The Moment Generating Function

One function, every moment

A catalog of MGFs

Existence, uniqueness, and what MGFs are for

The r-th derivative of the MGF evaluated at t = 0 equals what?

A random variable has MGF equal to e raised to the quantity 2 times e to the t, minus 2. Which distribution is it?

The Bernoulli(p) MGF is 1 - p + p e^t. Its first derivative at t = 0 is e^0 = 1 times p, plus zero. For p = 0.3, what is the mean? Enter a decimal like 0.3.

The Expo(lambda) MGF is lambda over (lambda minus t). For lambda = 4, M'(0) = lambda squared over lambda squared, or compute directly: what is the mean? Enter a decimal like 0.25.

The Uniform(0,1) MGF is (e^t minus 1) over t. Near t = 0 it limits to 1. Its series begins 1 + t/2 + t squared over 6. What is E(X)? Enter a decimal like 0.5.

In your own words, state the most important definition, theorem or method of this lesson and explain what it says.