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A random variable X has a distribution with probability function f(x) = K(nx)2x for x = 0,1,2,...,n where n is a positive integer. a. Find the constant k. b. Find the expected value M(S) = E(esX) as a function of the real numbers s. Compare the values of the derivative of this function M'(0) at 0 and the expected value of a random variable having the probability function above. c. What distribution has probability function f(x)? Let X1, X2 be independent random variables both...
11. Let n be a positive integer and [r], [y e Zn. Show that the following conditions are equivalent. (i) []= [v] (ii) - y nr for some integer r. (ii) n/(x-y)
* (9) Let n be a positive integer. Define : Z → Zn by (k) = [k]. (a) Show that is a homomorphism. (b) Find Ker(6) and Im(). yrcises (c) To what familiar group is the quotient group Z/nZ isomorphic? Explain.
Given X ~ N(1,22 ): a. P(X2− 2X < 3) =? b. E(X2 − 2X) =?
Let f(x)-12.2-2x-11 and a(z) = x2 + 12n, where Ir] is the largest integer less than or equal to r. (a) Evaluate the upper and lower sums U(f, P) and L(f, P) of f with respect to or if P is the partition {0、름, î,3.3.2) of [O, 2]. 4 42 (b) Explain why f є [0,2] and use results in part (a) to give a range of fda.
Let f(x)-12.2-2x-11 and a(z) = x2 + 12n, where Ir] is the...
QUESTION C.
(a) Let k be a field and let n be a positive integer. Define what is meant by a monomial ideal in k[x,...,zn]. 2. (b) State what it means for a ring R to be Noetherian. (c) State Hilbert's basis theorem. Give a proof of Hilbert's basis theorem using the fact if k is a field the polynomial ring kli,..., In] is Noetherian. 1S
(a) Let k be a field and let n be a positive integer. Define...
Let n be a positive integer and let F = {X 5 [n]: X|2|[n] \X]} Prove that F is a maximum intersecting family.
Use mathematical induction to prove that the statements are true for every positive integer n. 1 + [x. 2 - (x - 1)] + [ x3 - (1 - 1)] + ... + x n - (x - 1)] n[Xn - (x - 2)] 2 where x is any integer 2 1
a) Make a function which returns x/3 with a given integer x. double third(int x) { //Complete your code below. ______________________________________ return y; } b) Make a program which shows 2 ∗ n with a given user input n. #include<iostream> using namespace std; // Complete the blank. _______________________ int main(){ int n; cout<<"Enter:"; cin>>n; cout<<twice(n); return 0; } int twice(int x){ return 2*x; } c) Make a function which returns true if n is positive, otherwise returns false. ________ positive(int...
using the general power rule
Question 1 let y = (x2 +x)3 Find y' 2x+1 3(x2+x)2 3(x2+x)2 (2x+1) • (x2+x)2 (2x+1) recall general power rule formula has three parts: [u(x)" ]' = n u(x)" 1 u'(x) Question 2 let y = (x3 +x2) 1/3 Find y' (x3 +x2) 1/3 (1/3) (x3 +x2) 1/3 . (1/3)(x3 +x2)-2/3 (1/3)(x3 +x2-2/3 (3x2+2x) recall general power rule has three parts. [u(x)"l' = n u(x)n-1 u'(x) Question 5 let g(x) = 1/(x3+x2)3 find g'(x) (x²+x23...