E(X) means the expected value or the mean of a random variable X
For a discrete random variable, it is calculated by:
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
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
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
Exam Tip
Check if your answer makes sense. The mean should fit within the range of the values of X.