Let X be a random variable such that P(X ≤0)=0 and let μ=E(X) exist. Show that P(X ≥2 μ) ≤(1)/(2

step 1 So in this question, we have a random variable x whose support is always positive. So the probab

Let X be a random variable such that P(X ≤0)=0 and let...

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Nonnegative Random Variable

A nonnegative random variable is one that takes only values greater than or equal to zero almost surely. This quality is crucial when applying probability bounds like Markov's inequality, which rely on the variable having a nonnegative range to convert information about its mean into information about the probability of large deviations.

Expectation

Expectation, or the mean of a random variable, provides a measure of its central tendency. In this context, the existence of the expectation allows us to use it as a benchmark for the size of the random variable, under the premise that large deviations from the mean occur with limited probability.

Markov's Inequality

Markov's inequality is a fundamental tool in probability theory that gives an upper bound for the probability that a nonnegative random variable exceeds a certain value. Specifically, it states that for any a > 0, the probability that the random variable is at least a is at most the expectation divided by a. Here, setting a to twice the expectation directly yields the result P(X ? 2?) ? 1/2.

Transcript

Please give Ace some feedback