VARIATION AND PROBABILITY

The cost of drilling is $200,000 ($200K).

Drilling will lead to one of the following outcomes

Well type	Probability	Payoff
Dry	0.5	$0
Wet	0.4	$400K
Gusher	0.1	$1500K

If it sells now, the company can get $125K.

If it holds for a year and oil prices rise (probability =0.6) it can sell for $300K or if oil prices fall (probability = 0.4) it can get $100K. What should it do?

The best decision is to hold for a year and then sell. This is an example of using a tree diagram for Decision Analysis. It also illustrates the concept of the expected value of a random variable.

Expected Value of a Random Variable

If the probability distribution of a random variable X is

Values of X	x1	x2	...	xk
Probabilities	p1	p2	...	pk

its expected value is

e.g. Drilling for oil example

Well Type	Probability	Pay-off
Dry	0.5	0
Wet	0.4	$400K
Gusher	0.1	$1500K

Let random variable X be the financial gain

The probability distribution for X is

x	-200	200	1300
P(X=x)	0.5	0.4	0.1

so the expected value (average) of X is

E(X) = -200 x 0.5 + 200 x 0.4 + 1300 x 0.1 = $110K

This is directly analogous to the sample mean

E(X) can be regarded as an idealisation of, or a theoretical value for, the sample mean .

E(X) is often denoted by the Greek letter µ (pronounced "mu")

Variance of a Random Variable

The variance of a Random Variable X is defined as

It represents the theoretical limit of the sample variance s² as the sample size n becomes very large.

var(X) is often denoted by s ² (sigma squared).

A simpler formula for var(X) is

Example - Gender in a class of 5

Assume that the probability of a student in a class being male is a half. Let the random variable X be the number of male students in a group from the class of size 5

What is E(X), the expected number of males in the group, and what is the variance of X?

Assume X ~ binomial (5,0.5) .

Then the probability distribution of X is

x	0	1	2	3	4	5
P(X=x)	1/32	5/32	10/32	10/32	5/32	1/32

(Check this using the formula for binomial probabilities and/or draw a tree diagram to look at the structure of the outcomes.)

i.e. on average such groups have 2.5 males.

This is a measure of the variability of X.

Check the calculation using the alternative formula:

In general if X ~ binomial (n,p) it can be shown that

Check the values of E(X) and var(X) calculated above for X ~ binomial (5,0.5) using these formulas.

Expected Value and Variance for a Function of Random Variables

If Y = a X + b

where X is a random variable and a and b are known constant values, then

and

Similarly if T = a X + b Y + c where X and Y are random variables and a , b and c are known constants, then

and

In particular, if X and Y are independent then the covariance cov(X,Y) is zero, so

Proof: Follows from the definitions of E(X) and var(X).

Example - Estimated profit prediction

A company makes products for local and export markets.

The number of sales next year cannot be predicted exactly but estimates are as follows

local,X units	1,000	3,000	5,000	10,000
probability	0.1	0.3	0.4	0.2

export,Y units	300	500	700
probability	0.4	0.5	0.1

Hence E(X) = 1000 x 0.1 + 3000 x 0.3 + 5000 x 0.4 + 10000 x 0.2

The company makes a profit of $2000 on each unit sold on the local market and $3500 on each exported unit.

Hence total profit is

Using the above formula

- this is the estimated profit for next year.

Example - Metal component manufacturing

A component is made by cutting a piece of metal to length X and then trimming it by amount Y. Both of these processes are somewhat imprecise.

The net length is then

This is of the form T = a X + b Y with a = 1 and b = -1

so

i.e. var(T) is greater than either var(X) or var(Y), even though T = X - Y, because both X and Y contribute to the variability in T.