Real analysis. Please solve all questions thank you 1. Let h be a positive real number, a <c< d < b and let Sh c< x <d, J() = 1 0 r < c, x > d (a) Using the definition only, find ſº f(x)dx. In fact, given e > 0, you should find an explicit d > 0) which works in the definition. (b) For a given partition P of [a, b], find a good upper bound on S(P)...
2. Let a be a positive real number, let r be a real number satisfying r >1, let N be an integer greater than one, and let tR -R be the integrable simple function defined such that tr,N(r) = 0 whenver x < a or z > ar*, tr,N(a) = a-2 and tr,N(z) = (ar)-2 whenever arj-ıく < ar] for some integer j satisfying 1 < j < N. Determine the value of JR trN(x) dz.
1. Let p be a positive real number with 0 < p < 1, and let ni, N2, N3 be positive integers. Sup- pose that Xı has a Bin(ni,p) distribution, X2 has a Bin(n2, p) distribution, and X3 has a Bin(n3, p) distribution. Show that the random variable Z = X1 + X2 + X3 also has a binomial distribution, and find the parameters of that distribution.
Exercise 6 requires using Exercises 4 and 5. Exercise 4. Let a be any real number. Prove that the Euclidean translation Ta given by Ta(x, y)(a, y) is a hyperbolic rigid motion. *Exercise 5. Let a be a positive real number. Prove that the transformation fa: HH given by fa(x, y) (ax, ay) is a hyperbolic rigid motion Exercise 6. Prove that given any two points P and Q in H, there exists a hyperbolic rigid motion f with f(P)...
Translate to an equation and solve. Let x be the unknown number. 78 is 5% of what number?
5. For any real number L > 0, consider the set of functions fx(x) = cos ("I") and In(x) = sin (^) se hos e mais a positive in where n is a positive integer. Show that these functions are orthonormal in the sense that (a) 1 L È Lsu(w) m(e)dx = {if m=n. fn (2) fm(x) dx = {. if m En if m =n -L 1 L il fn(x)9m(x)dx = 0 (c) il 9.(X)gm()dx = {{ if m=n...
Complex variables Stoo proo Coslax) dx x(x++b%) Assuming that a and b are positive real number, solve the integral using "indented Contor"
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we would like to show that for all 0 SjSk-, the function satisfies the advancement operator equation (A -r)f0 (a) Show that this is true whenever J-0. You can use the fact that f(n) = crn satisfies (A-r)f = 0. (b) Suppose fm n) satisfies the equation when m s k-2 for every choice of c. Show that )...
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+