Let X ~ Geomeric(p). Using Chebyshev's inequality find an upper bound for P(|X – E[X]] >b).
6. If E(x) 16 and E(X?) -292, use Chebyshev's inequality to determine a) A lower bound for P(8< X < 24). (b) An upper bound for P(X 162 18)
Modify X and apply Markov's inequality to upper bound P(X > 3) when X > 2 and E[X] = 2.5.
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.
2. Suppose that is an exponential random variable with pdf f(y)= e), y>0. a. Use Chebyshev's Inequality to get an upper bound for the probability that takes on a value more than two standard deviations away from the mean. b. Use the given pdf to compute the exact probability that takes on a value more than two standard deviations away from the mean.
1. a) Let X ∼ Exponential(λ). Using Markov’s inequality find an upper bound for P(X ≥ a), where a > 0. Compare the upper bound with the actual value of P(X ≥ a). b) Let X ∼ Exponential(λ). Using Chebyshev’s inequality find an upper bound for P(|X − EX| ≥ b), where b > 0.
Apply Chebyshevs Inequality to lower bound P(O< X < 4) when E(X) 2 and E(X2)-5
8. A random variable X has a mean u = 10 and a variance o= 4. Using Chebyshev's theorem, find (a) P(X – 10 > 3); (b) P(X - 10 < 3); (c) P(5< X < 15); (d) the value of the constant c such that P(X – 10 > c) <0.04.
2. Let YBinomial (n, p) where p docs not depend on . Without using the WLLN (but you can use e.g. Chebychev's inequality) show that (a) Y/n-, p as n → oo (b) (1-Y/n)-> (1-p) as n → oo
Let X, Y ~ 10,11 independently. Find P(max(X, Y} > 0.8 1 min(X, Y} = 0.5)
Let X, Y be two independent exponential random variables with means 1 and 3, respectively. Find P(X> Y)