5. Suppose XExp(A). (a) [5 pts) Show that E(X) 1/A. Hint: You can directly use the...
5. Suppose X ~Exp(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+ |xs) P(x > t) for s,t>0. [Hint: You can directly use the tail probability P(X > x) = e-k for x > 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.]
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...
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.)
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
·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...
6, (6 pts.) Let > 0 be a constant. Show that the random variable X with probability density function f(x) = 0 If x < 0. "has no memory." More precisely, show that P(X > t|X > s} = P(X > t-s) for any 0 s t< oo.
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 =...
5. The Exponential(A) distribution has density f(x) = for x<0' where λ > 0 (a) Show/of(x) dr-1. (b) Find F(x). Of course there is a separate answer for x 2 0 and x <0 (c Let X have an exponential density with parameter λ > 0 Prove the 'Inemoryless" property: P(X > t + s|X > s) = P(X > t) for t > 0 and s > 0. For example, the probability that the conversation lasts at least t...