Give an example of a sequence x1, x2, ...... of real numbers such that Iim Σk=1n Xk exists in R, but ∫ x dm is not defined, where u is counting measure on Z+ and x is the function from Z+ to R defined by x(k)=xk.
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1) Show that if U is a non-empty open subset of the real numbers then m(U) > O. 2) Give an example of an unbounded open set with finite measure. Justify your answer, 3) If a is a single point on the number line show that m ( a ) = O. 4) Prove that if K is compact and U is open with K U then m(K) m(U). 5) show that the Cantor set C is compact and m(C)...
Real Analysis II
(Please do this only if you are sure)
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I am also providing the convex set definition
And key details from my book which surely helps
11. Show that K is a convex set by directly applying the definition. Sketch K in the cases n= 1, 2, 3. is a basis for E. This is the n-parallelepiped spanned by vı, vertex 1% with 0 as a Definition. Let K E". Then K is a convex set...
This is a Fourier Analysis Question
This is a Fourier Analysis Question
Exercise 21.1 Assume that f is in Li (R) and g(z) = e2i". Compute f*g. 20.1.1 Definition The convolution of two functions f and g from R to C is the function f g, if it exists, defined by f * g(x) = | f(x-t)g(t) dt f(u)g(z-u) du. If no assumptions are made about f and g, the convolution is clearly not defined. Take, for example, f =...
Suppose X = Exp(1) and Y= -ln(x)
(a)Find the cumulative distribution function of Y .
(b) Find the probability density function of Y .
(c) Let X1, X2, ... , Xk be i.i.d. Exp(1), and let Mk =
max{X1,..... , Xk)(Maximum of X1, ..., Xk). Find the probability
density function of Mk.(Hint: P(min(X1, X2, X3) > k) = P(X1
>= k, X2 >= k, X3 >= kq, how about max ?)
(d) Show that as k → 00, the CDF...
8. An important distribution in the multivariate setting is the multivariate normal distribution. Let X be a random vector in Rk. That is Xk with X1, X2, ..., xk random variables. If X has a multivariate normal distribution, then its joint pdf is given by f(x) = {27}</2(det 2)1/2 exp {=} (x – u)?g="(x-1)} is the covariant matrix. Note with parameters u, a vector in R", and , a matrix in Rkxk that det is the determinant of matrix ....
Q3
Preliminary material The homework assignment is found on the next page. Our goal in this homework is to develop an algorithm for solving equations of the form f (x) (1) = X where f is a function S S, for some S C R". This kind of problem is sometimes called fixed point problem, and a solution x of problem (1) is called a fixed point of f. The algorithm we will consider is the following: a Step 0....
6. (10 points) Suppose X – Exp(1) and Y = -In(X) (a) Find the cumulative distribution function of Y. (b) Find the probability density function of Y. (c) Let X1, X2,...,be i.i.d. Exp(1), and let Mk = max(X1,..., Xk) (Maximum of X1, ..., Xk). Find the probability density function of Mk (Hint: P(min(X1, X2, X3) > k) = P(X1 > k, X2 > k, X3 > k), how about max ?) (d) Show that as k- , the CDF of...
Question 1 [10 points] Suppose that the sequence xo, X1, X2... is defined by xo = 2, x1 = 1, and Xk+2 = Xk+1+2xk for k>0. Find a general formula for xk. Be sure to include parentheses where necessary, e.g. to distinguish 1/(2k) from 1/2k. . xk = 0 Official Time: 22:32:44 SUBMIT AND MARK SAVE AND CLOSE
real analysis
1,3,8,11,12 please
4.4.3
4.4.11a
Limits and Continuity 4 Chapter Remark: In the statement of Theorem 4.4.12 we assumed that f was tone and continuous on the interval I. The fact that f is either stric tric. strictly decreasing on / implies that f is one-to-one on t one-to-one and continuous on an interval 1, then as a consequence of the value theorem the function f is strictly monotone on I (Exercise 15). This false if either f is...
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x; 1) = 212x3 e-tz, x > 0. a. Find E(XK), where k > -4. Enter a formula below. Use * for multiplication, / for divison, ^ for power, lam for 1, Gamma for the function, and pi for the mathematical constant i. For example, lam^k*Gamma(k/2)/pi means ik r(k/2)/n. Hint 1: Consider u = 1x2 or u = x2....