Moments of Bernoulli random variables
The nth moment of a random variable X is defined to be the expectation E[X^n] of the nth power of X.
Recall that a Bernoulli random variable with parameter p is a random variable that takes the value 1 with probability p, and the value 0 with probability 1-p.
Let X be a Bernoulli random variable with parameter 0.7. Compute the expectation values of X^k, denoted by E[X^k], for the following three values of k: k = 1,4, and 3203.
So
1. E[X] =
2. E[X^4] =
3. E[X^3203] =
1. E[X] = p = 0.7
2. E[X^4] = p^4 = 0.7^4 = 0.2401
3. E[X^3203] = p^3203 = 0.7^3203.
The value of E[X^3203] cannot be determined without the use of a calculator or computer because it involves a large exponent. However, we know that since X is a Bernoulli random variable, it can only take on values of 0 or 1, so E[X^k] for any positive integer k will eventually become 0 as p^k becomes negligibly small.