Question

How do you find the probability density function of a random variable X?

Answer 1

We first describe the probability density function. We express the differences for discrete variables and continuous variables. We understand the concept with an example.
The probability distribution for a random variable describes the possible values and likelihoods over the values of the random variable. For a discrete random variable, $X$, the probability distribution is defined by a probability mass function, denoted by $f\left( x \right)$. This function provides the probability along with mean (average), standard deviation, skewness, and kurtosis for each value of the random variable.

Complete step by step solution:
For continuous variable we use the concept of probability density function instead of probability mass function. The probability density function gives the height of the function at any particular value of $X$; not giving the probability of the random variable taking on a particular value.
We take an example to understand the concept better.
Let $X$ be a continuous random variable with the following PDF.
{{f}_{X}}\left( x \right)=\left\\{ \begin{aligned} & c{{e}^{-x}}\text{ , }x\ge 0 \\\ & 0, \text{ otherwise} \\\ \end{aligned} \right.. Here c is constant.

Note: The area under the PDF curve must be equal to 1. We can see that this holds for the uniform distribution since the area under the curve is 1. The analogy tells us that just like a physical object is a collection of particles, a probability space is a collection of outcomes.