Probability Density Function (DP IB Analysis & Approaches (AA): HL): 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 f(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

$\mathrm{P}(a\le X\le b)={\int }_a^{b}f\left(x\right)\mathrm{d}x$ 

• For any continuous random variable (including the normal distribution), the following are true:

  • $\mathrm{P}\left(a\le X\le b\right)=\mathrm{P}(a<X<b)$

    • I.e. $\le$ and $<$ are interchangeable

  • $\mathrm{P}(X=a)=0$, for any single value $a$

    • Because $\mathrm{P}(X=a)={\int }_a^{a}f\left(x\right)\mathrm{d}x=0$

• For linear functions it can be easier to calculate the probability integral using areas of geometric shapes

  • Rectangle: A = bh

  • Triangle: A = ½(bh)

  • Trapezoid: A = ½(a+b)h

How do I determine whether a function is a valid pdf?

• Some questions may ask you to justify the use of a given function as a probability density function

  • In such cases check and confirm 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.  $f\left(x\right)=\{\begin{array}{cc}0.02x& 0\le x\le 10\\ 0& \mathrm{otherwise}\end{array}$

  • 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 ask you to find one of the integration limits of the integral given the value of a probability

  • e.g. Find the value of $a$ given that $\mathrm{P}\left(0\le X\le a\right)=0.09$