# Question: Show that if X is a random variable with the

Show that if X is a random variable with the mean µ for which f(x) = 0 for x < 0, then for any positive constant a, P(X ≥ a) ≤ µ/a This inequality is called Markov’s inequality, and we have given it here mainly because it leads to a relatively simple alternative proof of Chebyshev’s theorem.

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Use the inequality of Exercise 4.29 to prove Cheby-shev’s theorem. In exercise P(X ≥ a) ≤ µ / a If we let RX(t) = lnMX(t), show that R'X(0) = µ and R''X(0) = σ2. Also, use these results to find the mean and the variance of a random variable X having the moment- generating function MX(t) = e4(e4 – 1) With reference to Example 3.22 on page 94, find the covariance of X1 and X3. With reference to Exercise 3.71 on page 100, find the conditional expectation of the random variable U = Z2 given X = 1 and Y = 2. With reference to Exercise 3.96 on page 107, what is the city’s expected water consumption for any given day?Post your question