2. Let Z1,..., Z, be i.i.d. standard normal random variables and define X = Z;, k = 1,..., n. j=1 Determine the distribution of the vector X = [X1, X2, . . . , X,]'.
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![2. Let Z1,..., Z„ be i.i.d. standard normal random variables and define
X = Z;, k = 1, ..., n.
j=1
Determine the distribution of the vector X = [X1, X2, . .. , X„]".](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1a4d3cf4-4f2b-4627-a22e-664fd6c2491a%2F376f1f97-9d63-4a4f-8a6e-8e3b1ec2d90a%2Fvh9bd7p_processed.jpeg&w=3840&q=75)
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- Assume Z1, Z2, . . . , Zn are independent standard normal random variables. The random variable Y defined bySuppose that Z1, Z2, . . . , Zn are statistically independent random variables. Define Y as the sum of squares of these random variablesSuppose the random variable y is a function of several independent random variables, say x1,x2,...,xn. On first order approximation, which of the following is TRUE in general?
- If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?Consider X₁, X₂, . . . , Xn to be independent random variables from a Normal(μ,σ ² ) where both parameters are unknown.If X is a random variable having the standard normaldistribution and Y = X2, show that cov(X, Y) = 0 eventhough X and Y are evidently not independent.
- Let Y1, Y2, ... , Yn be a random sample of size n from a gamma distribution with parameters α = 1and β = 2. Derive the probability distribution of the sample mean Y̅ using moment-generatingfunctions.Let X1, X2, ... Xn random variables be independent random variables with a Poisson distribution whose parameters are l1, l2, ... ln, respectively. Which of the following is the moment generating function of the random variable Z defined as (the little image)?Consider random variables X1, · · · Xn, independent and identically distributed such that each Xi ∼N (0, 1). Write down an expression for the joint pdf of the n-dimensional random vector (X1, · · · Xn).(i.e. what is the distribution of a random sample of n standard normal random variables
- Let X1 and X2 be independent chi-square random variables with r1 and r2 degrees of freedom, respectively. Let Y1=(X1/r1)/(X2/r2) and Y2=X2. (a) Find the joint pdf of Y1 and Y2.Let Y be a discrete random variable. Let c be a constant. PROVE Var (Y) = E (Y2) - E (Y)2Let X1,...,Xn be iid random variables with expected value 0, variance 1, and covariance Cov [Xi,Xj] = ρ, for i≠j. Use Theorem of linearity of expectation to find the expected value and variance of the sum Y = X1 +...+Xn.