Probability Density Function (Edexcel International A Level (IAL) Maths: Statistics 2): 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 x  (such as f(x)  or g(x) ) and is usually given piecewise

e.g.

$\mathrm{f}\left(x\right)=\{\begin{array}{ll}x& 0\le x\le 1\\ 1& 1.5\le x\le 2\\ 0& \mathrm{otherwise}\end{array}$

• f(x) should be defined for all values of $x\in \mathrm{ℝ}$

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

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

• The graph does not have to be continuous

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

  • $\mathrm{f}\left(x\right)\ge 0$ for all values of x

    • This is the equivalent to $P(X=x)\ge 0$  for a discrete random variable

  • The area under the graph must total 1

${\int }_{-\infty }^{\infty }\mathrm{f}\left(x\right)\mathrm{d}x=1$

• This is equivalent to $\mathrm{\Sigma }$ $\mathrm{P}(X=x)=1$ 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 X lies in the interval
  $a\le X\le b$, where X has the probability density function f(x) , is given by

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

• As with the normal distribution $P(a\le X\le b)=P(a<X<b)$

  • for any continuous random variable, $\mathrm{P}(X=n)=0$  for all values of n

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

• Piecewise Function are often used as the p.d.f. is not often a single function of x

  • Finding a probability may involve splitting the area across more than one piece of the function

  • This will depend on the limits a and b in $P(a\le X\le b)$ that is being found  

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 $\mathrm{f}\left(\mathrm{x}\right)\ge 0$ for all values of $x$ and the total area under the graph is 1

• If asked to find a probability

  • STEP 1
    As the probability density function, f(x), is usually given piecewise make sure you are clear about the values of x for which each part applies

                                                e.g.  $\mathrm{f}\left(\mathrm{x}\right)=\{\begin{array}{l}x-11\le x\le 2\\ 3-x2\le x\le 3\\ 0\mathrm{otherwise}\end{array}$      

• STEP 2

If simple to do so, sketching the graph of y= f(x) 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

• STEP 3

  • Identify the range of X, particularly noting if it is split across different parts of the p.d.f.

  • Find the required area (probability), either by basic shapes or integrate f(x) and evaluate it between the two limits, splitting if necessary

• 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 $\mathrm{P}(0\le X\le a)=0.09$

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

$\mathbf{P}\mathbf{(}\mathbf{X}\mathbf{<}\mathbf{m}\mathbf{)}\mathbf{=}\mathbf{P}\mathbf{(}\mathbf{X}\mathbf{>}\mathbf{m}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{5}$

•  Since $\mathrm{P}(X=m)=0$ this can also be written as $P(X\le m)=P(X\ge m)=0.5$

• 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

 ${\int }_{-\infty }^m\mathrm{f}\left(x\right)\mathrm{dx}=0.5$

                                and

 ${\int }_m^{\infty }\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=0.5$

• If the graph of $y=\mathrm{f}\left(x\right)$ is symmetrical, symmetry may be used to deduce the median

• For piecewise functions, you will need to determine which part of the function the median lies within to determine which equation to use

  • If there are more than two (non-zero) parts to a function then the integration may need splitting

• You can also use the cumulative distribution function to find the median

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

• In a similar way to finding 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 find 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

• In all cases start by determining which part(s) of the function are involved

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 y= 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 $x$ 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