E(X) & Var(X) (Continuous)
What are E(X) and Var(X)?
- E(X)is the expected value, or mean, of a random variable X
- E(X) is the same as the population mean so can also be denoted by µ
- Var (X) is the variance of the continuous random variable X
- Standard deviation is the square root of the variance
How do I find the mean and variance of a continuous random variable?
- The mean, for a continuous random variable X is given by
- This is equivalent to
for discrete random variables
- If the graph of
has axis of symmetry, x = a , then E(X) = a
- The variance is given by
- This is equivalent to
for discrete random variables
- Be careful about confusing
and
“mean of the squares”
“square of the mean”
- If you are happy with the difference between these and how to calculate them the variance formula becomes very straightforward
How do I calculate E(g(X))?
- In particular:
as seen above
- If
(a linear function) then
Worked example
A continuous random variable, X, is modelled by the probability distribution function f(x), such that
(a)
Find
.
(b)
Find
.
(a)
Find
.
(b)
Find
.
![1-3-2-ial-fig1-we-solution-part-2](https://cdn.savemyexams.com/cdn-cgi/image/f=auto,width=1920/uploads/2022/05/1-3-2-ial-fig1-we-solution-part-2.png)
Exam Tip
- A sketch of the graph of y = f(x) can highlight any symmetrical properties which can help reduce the work involved in finding the mean and variance
- Take care with awkward values and negatives – use the memory features on your calculator and avoid rounding until your final answer (if rounding at all!)
- The formulae for
and
are given in the formulae booklet