Random Variables Part 3 in Statistics

There are measurements and data that may or may not be always available as discrete values. For example, the recorded daily temperatures of a city, the daily price of a stock, etc. 

Such data is typically represented by continuous values like 34.6 and 30.25. 

In the previous segments, you have seen a few examples of discrete random variables. In this segment, you will explore continuous random variables with the help of an example. 

In the upcoming video, you will be able to explore an example of continuous random variables.

In the video above, you learnt that the probability distribution of a continuous random variable is called a probability density function (PDF). It is important to understand that the value of a PDF at a given point is not the probability of getting that point. In fact, the probability of getting any single point is zero. Some properties of a PDF are as follows:

1. The probability of getting any single point is zero.

2. The probability of an interval is the area under the PDF curve in that interval.

3. The total area under the PDF curve is always equal to 1, as the total probability should be equal to 1.

Further, you need to know that the cumulative probability at a point is the probability of any value occurring less than or equal to that point. It is denoted by F(x) and can be represented in mathematical terms as follows:

$F\left(x\right)=P(X\le x)$

For a continuous random variable, the cumulative probability at a point is the area under the curve from -∞ up to that point. As the PDF never takes a negative value, the cumulative probability function is a non-decreasing function. This means that the value of the cumulative probability function either stays the same or increases as we move to the right in the graph.

In the next segment, you will be introduced to a distribution that is important and the most commonly occurring continuous probability distribution – normal distribution.