Let Y = X - 2, then E[Y] = E[X - 2] = E[X] - 2 = 2.5 - 2 = 0.5
As, X 2 => X - 2 0 => Y 0
Now, as Y is nonnegative random variable, then by Markov's inequality,
P(Y a) E(Y) / a for any a > 0
Now,
P(X 3) = P(X - 2 3 - 2) = P(Y 1)
From Markov's inequality,
P(X 3) = P(Y 1) 0.5 / 1
=> P(X 3) 0.5
Modify X and apply Markov's inequality to upper bound P(X > 3) when X > 2...
Let X ~ Geomeric(p). Using Chebyshev's inequality find an upper bound for P(|X – E[X]] >b).
Apply Chebyshevs Inequality to lower bound P(O< X < 4) when E(X) 2 and E(X2)-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.
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)
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.
If X follows Bernoulli distribution Bp,p > 0.5 and V(X) 0.24 . E(X)?
2 5. Suppose X is a discrete random variable that has a geometric distribution with p= a. Compute P(X > 6). [5] b. Use Markov's Inequality to estimate P(X > 6). [5] c. Use Chebyshev's Inequality to estimate P(X > 6). [5]
3. Let X be a geometric random variable with parameter p. Prove that P(X >k+r|X > k) = P(X > r). This is called the memoryless property of the geometric random variable.
this curve, with X - axis rotation, calculate surfacea area y=e-72, (x > 0)
2 of 3 01- 5. Suppose X is a discrete random variable that has a geometric distribution with p= a. Compute P(X > 6). [5] b. Use Markov's Inequality to estimate P(X > 6). [5] c. Use Chebyshev's Inequality to estimate P(X > 6). (5)