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

  • 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
  

  • This is also given in the exam formula booklet

  • In the formula note that

    • from above

    • 

    • So this is equivalent to

• The formula booklet also gives the formula

  • but this is usually not as practical for solving problems

• Be careful not to confuse and

  •   ^"mean of the squares"

  •   "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

and

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