9. Let x,y > 0 be real numbers and q, r E Q. Prove the following: (а) 29 > 0. 2"а" and (29)" (b) x7+r (с) г а — 1/29. 0, then x> y if and only if r4 > y (d) If q (e) For 1, r4 > x" if and only if q > r. For x < 1, x4 > x* if and only if q < r.
For all n E N prove that 0 <e- > < 2 k!“ (n + 1)! k=0 Hint: Think about Taylor approximations of the function e".
| Prove that for n e N, n > 0, we have 1 x 1!+ 2 x 2!+... tnx n! = (n + 1)! - 1.
Let Xi X, lid f(r 0) with f(r:0)-e ( e) for r > ? and ? e R. (a) Find the MLE of ? (c) Using the prior density ?(0)-e-91(0,0)( ?), find the Bayesian estimator of ?
IF Let x(t) Show that e 20" σ>0, and let (o) be the Fourier transform of x(t) .
·J (I) < 0 for all such y. (Hint: let g(x)--f(x) and use part (a)) 3. In this problem, we prove the Intermedinte Value Theorem. Let Intermediate Value Theorem. Let f : [a → R be continuous, and suppose f(a) < 0 and f(b) >0. Define S = {t E [a, b] : f(z) < 0 for allェE [a,t)) (a) Prove that s is nonempty and bounded above. Deduce that c= sup S exists, and that astst (b) Use Problem...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
4. The solution of the inequality x2 – 4 < 0 is (a) –2 < x or x > 2 (b) –2 < x < 2 (C) x>-2 (d) x < 2 (e) None of the above 5. The domain of the function f(x) = V2is (a) (-2,2) (b) (-0, -2) U (2,00) (c) (-0, -2] U (2,0) (d) (-20, -2] U (2,00) (e) None of the above 6. The range of the function f(x) = 2 sin(x) is (a)...
I. (5 points) Let X be a random variable with moment generating function M(t) = E [etx]. For t > 0 and a 〉 0, prove that and consequently, P(X > a inf etaM(t). t>0 These bounds are known as Chernoff's bounds. (Hint: Define Z etX and use Markov inequality.)
Prove that if X 20, Y 2 0 and 0 p1, then E(X +Y)] Show that for any real numbers x > 0 and y > 0, E(X)E(YP). HINT: Here is how you can show the above formula holds. Start off by letting 0y. Use the fact that the function g(z) - z is concave-down (i.e., "spills water") on (0, oo) and is thus bounded above by its tangent line at any particular point. Find the tanget line at the...