You know the population mean and variance of some random variable, but would like to know the population mean and variance of another random variable that depends on the first in some complicated way.

You want to know how close your sample mean is to the population mean for some given level of uncertainty.

A manufacturer claims something measurable about their product and you wish to check the truth of their claim. 

You want to know the sample size needed to ensure that the your sample mean has a certain probability of being within some given tolerance of the population mean.

You know the distribution of a random vector and wish to know the probability that an event takes on a value in a known region.

You have a dataset and know what distribution it can be modelled with, and want to know about the value of a parameter of the distribution.