Question

Let random variable X be uniformly distributed in interval [0, T].
a) Find the nth moment of X about the origin.
b) Let Y be independent of X and also uniformly distributed in [0, T]. Calculate the
second moment about the origin, and the variance of Z = X + Y

Answer 1

a.

The moments of a distribution are a set of parameters that summarize it. Given a random variable X, its first moment about the origin, denoted , is defined to be E[X].

We can extract all the moments of the distribution from the MGF as follows: If we differentiate M(t) once, the only term that is not multiplied by t or a power of t is . So, .

b.

Let X be a uniform random variable defined in the interval [0,1]. This is also called a standard uniform distribution. We would like to find all its moments. We find that . However, this function is not defined—and therefore not differentiable—at t = 0. Instead, we revert to the series:

which is differentiable term by term. Differentiating r times and setting t to 0, we find that . So, is the mean, and . Note that we found the expression for M(t) by using the compact notation, but reverted to the series for differentiating it. The justification is that the integral of the compact form is identical to the summation of the integrals of the individual terms.