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Probability Density Function

Posted on April 7, 2020April 22, 2020 By admin No Comments on 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.

Definition

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.

Probability Distributions

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