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 
.
c. Use part (b) to show that 
. 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 µ Y and variance 
. Show that 
.
e. Let k > 0 be a constant. Show that P
.
f. Use part (e) along with Markov’s inequality to prove Chebyshev’s inequality: 
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