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) .
To apply the Markov inequality, we need to use the fact that Z is a nonnegative random variable. The Markov inequality states that for any nonnegative random variable Y and any positive constant a, the probability of Y being greater than or equal to a is bounded by the expected value of Y divided by a.
In this case, we are interested in finding an upper bound on the probability that Z is greater than or equal to 2, denoted by P(Z ≥ 2). However, we are given information about the fourth power of Z, denoted by Z^4, as E[Z^4] = 4.
To apply the Markov inequality, we need to relate Z and Z^4. Since Z is nonnegative, we can say that Z^4 is greater than or equal to Z^2, which is greater than or equal to Z. We can use this relation to find an upper bound on P(Z^4 ≥ 2^4) or P(Z^4 ≥ 16).
Now, let's calculate the upper bound using the Markov inequality:
P(Z^4 ≥ 16) ≤ E[Z^4]/16
Given that E[Z^4] = 4, we can substitute this value into the inequality:
P(Z^4 ≥ 16) ≤ 4/16
Simplifying the expression:
P(Z^4 ≥ 16) ≤ 1/4
Therefore, we have found the tightest possible upper bound on P(Z ≥ 2) using the Markov inequality: P(Z ≥ 2) ≤ 1/4.