. (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,...
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...
Show all steps 1. Bounds A random variable X, not necessarily non-negative, has E(X) = 20...
Show all steps
1. Bounds A random variable X, not necessarily non-negative, has E(X) = 20 and SD(X) = 4. In each part below find the best bounds you can based on the information given. (a) Find upper and lower bounds for P(0 < X < 40). (b) Find upper and lower bounds for P(10 < X < 40). (c) Find an upper bound for P(X > 40). (d) Find an upper bound for P(X2 900).
3. Let X be a random variable with possible outcomes 11,..., In, and Y a random...
3. Let X be a random variable with possible outcomes 11,..., In, and Y a random variable with possible outcomes 1.... ym. Let Z be a random variable that corresponds to testing X followed by Y, so the possible outcomes are pairs (x,y) with P(x, y) = P(x) P(y) Use the definition of entropy to prove that H(Z) = H(X) + H(Y). This is a special case of property 3 from Shannon's theorem. Definition. The Key Equivocation of a cryptosystem...
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...
Show all steps
1. Bounds A random variable X, not necessarily non-negative, has E(X) = 20 and SD(X) = 4. In each part below find the best bounds you can based on the information given. (a) Find upper and lower bounds for P(0 < X < 40). (b) Find upper and lower bounds for P(10 < X < 40). (c) Find an upper bound for P(X > 40). (d) Find an upper bound for P(X2 900).
3. Let X be a random variable with possible outcomes 11,..., In, and Y a random variable with possible outcomes 1.... ym. Let Z be a random variable that corresponds to testing X followed by Y, so the possible outcomes are pairs (x,y) with P(x, y) = P(x) P(y) Use the definition of entropy to prove that H(Z) = H(X) + H(Y). This is a special case of property 3 from Shannon's theorem. Definition. The Key Equivocation of a cryptosystem...
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