E(X) & Var(X) (Continuous) (Cambridge (CIE) A Level Maths: Probability & Statistics 2): Revision Note

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

$\mathbf{E}\mathbf{\left(}\mathit{X}\mathbf{\right)}\mathbf{=}{\mathbf{\int }}_{\mathbf{-}\mathbf{\infty }}^{\mathbf{\infty }}\mathit{x}\mathbf{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{d}\mathit{x}$

• This is equivalent to $\mathrm{\Sigma }x\mathrm{P}(X=x)$ for discrete random variables

• If the graph of $y=\mathrm{f}\left(x\right)$has axis of symmetry, x = a , then  E(X) = a

• The variance is given by

$\mathbf{Var}\mathbf{\left(}\mathit{X}\mathbf{\right)}\mathbf{=}{\mathbf{\int }}_{\mathbf{-}\mathbf{\infty }}^{\mathbf{\infty }}{\mathit{x}}^{\mathbf{2}}\mathbf{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{d}\mathit{x}\mathbf{-}{\left[E\left(X\right)\right]}^{\mathbf{2}}$

• This is equivalent to $\mathrm{\Sigma }{x}^2\mathrm{P}(X=x)-{\left[E\left(X\right)\right]}^2$  for discrete random variables

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

  • $E\left(X^2\right)={\int }_{-\infty }^{\infty }{x}^2\mathrm{f}\left(x\right)\mathrm{d}x$                “mean of the squares”

  • ${\left[E\left(X\right)\right]}^2={\left[{\int }_{-\infty }^{\infty }x\mathrm{f}\left(x\right)\mathrm{d}x\right]}^2$       “square of the mean”

  • If you are happy with the difference between these and how to calculate them the variance formula becomes very straightforward

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

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

• $\mathrm{E}\left(g\left(X\right)\right)={\int }_{-\infty }^{\infty }g\left(x\right)\mathrm{f}\left(x\right)\mathrm{d}x$

• In particular:

  • $\mathrm{E}\left(X^2\right)={\int }_{-\infty }^{\infty }{x}^2\mathrm{f}\left(x\right)\mathrm{d}x$as seen above