1. The Mean of a Discrete Random Variable
We can find the mean of a discrete random variable in a similar way to that used for data.
Suppose we take our first example of rolling a die.
Number on die 1 2 3 4 5 6
Frequency 12 9 11 10 7
The mean is given by
1st x-value  1st frequency
But, can be replaced by , the probabilities of getting 1, 2, . . .
So, the mean

2. Notation for the Mean of a Discrete Random Variable
When dealing with a model, we use the letter m for the mean (the greek letter m).
pronounced “mew”
We write
or, more often, replacing p by ,
Instead of m, we can also write E(X).
This notation comes from the idea of the mean being the Expected value of the r.v. X.

3. Notation for the Mean of a Discrete Random Variable
When dealing with a model, we use the letter m for the mean (the greek letter m).
pronounced “mew”
We write
or, more often, replacing p by ,
Instead of m, we can also write E(X).
This notation comes from the idea of the mean being the Expected value of the r.v. X.
( Think of this as being what we expect to get on average ).

4. e.g. 1. A random variable X has the probability distribution
Find (a) the mean of X.
Solution:
(a) mean,
Tip: Always check that your value of the mean lies within the range of the given values of x. Here, or 5·25, does lie between 1 and 10.

5. Expectation of a continuous random variable
E(X) = m =
This formula is similar to that used in discrete random variables
E(X) = xP(X = x)
Replace by and P(X = x) by f(x)
Ex1 Find the expected value of
E(X) = m =
The boundaries are 16 and  as x can take any value greater than 16

6. E(X) = m =
E(X) =
A= 0 as
= if x = 
A16=
= -26
Area =  = 26
So the mean = 26

7. Symmetry
If a function is symmetrical then the expected value lies on the line of symmetry
Ex2
From the sketch the line of symmetry clearly lies on x = 2
So E(X) = 2

8. Variance of a Discrete Random Variable
The variance of a discrete random variable is found in a similar way to the one we used for the mean.
For a frequency distribution, the formula is
Replacing by etc. gives

9. The variance of X is also written as Var(X).
But we must replace by
So,
The variance of X is also written as Var(X).

10. x P(X = x ) e.g. 1 Find the variance of X for the following: Solution:
We first need to find the mean, m :

11. Variance of a continuous random variable
Var(X) = 2 =
This formula is similar to that used in discrete random variables Var(X) = x2P(X = x) – mean2
Replace by and P(X = x) by f(x)
Ex2
From the example above E(X) = 2 So m = 2

12. Var(X) = 2 =
A4 = A0 = 0
Area = = 4.8
So Var(X) =
= = 0.8
S.D =
= 0.9