ant Puperinn atememt 5. Prove the weak law of large numbers: Given any & > 0,...
15.12 Suppose that R>0 is given. Prove that, if N is sufficiently large, Σχη/n!*0 for all z ED(0; R). n-0
15.12 Suppose that R>0 is given. Prove that, if N is sufficiently large, Σχη/n!*0 for all z ED(0; R). n-0
3.5 Which of the following is implied from the law of large numbers? (a) If we have a sufficiently large number of observations, its sample mean is close to the population mean. (b) If we have a sufficiently large number of observations, its sample variance is close to the population variance (c) If we have a sufficiently large number of observations, these observations follow a normal distribution. (d) None of the above. 3.6 In a simple regression model, y =...
(2) Define the set A by (a) Prove that for any N 20 the set is compact. (b) Prove that for any e>0 there exists some N 2 0 so that for any x A we have (c) Prove that A is totally bounded. d) Prove that A is compact.
Suppose that A is diagonalizable and all eigenvalues of A are
positive real numbers. Prove that det (A) > 0.
(1 point) Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that det(A) > 0. Proof: , where the diagonal entries of the diagonal matrix D are Because A is diagonalizable, there is an invertible matrix P such that eigenvalues 11, 12,...,n of A. Since = det(A), and 11 > 0,..., n > 0,...
(2) Define the set A C 2 by s) n-0 (a) Prove that for any N 2 0 the set is compact. (b) Prove that for any є > 0 there exists some N > 0 so that for any x E A we have (c) Prove that A is totally bounded. (d) Prove that A is compact
(2) Define the set A C 2 by s) n-0 (a) Prove that for any N 2 0 the set is compact....
We write R+ for the set of positive real numbers. For any positive real number e, we write (-6, 6) = {x a real number : -e < x <e}. Prove that the intersection of all such intervals is the set containing zero, n (-e, e) = {0} EER+
We write R+ for the set of positive real numbers. For any positive real number e, we write (-6, 6) = {x a real number : -e < x <e}. Prove that the intersection of all such intervals is the set containing zero, n (-e, e) = {0} EER+
Prove the ratio test . What does this tell you if
exists?
(Ratio test) If
for all sufficiently large n and some
r < 1, then
converges absolutely; while if
for
all sufficiently large n, then
diverges.
lim |.1n+1/01 700 In+1/xn < We were unable to transcribe this image2x+1/2 > 1 We were unable to transcribe this image
Suppose that > 0 and consider Pn=. Prove that if Xn ~ Bin(n.pn) with n large enough then for all k € {0, 1, ...} it gets lim P(Xn = k) = P(X = k) where X Poisson (2) n-
Law of Large Number↓
Led tin eperaje Theorem 9.11. (Central limit theorem) Suppose that we have i.i.d. random variables Xi,X2. X3,... with finite mean EX and finite variance Var(X) = σ2. Let Sn-Xi + . . . + Xn. Then for any fixed - oo<a<b<oo we have lim Pax (9.6) Theorem 4.8. (Law of large numbers for binomial random variables) For any fixed ε > 0 we have (4.7) n-00