Generating functions

Objective. Use probability and moment generating functions.

1. Learn
2. Worked example
3. Practice
4. Feedback
5. Continue

Learn: the key idea

Generating functions encode an entire probability distribution in a single function, with the moment generating function M(t) = E[e^(tX)] whose derivatives at t = 0 give the moments: M′(0) = E[X], M″(0) = E[X²]. For discrete non-negative integer variables, the probability generating function G(s) = E[s^X] similarly encodes probabilities as its Taylor coefficients. For X ~ Poisson(λ), M(t) = exp(λ(e^t − 1)), and differentiating twice at t = 0 recovers mean λ and variance λ.

Worked example

For a random variable with moment generating function M(t) = e^(3t+2t²), find E[X] using generating functions.

1 $Differentiate M(t) = e^(3t+2t²) using the chain rule$
2 $M'(t) = (3+4t)e^(3t+2t²)$
3 $Evaluate at t = 0: M'(0) = (3+0)e^0$
4 $Simplify to obtain the mean$

Answer: E[X] = 3

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. A frequent mistake is computing E[X²] directly from M(0) rather than from M″(0), confusing the function's value with its derivatives. Remember moments come from differentiating the generating function then setting t = 0.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard