A Level Further Mathematics for AQA Statistics Student Book
Section 1: Average and spread of a discrete random variable Tip
The most commonly used measure of the average of a random variable is the expectation. It is a value representing the mean result if the variable were to be measured an infinite number of times.
The expectation of a random variable does not need to be a value which the variable can actually be.
Key point 1.1 The expectation of a random variable X is written E( X ) and calculated as
Tip
E(X ) = ∑ xi pi
pl e
The subscript i in the formula in Key point 1.1 is just a counter referring to each possible value and its associated probability.
You do not need to be able to prove this result, but you might find it helpful to see this proof.
Sa m
PROOF 1 The mean of n pieces of discrete data is 1∑ f x x=n i i f = ∑ i xi n
Start from the definition of the mean.
1 is constant you can take it into the sum. Since n
D ra ft
f If n is large, ni will tend towards the probability of xi happening, therefore x = µ = ∑ xi pi
When the sample size tends to infinity, the sample mean x becomes the true population mean, µ.
WORKED EXAMPLE 1.1
The random variable X has a probability distribution as shown in the table. Calculate E(X ). X
1
2
3
4
5
6
P( X = x )
1 10
1 4
1 10
1 4
1 5
1 10
E( X ) = 1× 1 + 2 × 1 + 3 × 1 + 4 × 1 + 5 × 1 + 6 × 1 10 4 10 4 5 10 7 =2
Use the values from the distribution in the formula in Key point 1.1.
As well as knowing the expected average, you may also be interested in how far away from the average you can expect an outcome to be. The variance, σ 2 , of a random variable is a value representing the degree of variation that would be seen if the variable were to be repeatedly measured an infinite number of times. It is a measure of how spread out the variable is. 2
Fast forward You will see in Section 2 how to find expectations of other functions of X.
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