Let Z be a nonnegative random variable that satisfies E[Z4]=4 . Apply the Markov inequality to the random variable Z4 to find the tightest possible (given the available information) upper bound on P(Z≥2) .
P(Z≥2)≤
1/4
P(Z≥2)≤ E[Z4] / (2^4)
P(Z≥2)≤ 4 / 16
P(Z≥2)≤ 1/4
So, the tightest possible upper bound for P(Z≥2) is 1/4. But let's not feel too down about it. After all, a quarter of a chance is not so bad, is it? Just keep your hopes up!
To apply the Markov inequality, we need to find the expectation of Z^4, denoted as E[Z^4].
Given that E[Z^4] = 4, we can use the Markov inequality to obtain an upper bound on P(Z ≥ 2).
The Markov inequality states that for any nonnegative random variable Y and any positive constant a:
P(Y ≥ a) ≤ E[Y]/a
In our case, we want to find an upper bound on P(Z ≥ 2), so we let Y = Z^4 and a = (2^4) = 16.
Using the Markov inequality, we have:
P(Z^4 ≥ 16) ≤ E[Z^4]/16
Since E[Z^4] = 4, we can substitute it into the inequality:
P(Z^4 ≥ 16) ≤ 4/16
Simplifying the expression:
P(Z^4 ≥ 16) ≤ 1/4
Finally, we can rewrite this in terms of P(Z ≥ 2) by taking the 4th root of both sides:
P(Z ≥ 2) ≤ (1/4)^(1/4) = 1/2
Therefore, the tightest possible upper bound on P(Z ≥ 2), based on the given information, is 1/2.
To find the upper bound on P(Z ≥ 2) using the Markov inequality, we need to know the expected value of Z4, denoted as E[Z4]. The Markov inequality states that for any nonnegative random variable Y and any positive constant a, P(Y ≥ a) ≤ E[Y] / a.
In this case, we are given E[Z4] = 4, which means the expected value of Z4 is 4. We want to find an upper bound on P(Z ≥ 2). To do this, we can apply the Markov inequality to the random variable Z4.
Using the Markov inequality, we have:
P(Z4 ≥ a) ≤ E[Z4] / a
Substituting the values we have, we get:
P(Z4 ≥ a) ≤ 4 / a
Since we want to find an upper bound on P(Z ≥ 2), we can substitute a = 2:
P(Z4 ≥ 2) ≤ 4 / 2
Simplifying the right-hand side:
P(Z4 ≥ 2) ≤ 2
Since Z4 is a nonnegative random variable, the event Z ≥ 2 implies Z4 ≥ 2 as well. Therefore, we can conclude that:
P(Z ≥ 2) ≤ P(Z4 ≥ 2) ≤ 2
So, the tightest possible upper bound on P(Z ≥ 2), given the available information, is 2.