Question

Let pi = P{X = i} and suppose that p₁+p₂+p₃ = 1. If E[X] = 2, what values of p₁,p₂,p₃

(a) maximize and,
(b) minimize Var(X)?

Answer 1

To maximize Var(X), assign a probability of 1 to the value farthest from the mean. To minimize Var(X), assign a probability of 1 to the value at the mean, thereby reducing variance to zero.

Step-by-step explanation:

Maximizing and Minimizing Variance

To maximize the variance (Var(X)) of a random variable X with probabilities p₁, p₂, and p₃ where p₁+p₂+p₃=1 and E[X]=2, we need to increase the spread of the distribution. Maximizing the variance would involve assigning probability as far from the mean as possible. Therefore, p₁ could be set to 1 (with p₂ and p₃ set to 0), or p₃ to 1 (with p₁ and p₂ set to 0), assuming X takes on values 1, 2, and 3 respectively.

To minimize the variance, we want to concentrate the probability distribution closer to the mean. This would be achieved by assigning p₂ a value of 1 (with p₁ and p₃ being 0). In this case, all the probability is concentrated at the mean, which is 2, so the variance would be zero.

Note that we have not been given specific values for X, but the principles above would apply to any distribution where the sum of probabilities equals 1, and the expected value is 2.