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
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).
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