8.33 Let X1, X2, L(X — м), for n 3D 0, 1, 2, .. (a) Show that Zo, Z\, ... is a martingale with respect to be i.d random variables with mean u < o0. Let Z, = ... Хо. X. 8.33 Let X1, X2, L(X — м), for n 3D 0, 1, 2, .. (a) Show that Zo, Z\, ... is a martingale with respect to be i.d random variables with mean u
6.9. A process (Z,, n z is said to be a reverse, or backwards, martingale if EZfor al n and EIZ.IZ.,. Z.., ]- Show that if X,,i 21, are independent and identically distributed random vanables with finite expectation, then Z" = (x, + . . + X,)/n, n 1, is a reverse martingale 6.9. A process (Z,, n z is said to be a reverse, or backwards, martingale if EZfor al n and EIZ.IZ.,. Z.., ]- Show that if X,,i...
Question 2. Suppose (X.,X) . FXY, for i = 1, , n. We collect sample data for n-100, obtain sz-2 and Sy-1, and would like to test H0 : Var(x)-Var(y) versus HA : Var(z) Var(y). (a) Using the F test, what is the observed statistic? (b) Derive the null distribution and write out the p-value. Question 2. Suppose (X.,X) . FXY, for i = 1, , n. We collect sample data for n-100, obtain sz-2 and Sy-1, and would like...
6.7. Let X,, be a sequence of independent and identically distributed X, and show Pl random variables with mean 0 and variance σ. Let 1-1 that {Z., n 2 1j is a martingale when 6.7. Let X,, be a sequence of independent and identically distributed X, and show Pl random variables with mean 0 and variance σ. Let 1-1 that {Z., n 2 1j is a martingale when
N(0, 1) and let S be a 4. Let Z random sign independent of Z, i.e., S is 1 with probability 1/2 and -1 with probability 1/2. Show that SZ N(0,1) 5. Let Z N(0, 1) and X = Z2. This distribution is called chi-square with degree of freedom. Calculate P(1 < X < 4) one N(0, 1) and let S be a 4. Let Z random sign independent of Z, i.e., S is 1 with probability 1/2 and -1...
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale. 2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
2. Let h(z, n) = 1 + x + x2 + +z" = Σ'=0a". Write an R program to calculate h(z, n) using a for loop
Please show every step, thank you. Let Xi ~ N(μ, σ?), where ơỈ are known and positive for i-1, are independent. Let /- (a) Find the mean and variance of μ. (b) Compare μ to X,-n-Σί.i Xi as an estimator of μ. , n, and Xi, X, , E-1(1/o .m be the MLE of μ. Let Xi ~ N(μ, σ?), where ơỈ are known and positive for i-1, are independent. Let /- (a) Find the mean and variance of μ....
2. Let 'n ,n > l be a sequence of r.v.s such that E[Xi] μί and Var(X) σ for i-: 1, 2, , and Cov(Xi, Χ.j) Ơij for i J. Let {an ,n 1) and (bn, n 1) be the sequences of real numbers. Write down the expressions for i-l (i,Xi, Xi), Cov every i and Ơij 0 for every i j, state Var(Σί ! així), Coy(Σ, aixi, xi),
In the simple linear regression with zero-constant item for (xi , yi) where i = 1, 2, · · · , n, Yi = βxi + i where {i} n i=1 are i.i.d. N(0, σ2 ). (a) Derive the normal equation that the LS estimator, βˆ, satisfies. (b) Show that the LS estimator of β is given by βˆ = Pn i=1 P xiYi n i=1 x 2 i . (c) Show that E(βˆ) = β, V ar(βˆ) = σ...