06 mean var

Hadley Wickham

Stat310Mean, Variance and Distributions

Saturday, 24 January 2009

1. Recap: random variables & pmf

2. Expectation

3. Mean & variance

4. Meet some random variables


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.

Random variable


Discrete r.v.

A discrete random variable has a countable sample space, typically a subset of the integers.


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)


To be a pmf

A function must satisfy two properties:

• f(x) ≥ 0, for all x

• ∑ f(x) = 1


x f(x)

1 0.35

2 0.25

3 0.2

4 0.1

5 0.1

x f(x)

10 -0.1

20 0.9

30 0.2

x f(x)

-1 0.3

0 0.3

2 0.3

x f(x)5 1

x f(x)

10 0.1

20 0.9

30 0.2


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


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!)


x

f(x)

0.0

0.1

0.2

0.3

0.4

1 2 3 4 5x

f(x)

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

1 2 3 4 5

x

f(x)

0.0

0.2

0.4

0.6

0.8

1.0

1.0 1.5 2.0 2.5 3.0x

f(x)

0.0

0.2

0.4

0.6

0.8

1.0 1.5 2.0 2.5 3.0

a) b)

c) d)


Expectation

• Allows us to summarise a pmf with a single number

• Definition

• Properties


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)


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


Meet the distributions

Discrete uniform

Bernoulli

Binomial


Discrete uniform

Equally likely events

f(x) = 1/m x = 1, ..., m

X ~ DiscreteUniform(m)

What is the mean?

What is the variance?


Useful facts

Sum of integers from i = 1 to m is

Sum of squared integers from i = 1 to m is


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?


Transformations

If X ~ Bernoulli(p)

What is 1 - X?

What is X2?

(Think about X, not f(x))


Binomial distributionn 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?


06 mean var

Education