E(X) & Var(X) (Discrete) (Cambridge (CIE) A Level Maths) : Revision Note
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E(X) & Var(X) (Discrete)
What does E(X) mean and how do I calculate E(X)?
E(X) means the expected value or the mean of a random variable X
For a discrete random variable, it is calculated by:
Multiplying each value of
with its corresponding probability
Adding all these terms together
Look out for symmetrical distributions (where the values of X are symmetrical and their probabilities are symmetrical) as the mean of these is the same as the median
For example if X can take the values 1, 5, 9 with probabilities 0.3, 0.4, 0.3 respectively then by symmetry the mean would be 5
How do I calculate E(X²)?
E(X²) means the expected value or the mean of a random variable defined as X²
For a discrete random variable, it is calculated by:
Squaring each value of X to get the values of X2
Multiplying each value of X2 with its corresponding probability
Adding all these terms together
In a similar way E(f(x)) can be calculated for a discrete random variable by:
Applying the function f to each value of to get the values of f(X)
Multiplying each value of f(X ) with its corresponding probability
Adding all these terms together


Is E(X²) equal to (E(X))²?
Definitely not!
They are only equal if X can take only one value with probability 1
if this was the case it would no longer be a random variable
E(X²) is the mean of the values of X²
(E(X))² is the square of the mean of the values of X
To see the difference
Imagine a random variable X that can only take the values 1 and -1 with equal chance
The mean would be 0 so the square of the mean would also be 0
The square values would be 1 and 1 so the mean of the squares would also be 1
In general E(f(X)) does not equal f(E(X)) where f is a function
So if you wanted to find something like
then you would have to use the definition and calculate:
What does Var(X) mean and how do I calculate Var(X)?
Var(X) means the variance of a random variable X
For any random variable this can be calculated using the formula
This is the mean of the squares of X minus the square of the mean of X
Compare this to the definition of the variance of a set of data
Var(X) is always positive
The standard deviation of a random variable X is the square root of Var(X)
Worked Example
The discrete random variable has the probability distribution shown in the following table:
2 | 3 | 5 | 7 | |
0.1 | 0.3 | 0.2 | 0.4 |
(a) Find the value of .
(b) Find the value of .
(c) Find the value of .



Examiner Tips and Tricks
Check if your answer makes sense. The mean should fit within the range of the values of X.
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