Let X and Y be a first countable spaces. Prove that f:XY
is continuous if whenever xnx
in X then f(xn
)f(x)
in Y
Let X and Y be a first countable spaces. Prove that f:XY is continuous if whenever...
3. (8 marks) Let be the set of integers that are not divisible by 3. Prove that is a countable set by finding a bijection between the set and the set of integers , which we know is countable from class. (You need to prove that your function is a bijection.) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous, Let X and Y be topological spaces, and let X × y be equipped with the product topology. Let yo E Y be fixed. Define the map f XXx Y by f(x) (x, yo) Prove that f is continuous,
Please show all work: Let If x is odd then If x is even then Prove that is true and then solve it. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
(b) 4 Let F: X Y be a linear map between two normed spaces. Prove that F is continuous at Ojf and only if F is uniformly continuous on X.
Let Y = Xβ + ε be the linear model where X be an n × p matrix with orthonormal columns (columns of X are orthogonal to each other and each column has length 1) Let be the least-squares estimate of β, and let be the ridge regression estimate with tuning parameter λ. Prove that for each j, . Note: The ridge regression estimate is given by: The least squares estimate is given by: We were unable to transcribe this...
Let X1, X2,.......Xn be a random sample of size n from a continuous distribution symmetric about . For testing H0: = 10 vs H1: < 10, consider the statistic T- = Ri+ (1-i), where i =1 if Xi>10 , 0 otherwise; and Ri+ is the rank of (Xi - 10) among |X1 -10|, |X2-10|......|Xn -10|. 1. Find the null mean and variance of T- . 2. Find the exact null distribution of T- for n=5. We were unable to transcribe this imageWe were...
Let ⊂ be a rectangle and let f be a function which is integrable on R. Prove that the graph of f, G(f) := {(x, f(x)) ∈ : x ∈ }, is a Jordan region and that it has volume 0 (as a subset of ). We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let be a sequence of random variables, and let Y be a random variable on the same sample space. Let An(ϵ) be the event that |Yn − Y | > ϵ. It can be shown that a sufficient condition for Yn to converge to Y w.p.1 as n → ∞ is that for every ϵ > 0, (a) Let be independent uniformly distributed random variables on [0, 1], and let Yn = min(X1, . . . , Xn). In class,...
Using FTLM. a) Let . Use linear algebra to prove that there is a polynomial such that p + p' - 3p'' = q. Hint: consider the map defined by Tp: p + p' - 3p'', and use FTLM. b) Let be distinct elements of . Let be any elements of . Use linear algebra to prove that there is a such that Hint: consider the map defined by . You can use any facts from algebra about the solution...
Let n, and let n be a reduced residue. Let r = odd(). Prove that if r = st for positive integers s and t, then old(t) = s. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image