Median & Mode of a CRV (DP IB Analysis & Approaches (AA)): Revision Note

Median & Mode of a CRV

What is meant by the median of a continuous random variable?

• The median, m, of a continuous random variable, X, with probability density function f(x) is defined as the value of X such that

• Since P(X = m) = 0 this can also be written as 

• If the p.d.f. is symmetrical (i.e. if the graph of y = f(x) is symmetrical) then the median will be equal to the x-coordinate of the line of symmetry

  • In such cases the graph of y = f(x) has an axis of symmetry in the line x = m

How do I find the median of a continuous random variable?

• The median, m, of a continuous random variable, X, with probability density function f(x) is defined as the value of X such that

or

• The equation that should be used will depend on the information in the question

  • If the graph of y = f(x) is symmetrical, symmetry may be used to deduce the median

  • This may often be the case if f(x) is linear and the area under the graph is a basic shape such as a rectangle

How do I find the median of a continuous random variable with a piecewise p.d.f.?

• For piecewise functions, the location of the median will determine which equation to use in order to find it

  • For example

    • if 

    • then  so the median must lie in the interval 2 ≤ x ≤ 5

    • so to find the median, m, solve 
      ('0.4 of the area' already used for 0 ≤ x ≤ 2)

    • Use a GDC to plot the function and evalutae integral(s)

What is meant by the mode of a continuous random variable?

• The mode of a continuous random variable, X, with probability density function f(x) is the value of x that produces the greatest value of f(x)

How do I find the mode of a continuous random variable?

• This will depend on the type of function f(x); the easiest way to find the mode is by considering the shape of the graph of y = f(x)

• If the graph is a curve with a maximum point, the mode can be found by differentiating and solving f’(x) = 0

  • If there is more than one solution to f’(x) = 0 then further work may be needed in deducing the mode

    • There could be more than one mode

    • Look for valid values of x from the domain of the p.d.f.

    • Use the second derivative (f’’(x)) to deduce the nature of each stationary point

    • Check the values of f(x) at the lower and upper limits of x, one of these could be the maximum value f(x) reaches