(book # 4.4) Let X denote the number of heads when ipping a fair coin n times, i.e., X Bin(n; p) with p = 1=2. Find a Cherno bound for Pr(X a). Find the sharpest (i.e., smallest) Cherno bound. Evaluate your answer for n = 100 and a = 68.
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Let X denote the number of heads when flipping a fair coin n times, i.e., X ~ Bin(n, p) with p = 1/2. Find a Chernoff bound for Pr(X ≥ a). Find the sharpest (i.e., smallest) Chernoff bound. Evaluate your answer for n = 100 and a = 68.
Ahmet Y.
Let X denote the number of heads when flipping a fair coin n times, i.e., X ~ Bin(n, p) with p = 1/2. Find a Chernoff bound for Pr(X >= a). Find the sharpest (i.e., smallest) Chernoff bound. Evaluate your answer for n = 100 and a = 55. Mark the number that comes closest to your answer: (a) 0.001 (b) 0.002 (c) 0.004 (d) 0.008 (e) 0.016
Qudsiya A.
3. (3 points) A fair coin is flipped 60 times, X denotes the number of Heads. Give an upper bound on the probability P(|X - 30| ≥ 20) using Chebyshev's inequality. 4. A fair coin is flipped 60 times, X denotes the number of Heads. In this exercise, we'll give an upper bound on the probability P(|X - 30| ≥ 20) using Chernoff's bound. (a) (2 points) Let Yt = etx for t > 0. Show that E(Yt) = 2-60(1 + et)60 (b) (2 points) Give an upper bound on the probability P(X ≥ 50) by applying Markov's inequality for the random variable Yt. (c) (3 points) Find the value of t that gives the lowest upper bound in part (b) (Hint: you will end up minimizing the convex function f(t) = ln(1 + et) - 5/6 t.) (d) (2 points) Prove that P(|X - 30| ≥ 20) ≤ 2 ∙ 3^60 ∙ 5^-50 < 10^-6 Compare this with the result of Problem 3.
Sri K.
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