
Just Question 2 1. Let xn 2 п a) To what value x does xn Converge?...
1. Let Xn = 2 - 3 a) To what value x does xn converge? b) Find the smallest n, such that n > n. = |xn – x] < 0.1. c) Find the smallest no such that n > no [xn – x] < 0.005. d) Find the smallest no such that n > no = |xn – x] < 10-6. e) Find the smallest no such that n >no = |xn – x] < E.
Let X1, X2, · · · be independent random variables, Xn ∼ U(−1/n, 1/n). Let X be a random variable with P(X = 0) = 1. (a) what is the CDF of Xn? (b) Does Xn converge to X in distribution? in probability?
Let X1, X2, .., Xn be a random sample from Binomial(1,p) (i.e. n Bernoulli trials). Thus, п Y- ΣΧ i=1 is Binomial (n,p). a. Show that X = ± i is an unbiased estimator of p. Р(1-р) b. Show that Var(X) X(1-X (п —. c. Show that E P(1-р) d. Find the value of c so that cX(1-X) is an unbiased estimator of Var(X): п
5. Let {xn} and {yn} be sequences of real numbers such that x1 =
2 and y1 = 8 and for n = 1,2,3,···
x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y .
nn nn
(a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all
positive integers n.
(xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive
integers n.
Hence, prove...
9 Let Xi, X2, ..., Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X (b) S2 =X1 + X2 . (c) Sn=X1+X2 + . . . + Xn. (d) An -Sn/n
9 Let Xi, X2, ..., Xn be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for (a) X (b) S2 =X1 + X2 . (c)...
QUESTION 16 For what values of x does the series converge conditionally? (-1)"(x + 5)" w n n=1 OA. x= -6 B. x = – 4 x= -6, x= -5 D. x=-6, x = -4 O E. x= -5,x= -4
Let X1, X2,..., Xn be a r.s. from f(x) = 0x0-1, for 0 < x <1,0 < a < 0o. (a) Find the MLE of 0. (b) Let T = -log X. Find the pdf of T. (c) Find the pdf of Y = DIT: (i.e., distribution of Y = - , log Xi). (d) Find E(). (e) Find E( ). (f) Show that the variance of 0 MLE → as n → 00. (g) Find the MME of 0.
just answer e through h
8. (11 pts) Let (Xn) be a sequence in Rº such that VnEN, Xn+1 = A· Xn+ where A = (5/8 5/3) and Xo = (-1) (a) (1 pt) Find X1. (b) (2 pts) Find the corresponding equilibrium point. (c) (1 pt) Determine the two eigenvalues 11 and 12 of A. (d) (1 pt) For each eigenvalue, find an eigenvector. (e) (1 pt) Is the equilibrium point a sink? Justify. (f) (1 pt) Deduce the...
Let X1, X2,...,Xn be a random sample from the exponential distribution with rate A Let c > 0 be a fixed and known number. For i 1,2 п, let ..1 -{: : if Xic 1 Y otherwise Suppose that you get to observe Yı, Y2,... , Y,n but you do not get to observe X1, X2,... , X,n п. Find the MLE for X based on this information
For n ≥ 1, let Xn be a continuous random variable with pdf fn(x) = {cn/xn+1, x ≥ cn, 0, otherwise.} Xn’s are called Pareto random variables and are used to study income distributions. (a) Calculate cn, n ≥ 1. (b) Find E(Xn), n ≥ 1. (c) For what values of m does E(X^m+1 n) exist?