LEt T be a non central t statistics with n df and noncentrality
parameter
. Find E(t) and Var(t)
LEt T be a non central t statistics with n df and noncentrality parameter . Find...
Let f(n) = 5 sin(31) (I). Find the one-sided df) Laplace transform off) and g(t) = initial conditions are zero.
The above result has some important consequences. Verify the following statement let X1, , Xn be iid r.v., X1 ~ N(μ, σ2), then the random variable follows Student tn-1 distribution with n- 1 degrees of freedom (df) "Studentisation : "let T be a statistics and V be an estimate of Var(T). The quantity T E(T) is referred to as "studentized Τ" (we have E(T) 0, Var(Т) 1)
Let X1, ..., X., be i.i.d random variables N(u, 02) where u is known parameter and o2 is the unknown parameter. Let y() = 02. (i) Find the CRLB for yo?). (ii) Recall that S2 is an unbiased estimator for o2. Compare the Var(S2) to that of the CRLB for
5. A light bulb has a lifetime that is exponentially distributed with rate parameter λ-5. Let L be a random variable denoting the sum of the lifetimes of 50 such bulbs. Assume that the bulbs are independent. (a) Compute E[L] and Var(L). b) Use the Central Limit Theorem to approximate P(8 < L < 12 ( ). (c) Use the Central Limit Theorem to find an interval (a,b), centered at ELLI, such that Pa KL b) 0.95. That is, your...
4. Let Xi,X2, , Xn be n i.id. exponential random variables with parameter λ > Let X(i) < X(2) < < X(n) be their order statistics. Define Yǐ = nX(1) and Ya = (n +1 - k)(Xh) Xk-n) for 1 < k Sn. Find the joint probability density function of y, . . . , h. Are they independent? 15In
. Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size n from an exponential distribution with parameter θ = 1. (a) Find the pdf of Yr. (b) Find the pdf of U = e −Yr .
Question 14 df Let f(x,y) = ln(+ + y). Given that a(t) = y(t) = #find as a function of t. Use "A" for exponents. dt df dt • Previous No new data to save. Last checked at 5:
(a) If var[X o2 for each Xi (i = 1,... ,n), find the variance of X = ( Xi)/n. (b) Let the continuous random variable Y have the moment generating function My (t) i. Show that the moment generating function of Z = aY b is e*My(at) for non-zero constants a and b ii. Use the result to write down the moment generating function of W 1- 2X if X Gamma(a, B)
(a) If var[X o2 for each Xi (i...
Let Y1,K,Y n denote a random sample from a Poisson distribution with parameter λ . a. Find a sufficient statistics for λ. b. Find the minimum variance unbiased estimator(MVUE) of λ2 .
3 lat (0.142.0 person eye sependent at the mom 8. Let {N(t),t > 0} be a Poisson process with intensity 1 that is independent of the non-negative random variable T with mean p and variance o2. Find that (a) COV(T, N(T)), 5 marks (b) var(N(T)). 5 marks Total: 10 marks