E(X) & Var(X) (Continuous) (Cambridge International A-Level Mathematics)

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

$E ( X ) = integral_{ } - \text{infinity} ^{ } \text{infinity} x f ( x ) d x$

• This is equivalent to $ΣxP(X=x)$ for discrete random variables

• If the graph of $y=f(x)$has axis of symmetry, x = a , then E(X) = a

• The variance is given by

Var ( X ) = integral _ - inf end _ ^ inf x ^ 2 f ( x ) d x - stretchy left square bracket E ( X ) stretchy right square bracket ^ 2

• This is equivalent to capital sigma x ^2 P ( X = x ) - stretchy left square bracket E ( X ) stretchy right square bracket ^2 for discrete random variables

• Be careful about confusing $E(X^{2})$ and $[ E ( X ) ]^{2}$

  • $E ( X ^{2} ) = integral_{ - } \text{infinity} ^{infinity} x ^{2} \mathrm{f} ( x ) \mathrm{d} x$ “mean of the squares”
  • $[ E ( X ) ] ^{2} = [ integral_{ - } \text{infinity} ^{infinity} x \mathrm{f} ( x ) \mathrm{d} x ] ^{2}$ “square of the mean”
  • If you are happy with the difference between these and how to calculate them the variance formula becomes very straightforward

$\text{Var} ( X ) = \mathrm{E} ( X^{2} ) - [ \mathrm{E} ( X ) ]^{2}$

How do I calculate E(g(X))?

• $\mathrm{E} ( g ( X ) ) = integral_{ - } \text{infinity} ^{infinity} g ( x ) \mathrm{f} ( x ) \mathrm{d} x$

• In particular:

  • $\mathrm{E} ( X ^{2} ) = integral_{ - } \text{infinity} ^{infinity} x ^{2} \mathrm{f} ( x ) \mathrm{d} x$as seen above

Worked Example A continuous random variable, $X$, is modelled by the probability distribution function $f(x)$, such that

f ( x ) = { | 1.5 x ^2 ( 1 - 0.5 x ) | 0 <= x <= 2 | | 0 | otherwise | close

(i) Find $E ( X )$

Answer:

(ii) Find $Var ( X )$

Answer:

Exam Hint

• 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!)