Probability Density Function (Cambridge (CIE) A Level Maths): Revision Note

Calculating Probabilities using PDF

What is a probability density function (p.d.f.)?

• For a continuous random variable , it is often possible to model probabilities using a function

  • This function is called a probability density function (p.d.f.)

  • For the continuous random variable,X , it would usually be denoted as a function of   (such as f()  or g() )

• The distribution (or density) of probabilities can be illustrated by the graph of f()

• The graph does not need to start and end on the x-axis

• For f() to represent a p.d.f. the following conditions must apply 

  • for all values of 

    • This is the equivalent to   for a discrete random variable

  • The area under the graph must total 1

• This is equivalent to  for a discrete random variable

How do I find probabilities using a probability density function (p.d.f.)?

• The probability that the continuous random variable  lies in the interval
  , where has the probability density function f() , is given by

• As with the normal distribution 

  • for any continuous random variable,   for all values of n

  • One way to think of this is that  in the integral above

How do I solve problems using the PDF?

• Some questions may ask for justification of the use of a given function for a probability density function

  • In such cases check that the function meets the two conditions  for all values of  and the total area under the graph is 1

• If asked to find a probability

  • STEP 1
    Identify the probability density function, f(), this may be given as a graph, an equation or as a piecewise function

                                                e.g.        

• STEP 2
  Identify the range of  for a particular problem

                            Remember that 

                            Question: Can you explain why this is so?
                            (Answer is at end of this section)

• STEP 3
  Sketching the graph of y = f() if simple may help to find the probability

  • Look for basic shapes such as triangles or rectangles; finding areas of these is easy and avoids integration

  • Look for symmetry in the graph that may make the problem easier

  • Integrate f() and evaluate it between the two limits for the required probability

• Trickier problems may involve finding a limit of the integral given its value

  • i.e. one of the values in the range of X, given the probability
    e.g.        Find the value of  given 

• Answer to question in STEP 2:
  Since 

Median and 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 the continuous random variable X, such that

•  Since  this can also be written as 

• If the p.d.f. is symmetrical (i.e. the graph of y = f(x) is symmetrical) then the median will be halfway between the lower and upper limits of x

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

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

• By solving one of the equations to find m

                                and

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

• If the graph of  is symmetrical, symmetry may be used to deduce the median

How do I find quartiles (or percentiles) of a continuous random variable?

• In a similar way, to find the median

  • The lower quartile will be the value L such that P(X ≤ L) = 0.25 or
    P(X ≥ L) = 0.75

  • The upper quartile will be the value U such that P(X ≤ U) = 0.75 or
    P(X ≥ U) = 0.25

• Percentiles can be found in the same way

  • The 15^th percentile will be the value k such that P(X ≤ k) = 0.15 or
    P(X ≥ k) = 0.85

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 PDF?

• 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 f(x)

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

  • If there is more than one solution to f'(x) =  0 , further work may be needed to deduce which answer is the mode

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

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

    • You may need to check the values of f(x) at the endpoints too

(a) Find the median of X, giving your answer to three significant figures

(b) Find the exact value of the mode of X