# E(X) & Var(X) (Continuous)(CIE A Level Maths: Probability & Statistics 2)

Author

Paul

Expertise

Maths

## 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

#### Worked example

A continuous random variable, , is modelled by the probability distribution function , such that

(i)
Find

(ii)
Find

(i)
Find

(ii)
Find

#### 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!)

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