Distributions of Sums
Distributions of Sums¶
This chapter provides some general methods for working with sums of random variables, whether discrete or continuous.
We will start with the continuous analog of the convolution formula for the distribution of a sum of two independent discrete random variables.
We will then develop a more powerful version of the probability generating function that we defined earlier to study sums of independent random variables with finitely many non-negative integer values. The new version, called the moment generating function, will apply to all random variables, discrete and continuous, with finite or infinite sets of values.
In the process, we will be develop sharper tail bounds than the ones based on the inequalities of Markov and Chebychev.