Mean & Variance of a CRV (DP IB Analysis & Approaches (AA): HL): Revision Note

Mean & variance of a CRV

What are the mean and variance of a continuous random variable?

• E(X) is the expected value, or mean, of the continuous random variable X

  • E(X) can also be denoted by μ

• Var(X) is the variance of the continuous random variable X

  • Var(X) can also be denoted by σ²

  • The standard deviation, σ, is the square root of the variance

How do I find the mean and variance of a continuous random variable?

• The mean is given by
  $\mu =\mathrm{E}\left(X\right)={\int }_{-\infty }^{\infty }xf\left(x\right)\mathrm{d}x$

  • This is given in the exam formula booklet

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

• The variance is given by
  ${\sigma }^2=\mathrm{Var}\left(x\right)={\int }_{-\infty }^{\infty }{x}^{2}f\left(x\right)\mathrm{d}x-{\mu }^2$

  • This is also given in the exam formula booklet

  • In the formula note that

    • $\mu =\mathrm{E}\left(X\right)$ from above

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

    • So this is equivalent to $\mathrm{Var}\left(X\right)=\mathrm{E}\left(X^2\right)-{\left[\mathrm{E}\left(X\right)\right]}^2$

• The formula booklet also gives the formula $\mathrm{Var}\left(x\right)={\int }_{-\infty }^{\infty }{(x-\mu )}^{2}f\left(x\right)\mathrm{d}x$

  • but this is usually not as practical for solving problems

• Be careful not to confuse $\mathrm{E}\left(X^2\right)$ and ${\left[\mathrm{E}\left(X\right)\right]}^2$

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

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

How do I find the mean and variance of a linear transformation of a continuous random variable?

• For the continuous random variable X, with mean E(X) and variance Var(X), you can find the mean and variance of the associated random variable aX + b by using

$\mathrm{E}(aX+b)=a\mathrm{E}\left(X\right)+b$

and

$\mathrm{Var}(aX+b)=a^2\mathrm{Var}\left(X\right)$

• Note that adding on a constant b affects the mean, but it doesn't affect the variance