Show all steps 1. Bounds A random variable X, not necessarily non-negative, has E(X) = 20...
1. If the p.g.f. of a random variable X assuming non-negative values is Gx(s), then find the p.g.f.'s of the following [8 points]: (a) YX3 (b) Y2 (c) YX3/2 (d) Y43X 2. If X follows a binomial distribution with parameters n, and p, find the p.g.f. of X. From the p.g.f. derive the mean and variance of X. Show all the steps for receiving full credit. [6 points 3. Let Y Geometric(p), then show that [3 points] P P (Y...
5. Doing (much) Better by Taking the Min Let X be a random variable that takes on the values in the set {1,...,n} that satisfies the inequality Pr( x i) Sali for some value a>0 and all i € {1,...,n}. Recall that (or convince yourself that) E(X) = P(X= i) = Pr(x2i). 1. Given what little you know so far, give the best upper bound you can on E(X). 2. Let X1 and X2 be two independent copies of X...
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 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,...
Question 5 15 marks] Let X be a random variable with pdf -{ fx(z) = - 0<r<1 (1) 0 :otherwise, Xa, n>2, be iid. random variables with pdf where 0> 0. Let X. X2.... given by (1) (a) Let Ylog X, where X has pdf given by (1). Show that the pdf of Y is Be- otherwise, (b) Show that the log-likelihood given the X, is = n log0+ (0- 1)log X (0 X) Hence show that the maximum likelihood...
Suppose that the density function of a continuous random variable is given by f(x)=c(e-2x-e-3x) for non-negative x, and 0 elsewhere a) Determine c b) Compute P(X>1) c) Calculate PX<0.5 X<1.0)
Please show your work with a brief but logical explanation.
Suppose X is a random variable with p(X 0) 4/5, p(X-1) 1/10, p(X-9) 1/10. Then (a) Compute Var [X] and B [X] (b) What is the upper bound on the probability that X is at least 20 obained by applying Markov's inequality? c) What is the upper bound on the probability that X is at least 20 obained by applying Chebychev's inequality'?
Suppose X is a random variable with p(X...
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2.6.4. A discrete random variable X is such that 2n-1 P(X = n) =-, n=1.2, . . . , n, . . . . Il Show that EX- 3. 2.6.5. A discrete random variable X is such that P(X=2") = 1 2" n=1.2, . , show that EX = oo. That is, X has no mathematical expectation.
PLEASE SHOW DETAILED STEPS. THANK YOU. 1. A random variable X has a normal distribution N(5,3.5). Find P(X>0) 2. A random variable Xhas an exponential distribution Exponential (2.5). Find P(X < 0.75) Show the calculator input for your answer. 3. Mary is looking for someone with change of $1. She estimates that each person she asks has a 25% probability of having the right change. What is the probability that Mary will have to ask at least four people in...
. (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...