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Probabilty Density Function (PDF) – Properties,Definition and solved examples

Probabilty Density Function (PDF) Definition

Probabilty Density Function (PDF) is the more convenient representation for random variables. Each random variable has an associated PDF ( f(x) ). It records the probability associated with X as areas under its graph.

The Probabilty Density Function (PDF) is defined in terms of Cumulative Distribution Function (CDF) as

fx(x) = d Fx(x)/dx

or

Probabilty Density Function (PDF) is the differentiation of Cumulative Distribution Function (CDF).

Let us understand this PDF by example.

Let us take an example of a dice which has 6 outcomes, it mat be either 0,1,2,3,4,5 or 6. The probability of any outcome will be 1/6. But when we talk about CDF, then the probability of coming 1 is 1/6. But for outcome 2, the probability will not be same i.e 1/6. It will be 1/6 plus previous value of probability i.e probability of outcome 1 or simply individual probabilty of outcome 2 + previous probability of outcome 1 = 1/6 + 1/6 = 1/3

Similarly CDF of outcome 3 = CDF of outcome 2 + probability of 3

= 1/3+1/6 = 1/2

CDF of outcome 4 = CDF of outcome 3 + probability of 4

= 1/2 + 1/6 = 2/3

CDF of outcome 5 = CDF of outcome 4 + probability of 5

= 2/3 + 1/6 = 5/6

CDF of outcome 6 = CDF of outcome 5 + probability of 6

= 5/6 + 1/6 = 1

The CDF graph is shown below.

Now if we want to calculate the Probabilty Density Function (PDF) or if we want to calculate what will be the probability of coming 5 after looking in the graph above then it will be 1/6+1/6 +1/6+1/6+1/6 = 5/6. But that is CDF, we are calculating individual probability.

To find this we will do

Fx(5) – Fx(4) = P(5)

or we can differentiate it

fx(x) = d Fx(x)/dx

The graph shown above looks like a ramp and if differentiate ramp signal,then it becomes unit step as shown in figure below.

Properties of Probabilty Density Function (PDF) 

(1) fx(x) ≥ 0 for all x

(2) integration of PDF over an interval of – ∞ to ∞ is 1.

Solved examples of Probabilty Density Function (PDF) 

Question : A Random Variable has probability Density Function

fx(x)  = x/4 for 1≤ x ≤ 3

0 for elsewhere

Find Cumulative Distribution Function (CDF).

Solution :

Now we have got the maximum value of PDF over the range of 1 to 3 but we need a function value which covers all values. For this purpose we will replace 3 by any value i.e let’s say x which lies somewhere in the range of 1 to 3.

so integrating x/4 in the range of 1 to x we get 1/8 ( x2 – 1)

so Cumulative Distribution Function (CDF) we get

Fx(x) = 0 for x< 1

1/8 ( x2 – 1) for 1≤ x ≤ 3

1 for  x > 3