Summary: Continuous Probability Distributions

You started this session by learning that for a continuous random variable, the probability of getting an exact value is very low, almost zero. Hence, when talking about the probability of continuous random variables, you can only talk in terms of intervals. For example, for a particular company, the probability of an employee’s commute time being exactly equal to 35 minutes was zero, but the probability of an employee having a commute time between 35 and 40 minutes was 0.2.

Hence, for continuous random variables, probability density functions (PDFs) and cumulative distribution functions (CDFs) are used instead of the bar chart type of distribution used for the probability of discrete random variables. These functions are preferred because they talk about probability in terms of intervals.

Then, you understood that the major difference between a PDF and a CDF is that in a CDF, you can find the cumulative probability directly by checking the value at x. However, for a PDF, you need to find the area under the curve between the lowest value and x to find the cumulative probability.

You also learnt that PDFs are still more commonly used, mainly because it is very easy to see patterns in them. For example, for a uniformly distributed variable, the PDF and CDF look like this:

While the PDF clearly shows that the variable is uniformly distributed, the CDF does not offer any such quick insights.

Next, you learnt about a very famous probability density function: the normal distribution. You saw that it is symmetric, and its mean, median and mode lie at the centre.

You also learnt the 1-2-3 rule, which states that there is a:

1. 68% probability of the variable lying within 1 standard deviation of the mean,

2. 95% probability of the variable lying within 2 standard deviations of the mean, and

3. 99.7% probability of the variable lying within 3 standard deviations of the mean.

Then, you learnt that to find the probability, you do not need to know the value of the mean or the standard deviation; just knowing the number of standard deviations away from the mean your random variable is suffices. That is given by:

$Z=\left(\frac{X-\mu }{\sigma }\right)$

This is called the Z score, or the standard normal variable.

Finally, you learnt how to find the cumulative probability for various values of Z using the Z table. For example, you found the cumulative probability for Z = 0.68 using the Z table.

The intersection of row '0.6' and column '0.08', i.e., 0.7517, is your answer.

Also, you learnt how to use Excel to find this probability. For example, the cumulative probability for Z =1.5 can be found using Excel by typing:

= NORM.S.DIST(1.5, TRUE)

Also, you can find the probability without standardising. The syntax for that is:

= NORM.DIST(x, mean, standard_dev, TRUE)

A normal distribution finds use in many statistical analyses. In the next session, you will learn about its use in the central limit theorem, which is, in turn, useful for understanding the next module on hypothesis testing.