Hadley Wickham

Stat310Random variables

1. Feedback

2. Recap

3. Introduction to random variables

4. Expectation

Feedback

Reading through text

Did homework early

Did homework

Taking good notes

Attending class

0 4 8 12 16 20

Reading through text

Did homework early

Did homework

Taking good notes

Attending class

0 4 8 12 16 20

Start homework earlier

More work outside of class

Read book

Recaps

Powerpoints

Website

Clear presentation/explanations

Interactivity & Feedback

Examples

0 5 10 15 20 25

Recaps

Powerpoints

Website

Clear presentation/explanations

Interactivity & Feedback

Examples

0 5 10 15 20 25

T shirts

Accent

Improvements

Board skills: mistakes, more details & structure, straight lines

Pace: 5 good, 5 too fast, 2 too slow

Office hours. Homework answers online.

Connection to text/more practice.

Class mailing list / forums?

Notation

!

i!{1,...,k}

xi

k!

i=1

xi

!xi

Recap

What is the law of total probability?

What is the multiplication rule?

What is Bayes rule?

Recap

A, B and C are independent events.

What is P(A ∪ B ∪ C)?

Random variables

Why?

Probability is a set function. Kind of tricky to deal with. Easier to deal with functions of numbers.

Want to ignore details of problem (e.g. specific events) and focus on essence.

Real world ➙ mathematical world

Definition

A random variable is a function from the sample space to the real line

Usually given a capital letter like X, Y or Z

The space (or support) of a random variable is the range of the function (analogous to the sample space)

(Usually just call the result a random variable)

Discrete vs. continuous

Space of X is countable = can be mapped to integers = discrete

Space of X is uncountable = can be mapped to real numbers = continuous

(We’ll focus on discrete to start with)

Example

Select a family at random and observe their children. What is the sample space?

What random variables could we create from this experiment?

Example

Pick someone at random out of this class. Measure their height.

What random variables could we create from this experiment?

Random variables

For a countable sample space, usually a count. (But many things we could count)

For a uncountable sample space, usually just the value. (Typically fewer logical possibilities)

Random event Random variable

Anything Numbers

ProbabilityProbability mass

function

Discrete pmf

f(x) > 0 !x " S

!

x!S

f(x) = 1

P (X ! A) =!

x!A

f(x)

Example

• If X=1, f(x) = 0.9

• If X=2,3,4,5 or 6, f(x) = c/x

• (How to write and read more mathematically)

• Is this function is a pmf? What is c?

Why?

Once we have random variable + pmf, we don’t need any more information about the original experiment.

Means we can apply the same tools to completely different types of experiments.

Example

Draw two cards (with replacement) out of a shuffled pack. Let X be the number of hearts and clubs. What is the pmf of X?

Pick two people at random. Let Y be the number of males. What is the pmf of Y?

How are these pmfs related?

Expectation

E[u(X)] =!

x!S

u(x)f(x)

Summarises a function of a random number with a single number

Properties

E[c] = c

E[cu(X)] = cE[u(X)]E[au1(X) + bu2(X)] = aE[u1(X)] + bE[u2(X)]

What are c and u?

All conditions together imply E is a linear operator