Probability Theory

Sums of Independent Random Variables—Summary of Distributional Results

You can and should use these results when needed on homework and exams (you do not need to re-derive the results unless specifically asked).

Gamma Random Variables (shown in class)

Suppose are independent gamma random variables with respective parameters , i = 1, 2,..., n. Then has a gamma distribution with parameters .

Recall the exponential distribution is a special case () of the gamma distribution. Therefore, if are independent exponential () random variables, has a gamma distribution with parameters .

Normal Random Variables (shown in the textbook)

Suppose are independent normal random variables with respective parameters , i = 1, 2,..., n. Then has a normal distribution with parameters, .

Poisson Random Variables (shown in class)

Suppose are independent Poisson random variables with respective parameters , i = 1, 2,..., n. Then has a Poisson distribution with parameter .

Binomial Random Variables (shown in textbook)

Suppose are independent binomial random variables with respective parameters , i = 1, 2,..., n. Then has a binomial distribution with parameters,

Geometric Random Variables (shown in textbook—similar to what we did in class for Poisson r.v.s)

Suppose are independent geometric random variables, all with parameter . Then has a negative binomial distribution with parameters .

Squares of Standard Normal Random Variables (shown in class)

Suppose are independent standard normal random variables (recall a standard normal r.v. has mean 0 and standard deviation 1). Then has a chi-squared distribution with parameter n. (This parameter is called the “degrees of freedom.” More on this in Math 445.)

As we discussed in class, the chi-squared distribution is a special case, , of the gamma distribution.