Let Z be a nonnegative random variable that satisfies E[Z^4]=4. Apply the Markov inequality to the random variable Z^4 to find the tightest possible (given the available information) upper bound on P(Z≥2).
P(Z>=2)<= ?E[Z^4]/2 = 2 But this is not the right answer
P(Z>=2) <= 0.25
To find the tightest possible upper bound on P(Z ≥ 2) using the Markov inequality, we start by applying the inequality itself.
The Markov inequality states that for any non-negative random variable X and any a > 0, we have P(X ≥ a) ≤ E[X]/a.
In this case, we want to apply the Markov inequality to the random variable Z^4, where Z is a non-negative random variable that satisfies E[Z^4] = 4.
Using the Markov inequality, we can write:
P(Z^4 ≥ a^4) ≤ E[Z^4]/a^4
We want to find the tightest possible upper bound for P(Z ≥ 2), which means we want to find the smallest possible value for a^4 such that Z^4 ≥ a^4 implies Z ≥ 2.
Since Z is nonnegative, this means that Z^4 ≥ a^4 implies Z ≥ a.
Therefore, we need to find the smallest possible value for a such that Z ≥ a implies Z ≥ 2.
It follows that the smallest possible value for a is 2.
Substituting a = 2 into the Markov inequality, we have:
P(Z^4 ≥ 2^4) ≤ E[Z^4]/2^4
Simplifying:
P(Z^4 ≥ 16) ≤ 4/16
P(Z^4 ≥ 16) ≤ 1/4
Therefore, the tightest possible upper bound on P(Z ≥ 2) is given by P(Z^4 ≥ 16) ≤ 1/4.
To find the tightest possible upper bound on P(Z ≥ 2) using the Markov inequality, we will first need to calculate the expected value of Z^4.
Given that E[Z^4] = 4, we know that the fourth moment of Z is equal to 4.
The Markov inequality states that for a nonnegative random variable X and any positive constant a:
P(X ≥ a) ≤ E[X]/a
In this case, we want to find an upper bound for P(Z ≥ 2), so we will use the Markov inequality with X = Z^4 and a = 2^4 = 16.
P(Z^4 ≥ 16) ≤ E[Z^4]/16
Substituting the given information, we have:
P(Z^4 ≥ 16) ≤ 4/16 = 1/4
However, this upper bound is not tight. To find the tightest possible upper bound, we need to consider the fact that Z is nonnegative.
Since Z is nonnegative, we know that Z^4 ≥ 0. This means that Z^4 ≥ 16 implies Z ≥ 2. Therefore, we can revise the inequality as follows:
P(Z ≥ 2) ≤ P(Z^4 ≥ 16)
But since Z^4 ≥ 16 implies Z ≥ 2, we can also write it as:
P(Z ≥ 2) ≤ P(Z^4 ≥ 16) = 1/4
So, the tightest possible upper bound on P(Z ≥ 2) is 1/4, given the available information.