(μ2,Σ2)
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3).
3. (25 pts.) Let X1,...
Let x1, x2,x3,and x4 be a random sample from population with normal distribution with mean ? and variance ?2 . Find the efficiency of T = 1/7 (X1+3X2+2X3 +X4) relative to x= x/4 , Which is relatively more efficient? Why?
2. The random variables X1, X2 and X3 are independent, with Xi N(0,1), X2 N(1,4) and X3 ~ N(-1.2). Consider the random column vector X-Xi, X2,X3]T. (a) Write X in the form where Z is a vector of iid standard normal random variables, μ is a 3x vector, and B is a 3 × 3 matrix. (b) What is the covariance matrix of X? (c) Determine the expectation of Yi = Xi + X3. (d) Determine the distribution of Y2...
X1, X2, X3, X4,X5,X6,X7,X8 are independent identically distributed random variables. Their common distribution is normal with mean 0 and variance 4. Let W = X12+ X22 + X32 + X42+X52+X62+X72+X82 . Calculate Pr(W > 2)
Let X1, X2, and X3 be three independent, continuous random variables with the same distribution. Given X2 is smaller than X3, what is the conditional probability that X1 is smaller than X2?
If X1 and X2 are independent and identically distributed normal random variables with mean m and variance s2, find the probability distribution function for U=X1-3X2/2.
Suppose X1, X2,..., X10 are independent normal random variables with mean O and variance 1. Let max{X1, X2, ..., X10} What is the largest value of t so that P(M <t) < 0.90? That is, find the 90th percentile of M.