Probability Density Functions in Statistics- I

In the last section, you saw how to find the probability of certain events using the multiplication and addition rules of probability. Also, for some specific cases, you saw that probability distributions like binomial distribution and uniform distribution can be used to calculate probability.

So far, we have only talked about discrete random variables, e.g., number of balls, number of patients, cars, wickets, pasta packets, etc. What happens when we talk about the probability of continuous random variables, such as time, weight etc.? Is there any difference? Let’s find out.

Now you know what CDF and PDF are. Since these two functions talk about probabilities in terms of intervals rather than the exact values, it is advisable to use them when talking about continuous random variables, not the bar chart distribution that we used for discrete variables.

Recall that a CDF, or a cumulative distribution function, is a distribution that plots the cumulative probability of X against X.

A PDF, or a Probability Density Function, however, is a function in which the area under the curve gives you the cumulative probability.

For example, the area under the curve between 20, the smallest possible value of X, and 28 gives the cumulative probability for X, which is equal to 28.

The main difference between the cumulative probability distribution of a continuous random variable and a discrete one lies in the way you plot them. While a continuous variables’ cumulative distribution is a curve, a distribution for discrete variables looks more like a bar chart.

The reason for the difference is that for discrete variables, the cumulative probability does not change very frequently. In the discrete variable example, we only care about what the probability is for 0, 1, 2, 3 and 4. This is because the cumulative probability will not change between, say, 3 and 3.999999. For all values between these two, the cumulative probability is equal to 0.8704.

However, for the continuous variable, i.e., the daily commute time, you have a different cumulative probability value for every value of X. For example, the value of cumulative probability at 21 will be different from its value at 21.1, which will again be different from the one at 21.2, and so on. Hence, you would show its cumulative probability as a continuous curve, not a bar chart.