Suppose that Y ~ Normal(0, 1) and define new random variable X by X=Y². Compute ELYI
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![Suppose that
Y ~ Normal(0, 1)
and define new random variable X by X=Y2. Compute
E[X].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F457222b6-3da3-440b-a3f4-146e4d50c2f1%2F224be14a-6601-424f-836d-ff7b389b7e70%2F1194ch_processed.png&w=3840&q=75)
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- A scheme of random number generator is described as follows. First, one generates a normal random number with mean 0 and variance 16, a uniform random number on [0,24], then a geometric random number of parameter p = 0.25 (each independently). Then we define the random number to be the average of these three numbers. 30 random numbers are generated with the scheme above, denoted as {X}31. Estimate/approximate the probability that the average 1 X is deviated from μ = E[X₁] by a distance of more than 0.2 using: (a) Chebyshev's inequality; (b) central limit theorem. k=1Suppose that W is a (a = 3, 3 = })-gamma random variable and N is a µ = -Poisson random variable independent of W. What is E (w*]? un for u-Poisson random variable N. n! Note: PN (n) = P (N = n) = e Hint: pmf for a (r, p)-negative binomial random variable is given by n - n = r,r + 1,... r - 1"" please answer alll
- Show that two random variables X, and X, with joint pdf. Sx, x, (*1 »*2) = 16 Xl< 4, 2< X2<4 are independent and orthogonal.Exercise 3: Find MVUE for 0, in each of the following cases: 1. (X1, X2, ... ,Xx,} is a random sample from Gamma(4,0). 2. (X1, X2, ... ,X,} is a random sample from Binomial (10, 0). 3. (X1, X2, ...,Xx,} is a random sample from Poisson(0).Let U1 and U2 be two iid U(0,1) random variables. Show that U1 and 1 - U2 have the same distribution by showing their CDFs are the same
- 9. Let X be a random variable with expected value 2 and variance 3. Compute the following quantities: a) E(2X+3) b) Var (2X) c) Var (X+3)13) (3 points each) Suppose X and Y are independent random variables each having expected value 5 and standard deviation 3. Calculate d) If you were told that Cov(X,Y) = 4, and that X and Y are not independent, what would Var(X-Y) be?Suppose that X is a random variable * ? having the following m.g.f., Find Var(x) 7t_e4t M, (t) = eT 173 O 5 O 413 O 3148 O 55 O
