This document discusses topics for an upcoming exam, including: 1. The exam will be 80 minutes, allow 2 pages of notes, and all information will be on the course website. 2. When taking the exam, students should read all questions first and start with the easiest, they can come back to harder questions later. 3. The recap covers expectation, mean, variance, Bernoulli and discrete uniform distributions. 4. The binomial distribution is defined as the number of successes in n independent Bernoulli trials with constant probability p of success. Its probability mass function and mean and variance are discussed. Tables can be used to calculate binomial distribution probabilities.

1. Stat310
Moment generating function
Hadley Wickham
Tuesday, 27 January 2009

2. 1. Test info.
2. Recap
3. Binomial distribution
4. Moment generating function
5. Poisson distribution
6. Negative binomial (if time)
Tuesday, 27 January 2009

3. Exam
80 minutes in class on Thursday.
2 (double sided) pages of notes.
All info on website.
(Homework grade - out of 12.
But two are extra credit)
Tuesday, 27 January 2009

4. Exam technique
• Read all the questions first. Start with the
easiest question!
• If you get stuck, try another question and
come back to it.
• If you get really stuck, writing down the
essential elements of the question and
describing the approaches you think might
be successful will get you partial credit
Tuesday, 27 January 2009

5. Recap
• Expectation
• Mean & variance
• Bernoulli distribution
• Discrete uniform distribution
Tuesday, 27 January 2009

6. Binomial distribution
• A Bernoulli experiment is
performed n times
• The trials are independent
• The probability of success
is a constant p
• The random variable X is the
number of successes in the n trials
Tuesday, 27 January 2009

7. Binomial distribution
X1, X2, ..., Xn ~ Bernoulli(p)
Y = X1 + X2 + ... + Xn
Y ~ Binomial(n, p)
What is the pmf?
What is the mean? What is the variance?
Tuesday, 27 January 2009

8. Using tables
Tedious to calculate by hand.
Use computer or calculator if available.
If not use tables.
Tables typically are for F(x) = P(X ≤ x)
P(X > x) = 1 - P(X ≤ x)
P(a ≤ X ≤ b) = P(X ≤ b) - P(X ≤ (a - 1))
Tuesday, 27 January 2009

9. p=0.1 p=0.5
0.35
0.20
0.30
0.25
0.15
0.20
f(x)
f(x)
0.10
0.15
0.10
0.05
0.05
0.00 0.00
0 2 4 6 8 10 0 2 4 6 8 10
x x
p=0.7 p=0.9
0.25
0.35
0.30
0.20
0.25
0.15
0.20
f(x)
f(x)
0.15
0.10
0.10
0.05
0.05
0.00 0.00
0 2 4 6