This document provides an introduction to random variables. It defines random variables as functions that assign real numbers to outcomes of an experiment. Random variables can be either discrete or continuous depending on whether their possible values are countable or uncountable. The document also defines probability mass functions (pmf) which describe the probabilities of discrete random variables taking on particular values. Expectation is introduced as a way to summarize random variables using a single number by taking a weighted average of all possible outcomes.

1. Stat310
Random variables
Hadley Wickham

2. 1. Feedback
2. Recap
3. Introduction to random variables
4. Expectation

3. Feedback

4. Attending class
Taking good notes
Did homework
Did homework early
Reading through text
0 4 8 12 16 20

5. Attending class
Taking good notes
Did homework
Did homework early
Reading through text
0 4 8 12 16 20
Read book
More work outside of class
Start homework earlier

6. Examples
Interactivity & Feedback
Clear presentation/explanations
Website
Powerpoints
Recaps
0 5 10 15 20 25

7. Examples
Interactivity & Feedback
Clear presentation/explanations
Website
Powerpoints
Recaps
0 5 10 15 20 25
Accent
T shirts

8. 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?

9. Notation
k
xi
i=1
xi
i∈{1,...,k}
xi

10. Recap
What is the law of total probability?
What is the multiplication rule?
What is Bayes rule?

11. Recap
A, B and C are independent events.
What is P(A ∪ B ∪ C)?

12. Random variables

13. 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

14. 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)

15. 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)

16. Example
Select a family at random and observe
their children. What is the sample space?
What random variables could we create
from this experiment?

17. Example
Pick someone at random out of this class.
Measure their height.
What random variables could we create
from this experiment?

18. 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)

19. Random event Random variable
Anything Numbers
Probability mass
Probability
function

20. Discrete pmf
f (x) > 0 ∀x ∈ S
f (x) = 1
x∈S
P (X ∈ A) = f (x)
x∈A

21. 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?

22. 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.

23. 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?

24. Expectation
E[u(X)] = u(x)f (x)
x∈S
Summarises a function of a random
number with a single number

25. W
ha
ta
an re
Properties du c
?
E[c] = c
E[cu(X)] = cE[u(X)]
E[au1 (X) + bu2 (X)] = aE[u1 (X)] + bE[u2 (X)]
All conditions together imply
E is a linear operator