-. Show that if x ~ Unif(a, β) and μ-E(X) (so μ-(a + β)/2), then for...
7. Show all work to answer the following question. If the area enclosed by x = y2 – 4 and x = k where k > 0 is equal to 12, find the value of k. To earn any credit for this question you must use strategies
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3. Show that if n e Z so that n is odd then 8|n2 + (n + 6)2 +6. 4. (a) Let a, b, and n be integers so that n > 2. Define: а is congruent to b mod n. The notation here is a = b (modn). (b) Is 12 = 4 (mod 2)? Explain. (c) Is 25 = 3 (mod 2)? Explain. (d) Is 27 = 13 (mod8)? Explain. (e) Find 6 integers x...
8) Assume that X ~ N(μ = 4,02-1). Find c >0 such that P(-c 〈 X 〈 c) Find P(2 〈 X 〈 6) a. 0.95 b.
Suppose that X ~ unif(0,1). Find the distribution of Y = (1 – X)-B – 1 for some fixed B> 0. (Name it!)
3. Let f : [0, 1] → R be uniformly continuous, so that for every e > 0, there exists >0 such that |x – y < =\f(x) – f(y)] < e for every x, y € [0, 1]. The graph of f is the set Gf = {(x, f(x)) : € [0, 1]}. Show that Gf has measure zero (9 points).
Given the logistic map Xn+1 = run(1 – Xn) with r > 0. Show the 2-cycle is stable for 3 <r <1+V6.
Exercise 6. Show that if f(x) > 0 for all x e [a, b] and f is integrable, then Sfdx > 0.
(2) Prove that the following are equivalent for x ER and A CR. (a) X E A. Here A denotes the closure of A. (b) For every e > 0, N(x; e) n A +0. (c) For every open set U, if r EU then UNA+.
Compute E[X] if X has a density function given by 0 Otherwise If X is a normal random variable with parameters μ = 10 and σ = 6, compute 1. P(X > 5) 2. P (4 < X < 16) 3. P (X < 8) 4. P (IXI> 16)
A quantum particle of mass m is in the 1D potential: V(2) = <0, mw?z?, < > (1) Find the energy eigenvalues for the lowest three eigenstates.