Question

Find the moment generating function of a random variable X, which follows a binomial distribution with parameters n and p. Then, calculate E[X^3].

Answer 1

The moment generating function (MGF) of a binomial distributed random variable X with parameters n and p is given by:

`MGF(t) = (pe^t + q)^n`, where q = 1 - p

To find E[X^3], we first find the third derivative of the MGF with respect to t and evaluate it at t=0:

`E[X^3] = [(n(n-1)(pe^t + q)^(n-3)p(pe^t + q)^2 + 3n(n-1)(pe^t + q)^(n-2)p^2(pe^t + q) + 6n(pe^t + q)^(n-1)p^3)]_((t=0))`

Simplifying this expression, we get:

`E[X^3] = np(np-1) + 3np^2 + 6p^3 = np(np^2 + 3p)`

Step-by-step explanation:

The moment generating function (MGF) is a powerful tool in probability theory that allows us to find the moments of a random variable by taking derivatives of its MGF. In this case, we are interested in finding the third moment of a binomial distributed random variable X with parameters n and p. To do this, we first find the MGF of X, which is given by:

`MGF(t) = (pe^t + q)^n`, where q = 1 - p. This can be derived using the definition of the MGF and the binomial distribution formula. Next, we find the third derivative of the MGF with respect to t and evaluate it at t=0 to get E[X³]. The formula for E[X³] is obtained by substituting this expression into the general formula for the nth moment of a random variable using its MGF.

The resulting expression can be simplified using algebraic manipulations to obtain the final answer. The expression for E[X³] involves n, p, and their products, which are characteristic parameters of the binomial distribution. This formula can be used to calculate the third moment of a binomially distributed random variable for any values of n and p.