Question

if two random variable have the exact same expected value and variance, do they always have the same distribution?

Answer 1

No, two random variables with the same expected value and variance do not always have the same distribution.

Step-by-step explanation:

The answer to the question is no, two random variables with the same expected value and variance do not always have the same distribution.

A distribution is determined by the probabilities assigned to each value of the random variable. While the expected value and variance give us information about the central tendency and variability of the distribution, they do not provide information about the specific shape and probabilities assigned to each value.

For example, consider two random variables, X and Y, with the same expected value of 5 and variance of 9. X could follow a normal distribution with most values close to 5, while Y could follow a uniform distribution with equal probabilities assigned to each value from 1 to 9.