Question

Given X is a p-variate random vector with mean vector μ and covariance matrix Σ. Write down the necessary and sufficient condition (without proof) for X to be a multivariate normal random vector

Answer #1

X is a normal random variable with mean μ and standard
deviation σ. Then P( μ− 1.2 σ ≤ X ≤ μ+ 1.9 σ) =?
Answer to 4 decimal places.

given normal random variable x with mean μ= 57.1 and standard
deviation σ=13.2, what is P (46 < x̄ < 69)
for a sample of size n= 16?

Let (X,Y) be a two dimensional Gaussian random vector with zero
mean and the covariance matrix {{1,Exp[-0.1},{Exp[-0.1],1}}.
Calculate the probability that (X,Y) takes values in the unit
circle {(x,y), x^2+y^2<1}.

Consider a multivariate random sample X1, . . . ,
Xnwhich comes from p-dimensional multivariate
distribution N(µ, Σ), with mean vector µ ∈ R p and the
variance-covariance positive definite matrix Σ. Find the
distribution and its parameters for the matrix nXTHX
where H is the idempotent centering matrix H = In −1/n
(1n ⊗ 1n ) and 1n is the
n-dimensional vector of all 1’s.

Let X be a Gaussian random variable with mean
μ and variance σ^2. Compute the following
moments:
Remember that we use the terms Gaussian random
variable and normal random variable
interchangeably.
(Enter your answers in terms of μ and σ.)
E[X^2]=
E[X^3]=
E[X^4]=
Var(X^2)=
Please give the detail process of proof.

given normal random variable Z with mean μ= 57.1 and standard
deviation σ=13.2, what is P (Z > 46)?

Given that x is a normal variable with mean μ = 48 and standard
deviation σ = 6.3, find the following probabilities. (Round your
answers to four decimal places.) (
a) P(x ≤ 60)
(b) P(x ≥ 50)
(c) P(50 ≤ x ≤ 60)
Please break down explanation a-c.

Suppose x has a distribution with μ = 22 and
σ = 20.
(a) If random samples of size n = 16 are selected, can
we say anything about the x distribution of sample
means?
Yes, the x distribution is normal with mean μ x = 22
and σ x = 1.3
No, the sample size is too small.
Yes, the x distribution is normal with mean μ x = 22
and σ x = 5
Yes, the x distribution...

Given that x is a normal variable with mean μ = 108 and standard
deviation σ = 14, find the following probabilities.
(a) P(x ≤ 120)
(b) P(x ≥ 80)
(c) P(108 ≤ x ≤ 117)

If X is distributed normally with mean μ and standard deviation
σ find P(μ−σ≤X≤μ+2σ)

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