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 formula booklet

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

• The variance is given by

  
  

  
  where 
  
  
  

  • This is given in the formula booklet

  • Another version of the variance is given in the formula booklet

    
    
    

  • but the first version above is usually more practical for solving problems

• Be careful about confusing E(X²) and [E(X)]²

  •    ^"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) then

and