information:filtering
Moment-generating function

The moment-generating function of the continuous uniform distribution is:

from which we may calculate the raw moments m k: {\displaystyle m_{k}:}

For a random variable following the continuous uniform distribution, the expected value is m 1 = a + b 2, {\displaystyle m_{1}={\tfrac {a+b}{2}},} and the variance is m 2 − m 1 2 = (b − a) 2 12. {\displaystyle m_{2}-m_{1}^{2}={\tfrac {(b-a)^{2}}{12}}.}

For the special case a = − b, {\displaystyle a=-b,} the probability density function of the continuous uniform distribution is:

the moment-generating function reduces to the simple form: