Question

Suppose we have a non-negative random variable with expectation value 50. Then:

1. Derive an upper bound for the probability that X is at least 100.

Answer 1

Markov's inequality states that for a non-negative random variable X with an expected value of 50, the probability that X is at least 100 cannot exceed 0.5. This provides an upper bound, although the actual probability can be lower.

Step-by-step explanation:

To derive an upper bound for the probability that a non-negative random variable X, with an expectation value (mean) of 50, is at least 100, we can invoke Markov's inequality. This is a theorem in probability theory that gives an upper bound for the probability that a non-negative random variable is greater than a certain positive value.

According to Markov's inequality:

P(X ≥ a) ≤ E(X) / a.

Here, E(X) is the expected value of X, and a is the value we are interested in, which in this case is 100.

Given that E(X) = 50, we can calculate:

P(X ≥ 100) ≤ 50 / 100 = 0.5.

This means that the probability of the random variable being at least 100 is less than or equal to 0.5. This is a theoretical upper bound, and the actual probability could be significantly lower, depending on the actual distribution of X.