Maths For Economics 7: Random Variables Topic 1 - One Random Variable and Its Probability Distribution

Question 1

Explain what random variables are and their importance

Answer 1

There’s conditional probability: how
information changes probabilities (by changing the sample space).
So far, we have allowed as simple outcomes and events any
occurrence that can be described by a sentence like ‘We win the
German contract …’
However, in economics and finance we are interested not only in the
true/false value of ‘We win the German contract.’
We are also interested in such numerical information as the value of the
contract. An unknown number with a probability distribution attached
is called a random variable.
With random variables, we can ask interesting new questions such as
‘On average, what will be the size of the value of the contracts we win?’
‘Random variable’ language about numerical outcomes is a
picturesque and useful alternative to ‘probability distribution’
language.
We flip between the two ways of speaking.
It is possible, but not usually desirable, to eliminate random variables
from the discussion, and speak only of probability distributions (to be
described below).

Question 2

Describe the simplest non-trivial random variable

Answer 2

A two-element sample space provides the simplest non-trivial example
– but a highly informative one - of what happens when there are
numerical outcomes.
Suppose that the sample space is omega = {0, 1), so the two simple
outcomes are numbers. (Previously we have allowed, for instance, omega = {heads, tails}.)
In mathematics in general, we use symbols like to stand for
unknown quantities.
In probability theory we use ‘random variables’ to stand for simple
outcomes. If omega= {0, 1}, let X (capital letter) stand for the actual
outcome.
So X=0 or X=1, but we don’t initially (or, perhaps, ever) know
which. X is a typical random variable.

Question 3

Describe the simplest Random Variable

Answer 3

For X to be a proper random variable, its values (which are simple
outcomes in omega must be numerical and have probabilities.
For example, suppose we believe that P(X=)0) = 0.37 and P(X=1) = 0.63. Now omega is called a probability space (not just a
sample space) and has the following probability distribution:
Every random variable has a probability distribution: simple outcomes
in row 1, probabilities in row 2.
This simplest (non-trivial) case of a random variable, with just two
simple outcomes omega = {0, 1}, is called the Bernoulli distribution

Question 4

I roll a fair 4-sided die, obtaining a value 𝑋 and a prize of 𝑋^2
dollars. What is the mathematical expectation of my prize?

Answer 4

𝐸(𝑋^2) ≡ ∑^4subscript‘r=1’r^2𝑃(𝑋 = 𝑟) / 4
= (1 + 4 + 9 + 16) / 4 = 7.5