Probability Density Function (DP IB Analysis & Approaches (AA)): Revision Note

Calculating Probabilities using PDF

A continuous random variable can take any value in an interval so is typically used when continuous quantities are involved (time, distance, weight, etc)

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

• For a continuous random variable, a function can be used to model probabilities

  • This function is called a probability density function (p.d.f.), denoted by f(x)

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

  • f(x) ≥ 0 for all values of x

  • The area under the graph of y = f(x) must total 1

• In most problems, the domain of x is restricted to an interval, a ≤ X ≤ b say, with all values of x outside of the interval having f(x)=0

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

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

  
  
  

  • P(a ≤ X ≤ b) = P(a < X < b)

    • For any continuous random variable (including the normal distribution) P(X = n) = 0

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

• For linear functions it can be easier to find the probability using the area of geometric shapes

  • Rectangles: A = bh

  • Triangles: A = ½(bh)

  • Trapezoids: A = ½(a+b)h

How do I determine whether a function is a 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

    • f(x) ≥ 0 for all values of x

    • total area under the graph is 1

How do I use a pdf to find probabilities?

STEP 1

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

e.g.  

Identify the limits of X for a particular problem

Remember that P(a ≤ X ≤ b) = P(a < X < b)

STEP 2

Sketch, or use your GDC to draw, the graph of y = f(x)
Look for basic shapes (rectangles, triangles and trapezoids) as finding these areas is easier without using integration
Look for symmetry in the graph that may make the problem easier
Break the area required into two or more parts if it makes the problem easier

STEP 3

Find the area(s) required using basic shapes or integration and answer the question

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

  • i.e. Find one of the boundaries in the domain of X, given the probability

    • e.g. Find the value of a given that P(0 ≤ X ≤ a) = 0.09