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
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 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.
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
8. Let X, X2, , xn all be be distributed Normal(μ, σ2). Let X1, X2, , xn be mu- tually independent. a) Find the distribution of U-Σǐ! Xi for positive integer m < n b) Find the distribution of Z2 where Z = M Hint: Can the solution from problem #2 be applied here for specific values of a and b?