5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that E...
5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that E X2] (Hint: Use the following special case of the Cauchy-Schwarz Inequality: First, make sure you see why this is a special case of the Cauchy-Schwarz Inequality; then apply it to get one of the inequalities of this problem.) 5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that E X2] (Hint: Use the following special case of the Cauchy-Schwarz Inequality: First,...
5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that (Hint: Use the following special case of the Cauchy-Schwarz Inequality: First, make sure you see why this is a special case of the Cauchy-Schwarz Inequality; then apply it to get one of the inequalities of this problem.) 5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that (Hint: Use the following special case of the Cauchy-Schwarz Inequality: First, make sure you see...
The geometric distribution is memoryless. A random variable that is positive and integer valued satisfies the memoryless property. Prove that this random variable must have a geometric distribution.
Problem 2 Prove the following bound known as the Chemoff bound: Let X be a random variable with moment generating function X (s) defined for s > 0, Then for any a and any s > 0, Hint: To prove the bound apply Markov's inequality with X replaced by e) Apply the се Chemoff bound in case X is a standard normal random variable and a > 0. Find the value of s >0 that gives the sharpest bound, i.e,...
. (Markov’s Inequality) Let X be a non-negative random variable defined on the sample Ω i.e. X(s) ≥ 0 for all s ∈ Ω. Let a be some fixed positive number. (a) If you know nothing about the probability distribution of X, what can you say about P(X ≥ a)? (b) Now, if you know what the value of (E(X) = µ) is, can you say anything about P(X ≥ a)? Turns out something non-trivial can be said about this...
3. Suppose that X is a nonegative integer valued random variable. Show that E[X] = P(X ). Hint: Start with the formula EX= k= k, Now try to rearrange the terms. P(X = k) and for all positive integers k write
Problem 3. Let X be a discrete random variable, gx) - a+ bX+ cX, and let a. b, c be constants. Prove, using the definition of expectation of a function of a random variable, namely , that E(a + bX + cx?) = a + bE(X) + cE(X2)
Let ? be a positive integer random variable with PMF of the form ??(?)=12⋅?⋅2−?,?=1,2,…. Once we see the numerical value of ?, we then draw a random variable ? whose (conditional) PMF is uniform on the set {1,2,…,2?}. 1.1 Write down an expression for the joint PMF ??,?(?,?). For ?=1,2,… and ?=1,2,…,2?: ??,?(?,?)=? 1.2 Find the marginal PMF ??(?) as a function of ?. For simplicity, provide the answer only for the case when ? is an even number. (The...
5. A non-negative valued continuous random variable X satisfies P(X > x +y|X > x) = P(X > y) > 0 for any x,y > 0. (a) Show that P(X > nx) = [P(X > x)]" and P(X > x/m) = [P(X > x)]1/m for positive integers n, m. (b) Show that X~ exponential() for some A > 0.
prove that A is non singular 5.(25 pts) For each positive integer n, let f()(+2)(1)(0,1. Let f()-0, (1) Prove that (fn) converges to fpointwisely on (0, 1) (2) Does (n) converges to f uniformly on (0, 1]?