1.
Stat310
Mean, Variance and Distributions
Hadley Wickham
Saturday, 24 January 2009

2.
1. Recap: random variables & pmf
2. Expectation
3. Mean & variance
4. Meet some random variables
Saturday, 24 January 2009

3.
Random variable
A random variable is a random
experiment with a numeric sample
space. (Can make many different random variables from
a single random experiment)
More formally, a random variable is a
function that converts elements of non-
numeric sample space to numbers.
Saturday, 24 January 2009

4.
Discrete r.v.
A discrete random variable has a
countable sample space, typically a
subset of the integers.
Saturday, 24 January 2009

5.
pmf
Every random variable has an
associated probability mass function
(pmf).
The random variable says what is
possible. (the sample space)
The pmf says how likely each
possibility is. (the probability)
Saturday, 24 January 2009

6.
To be a pmf
A function must satisfy two properties:
• f(x) ≥ 0, for all x
• ∑ f(x) = 1
Saturday, 24 January 2009

7.
x f(x) x f(x) x f(x)
-1 0.3 10 -0.1 1 0.35
0 0.3 20 0.9 2 0.25
2 0.3 30 0.2 3 0.2
4 0.1
x f(x) x f(x)
5 0.1
5 1 10 0.1
20 0.9
30 0.2
Saturday, 24 January 2009

8.
Notation
Normally call pmf f
If we have multiple rv’s and want to make
clear which pmf belongs to which rv, we
write:
fX(x) fY(y) fZ(z) for X, Y, Z
f1(x) f2(x) f3(3) for X1, X2, X3
Saturday, 24 January 2009

9.
Notation
Can give pmf in two ways:
• List of numbers (for small n)
• Function (for large n)
These are equivalent!
Also useful to display visually, with a bar
plot (not a histogram: the book is wrong!)
Saturday, 24 January 2009

10.
a) b)
0.8
1.0
0.6 0.8
0.6
0.4
f(x)
f(x)
0.4
0.2
0.2
0.0 0.0
1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0
x x
c) d)
0.5
0.4
0.4
0.3 0.3
0.2
f(x)
f(x)
0.2
0.1
0.1
0.0
0.0 −0.1
1 2 3 4 5 1 2 3 4 5
x x
Saturday, 24 January 2009

11.
Expectation
• Allows us to summarise a pmf with a
single number
• Definition
• Properties
Saturday, 24 January 2009

12.
Mean
• Summarises the “middle” of the
distribution
• If you imagine the number line as a
beam with weights of f(x) at position x,
the balance point is the mean
• mean = E(X)
Saturday, 24 January 2009

13.
Variance
• Summarises the “spread” of a
distribution
• Var[X] = E[ (X - E[X])2] = ...
• Expected squared distance from centre
• sd[X] = Var[X]0.5
Saturday, 24 January 2009

14.
Meet the distributions
Discrete uniform
Bernoulli
Binomial
Saturday, 24 January 2009

15.
Discrete uniform
Equally likely events
f(x) = 1/m x = 1, ..., m
X ~ DiscreteUniform(m)
What is the mean?
What is the variance?
Saturday, 24 January 2009

16.
Useful facts
Sum of integers from i = 1 to m is
Sum of squared integers from i = 1 to m is
Saturday, 24 January 2009

17.
Bernoulli distribution
Single binary event: either happens (with
probability p) or doesn’t happen
X ~ Bernoulli(p)
What is the mean of X?
What is the variance of X?
Saturday, 24 January 2009

18.
Transformations
If X ~ Bernoulli(p)
What is 1 - X?
What is X2?
(Think about X, not f(x))
Saturday, 24 January 2009

19.
Binomial distribution
n independent Bernoulli trials with the
same probability of success. X is the
number of success
X ~ Binomial(n, p)
What is the mean?
What is the variance?
Better tools?
Saturday, 24 January 2009