Probability Density Function

Earlier we used Probability Mass Function to describe how the total probability of 1 is distributed among the possible values of the Discrete Random Variable X. But we cannot define Probability Mass Function for a Continous Random Variable. We use the Probability Density Function to show the distribution of probabilities for a continuous random variable.


Let X be a continuous random variable. Then a probability distribution function (pdf) of X is a function f(x); such that for any two numbers a and with a ≤ b;

probability density function

f(x) should satisfy the following 2 conditions:

  1. f(x) > 0 for all x
  2. Integral \int_{a}^{b} f(x) dx is equal to the area under the graph of f(x) which is equal to 1

Detailed Example

Let X be a continous random variable whose probability density function is;

f(x) = 2x2 for 0< x <1

But f(x) ≠ P(X = x)

ex : f(2) = 2(2)2 = 4 this is clearly not a probability.

f(x) is the height of the curve at X = x so that the area under the curve is 1.

In our future posts, we will be discussing about several probability density functions such as Uniform Distribution, Normal Distribution, Gamma Distribution etc.

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