10. Problem 10 (extra 10 pts.) Prove that functions/(x) and g(x) are positive and monotone increasing,...
8. Suppose f : la,b] → R is monotone increasing. Prove that f is integrable. Of course you may not use the theorem that monotone functions are integrable!
8. Suppose f : la,b] → R is monotone increasing. Prove that f is integrable. Of course you may not use the theorem that monotone functions are integrable!
6-4 (a) The function g(x) is monotone increasing and y = g(x). Show that F(x) if xy(x, y) =İF,(y) if y>g(x) y<g(x) xytty (b) Find Fxy(x, y) if g(x) is monotone decreasing.
Part I. (30 pts) (10 pts) Let fin) and g(n) be asymptotically positive functions. Prove or disprove each of the following statements T a、 f(n) + g(n)=0(max(f(n), g(n))) 1. b. f(n) = 0(g(n)) implies g(n) = Ω(f(n)) T rc. f(n)- o F d. f(n) o(f(n)) 0(f (n)) f(n)=6((f(n))2)
Problem 10. Let f,g: [a,b] -R be Riemann integrable functions such that f(x) < g(x) for all x E [a,b]. Prove that g(x)
(e) For subsets {A,Jael, prove that2 I) Evaluate (g) Prove that XAAB (XA-X) (h) Use characteristic functions to prove the distributive law: AU(BnC) (AUB)n (AUC) Hint: start with the right-hand side. 1In this problem, the product of two functions and g is defined by (Jg)(x)-f() and the sum is defined by (f +g)(x) :-f(x) + g(x), as usua 2Here, Π denotes the product of an indexed set of numbers. For example: rL TL TL i n! i-1 -1
(e) For...
4. Let fín) and g(n) be asymptotically positive functions. Prove each of the following statements A. fin)-O(g(n)) if and only if fin) *gn)g(n)) B. fn) - Og(n if and only if fin)2- O(g(n)?)
4. Let fín) and g(n) be asymptotically positive functions. Prove each of the following statements A. fin)-O(g(n)) if and only if fin) *gn)g(n)) B. fn) - Og(n if and only if fin)2- O(g(n)?)
1. (10 pts) For each of the following pairs of functions, indicate whether f = 0(g), f = Ω(g), or both (in which case f-6(1). You do not need to explain your answer. f(n) (n) a) n (b) n-1n+1 (c) 1000n 0.01n2 (d) 10n2 n (lg n)2 21 е) n (f) 3" (g) 4" rl. 72 i-0 2. (12 pts) Sort the following functions by increasing order of growth. For every pair of consecutive functions f(n) and g(n) in the...
9. (9 pts) The random variable r-Gamma(x-2, β-4). functions to prove that the moment generating function for the random variable W mw(t) (1-12t)2. Use the method of moment-generating 3Y +5is eSt 10, (9 pts) Suppose that Y has a gamma distribution with α-n/2 for some positive integer n and β equal to some specified value. Use the method of moment-generating functions to prove that W- 2Y /g has a Chi-squared distribution with n degrees of freedom. Make sure you show...
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. Problem 8, (10 pts.) Prove that on the interval [0,0.8) -2n lim dx Problem 9, (10 pts.) na(1-z)". Let fn (z) Prove that . Problem 10, (10 pts.) Using Method of mathematical induction prove that: If function u(x) is such that a,--u then a ,u u, 2n1
. Problem 8, (10 pts.) Prove that on the interval [0,0.8) -2n lim dx Problem 9, (10 pts.) na(1-z)". Let fn (z) Prove that . Problem 10, (10 pts.)...
Let g, h be two real-valued convex functions on R. Let m(x) = max{h(x), g(x)). Prove that m(x) is also convex 3.