This exercise will lead you through a proof of Chebyshevs inequality. Let X be a continuous random
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This exercise will lead you through a proof of Chebyshev’s inequality. Let X be a continuous random variable with probability density function f (x). Suppose that P(X < 0) = 0, so f (x) = 0 for x ≤ 0.
a. Show that
b. Let k >0 be a constant. Show that μX ≥
c. Use part (b) to show that P(X ≥ k) ≤ μX /k. This is called Markov’s inequality. It is true for discrete as well as for continuous random variables.
d. Let Y be any random variable with mean μY and variance σ2Y. Let X = (Y − μY)2. Show that μX = σ2Y.
e. Let k > 0 be a constant. Show that P(|Y −μY| ≥ kσY ) = P(X ≥ k2σ2Y).
f. Use part (e) along with Markov’s inequality to prove Chebyshev’s inequality: P(|Y − μY| ≥ kσY ) ≤ 1/k2.
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