. (Markov’s Inequality) Let X be a non-negative random variable defined on the sample Ω 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 quantity. In particular, comment (and justify) on whether it is possible to have : P(X ≥ a) ≥ µ/a
(c) Conclude that if X is a non-negative random variable, then P(X ≥ a) ≤ µ/a
Homework Answers
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. (Markov’s Inequality) Let X be a non-negative random variable
defined on the sample Ω i.e....
5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that (...
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...
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 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,...
Let X be a positive random variable with E(X) = 2 and VarX= 20: (a) Use...
Let X be a positive random variable with E(X) = 2 and VarX= 20: (a) Use Markov’s inequality to obtain an upper bound onP(X≥25). (b) Use Chebyshev’s inequality to obtain an upper bound onP(X≥25).
Which of the following are correct versions of Markov’s Inequality for a nonnegative random variable X:...
Which of the following are correct versions of Markov’s Inequality for a nonnegative random variable X: A. P(X≥αμ)≤1/α B. P(X≥αμ)≤μα C. P(X≥μ)≤1/α D. P(X≥α)≤μ/α One or more answers are correct.
1. If the p.g.f. of a random variable X assuming non-negative values is Gx(s), then find...
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...
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...
Problem 4 (20p). Let α > 0, and for each n E N let Xn : Ω → R be a random variable on a probability space (Q,F,P) with the gamma distribution「an. Does there exist a random variable X:82 → R such...
Problem 4 (20p). Let α > 0, and for each n E N let Xn : Ω → R be a random variable on a probability space (Q,F,P) with the gamma distribution「an. Does there exist a random variable X:82 → R such that Xn-,X as n →oo?
Problem 4 (20p). Let α > 0, and for each n E N let Xn : Ω → R be a random variable on a probability space (Q,F,P) with the gamma distribution「an. Does...
5. Let X > 0 be a random variable with EX = 10 and EX2 =...
5. Let X > 0 be a random variable with EX = 10 and EX2 = 140. a. Find an upper bound on P(X > 14) involving EX using Markov's inequality. b. Modify the proof of Markov's inequality to find an upper bound on P(X > 14) in- volving EX? c. Compare the results in (a) and (b) above to what you find from Chebyshev's inequality.
3. Let X be the random variable characterized by a probability distribu tion p (P, ...p)...
3. Let X be the random variable characterized by a probability distribu tion p (P, ...p) and it can assume one of the values r1,....In with probabilities pi, , pni 0 < pi 1, Σǐpí = 1 The Shannon entropy of the random variable X is defined as Show that 0 s H(X) S log2 so that the maximal value in this inequality is saturated by unifornm probability distribution. What is the level of information for this state of the...
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...
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,...
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...
Problem 4 (20p). Let α > 0, and for each n E N let Xn : Ω → R be a random variable on a probability space (Q,F,P) with the gamma distribution「an. Does there exist a random variable X:82 → R such that Xn-,X as n →oo?
Problem 4 (20p). Let α > 0, and for each n E N let Xn : Ω → R be a random variable on a probability space (Q,F,P) with the gamma distribution「an. Does...
5. Let X > 0 be a random variable with EX = 10 and EX2 = 140. a. Find an upper bound on P(X > 14) involving EX using Markov's inequality. b. Modify the proof of Markov's inequality to find an upper bound on P(X > 14) in- volving EX? c. Compare the results in (a) and (b) above to what you find from Chebyshev's inequality.
3. Let X be the random variable characterized by a probability distribu tion p (P, ...p) and it can assume one of the values r1,....In with probabilities pi, , pni 0 < pi 1, Σǐpí = 1 The Shannon entropy of the random variable X is defined as Show that 0 s H(X) S log2 so that the maximal value in this inequality is saturated by unifornm probability distribution. What is the level of information for this state of the...
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