What is CDF?

CDF stands for Cumulative Distribution Function. In the field of probability theory and statistics, the CDF is a function that describes the probability that a random variable falls within a certain range.

The CDF is defined for both continuous and discrete random variables.

For a continuous random variable X, the CDF is defined as:

F(x) = P(X <= x)

This is the probability that the random variable X takes on a value less than or equal to x.

For a discrete random variable, the CDF is the sum of the probabilities of all outcomes less than or equal to x.

CDF has the following properties:

1. It's non-decreasing: As you move from left to right (i.e., as you increase x), the CDF doesn't decrease.

2. It approaches 1 as you go to positive infinity: This is because the CDF at a point x is the probability that the random variable is less than or equal to x. As x becomes large, it becomes more likely that the random variable is less than or equal to x.

3. It approaches 0 as you go to negative infinity: For the same reason, as x becomes very negative, it becomes less likely that the random variable is less than or equal to x.

4. It's right-continuous: This is a technical condition that is required for some mathematical reasons.

The CDF is a fundamental tool in the field of probability theory and statistics and is used to describe the distribution of random variables. You can also derive many other useful functions from the CDF, such as the probability density function (PDF) for continuous variables or the probability mass function (PMF) for discrete variables.

Explanation of CDF in simples Words:

Imagine you have a bag full of marbles, and each marble has a number on it. The CDF, or Cumulative Distribution Function, tells you how many marbles have a number less than or equal to a certain number.

For example, if you pull a marble from the bag and it has the number 10 on it, the CDF tells you the chance that you will pull out a marble with a number that is 10 or less.

Let's consider another example. Let's say you have a bag with 100 marbles and their numbers are from 1 to 100. If you wanted to know the chance of drawing a marble with the number 50 or less, you would see that there are 50 such marbles, so the chance (or CDF for 50) would be 50/100, or 50%.

So, in simple terms, the CDF is like a rule or function that tells you the chance (or probability) of getting a certain number or less from a bag of numbered marbles.