5. Suppose X ~Exp(A). (a) [5 pts] Show that E(X) 1/A. [Hint: You can directly use...
5. Suppose XExp(A). (a) [5 pts) Show that E(X) 1/A. Hint: You can directly use the definition and properties of a gamma function.] (b) [5 pts] Prove that P(X > t +s | X > s) = P(X > t) for s, t > 0, [Hint: You can directly use the tail probablity P(X >) e for 0.]
5. Suppose XExp(A). (a) [5 pts] Show that E(X) 1/A. [Hint: You can directly use the definition and properties of a gamma function.]
Extra Credit Question:[4+4=8 pts) If E [exp(aX)] exists for a given constant a, then show that for t > 0 (a) exp(-at)P(X >t) <E [exp(aX)], if a > 0. (b) exp(-at)P(X<t) <E [exp(aX)], if a <0.
Extra Credit Question:[4+4=8 pts) If E [exp(aX)] exists for a given constant a, then show that for to (a) exp(-at)P(x >t) <E (exp(aX)], if a > 0. (b) exp(-at)P(X <t) <E (exp(aX)], if a < 0.
hint
This exercise 5 to use the definition of Riemann integral
F. Let f : [a, b] → R be a bounded function. Suppose there exist a sequence of partitions {Pk} of [a, b] such that lim (U(Pk, f) – L (Pk,f)) = 0. k20 Show that f is Riemann integrable and that Så f = lim (U(P«, f)) = lim (L (Pk,f)). k- k0 1,0 < x <1 - Suppose f : [-1, 1] → R is defined as...
1. [15 pts] Use Definition 1.5 (definition of probability function) to prove Propo- sition 1.3 () 15 pts) & (iv) [10 pts). You do not need to prove (i) and (ii). [Definition 1.5/ Let Ω be a set of all possible events. A probability function P : Ω → 0,11 satisfies the follouing three conditions (i) 0s P(A) S 1 for any event A; (iii) For any sequence of mutually exclusive events A1, A2 ,A", i.e. A, n Aj =...
real analysis
hint
13 Suppose fis a continuous function on R', with period 1. Prove that lim Σ f(a)-| f(t) dt 0 for every irrational real number α. Hint: Do it first for f(t)= exp (2nikt), k = 0,±1, ±2, 4.13 Let 2 be the set of functions of form P(t)-Σ_NQC2nikt. The equality holds for functions in . For given ε > 0, there is a P E 2 such that llf-Plloo < ε. Then
6. (a) [5 pts] Show that the moment generate function of a Poisson distribution with parameter λ > 0 is M (t) eA(et-1) [Hint: You might need Σ¡ o a- ea, where k takes non-negative integer values.] (b) [5 pts) Use moment generating functions to prove that if Xi ~ Poisson(A1), X2 Poisson(A2), and Xi and X2 are independent, Xi+X2 Poisson(Ai+ A2).
we use this definition
5. [3 points Prove that the function f(x) = - , is continuous at := -1. You should give a proof that is directly based on the definition of continuity. Solution: You can type your solutions here. teso Isso sit & lx-xokę => 1 F(X) - F(Xoll LE
Advanced Calculus
(3) Let the function f(x) 0 if x Z, but for n e z we have f(n) . Prove that for any interval [a3] the function f is integrable and Ja far-б. Hint: let k be the number of integers in the interval. You can either induct on k or prove integrability directly from the definition or the box-sum criterion.
(3) Let the function f(x) 0 if x Z, but for n e z we have f(n) ....