What is the cumulative distribution function?

The cumulative distribution function (CDF) of a random variable X is defined as the probability that X takes on a value less than or equal to a given value.

The following function is used for calculating the cumulative distribution function:

Where $x$ represents a specific value of the random variable $X$ and $P$ represents the probability.

Let's take an example of a fair coin that is tossed twice. The random variable $X$ denotes the number of heads. Our task is to find the CDF of $X.$

Possible outcomes when we toss the fair coin twice are:

Let's look into the probability distribution for getting the heads.

To determine the CDF of $X$, we need to find the cumulative probabilities for each value of $X.$

Hence, the CDF of $X$ is:

Characteristics

Non-decreasing: F(x) is a non-decreasing function. $F(x)$ increases as $x$ increases.

Range: $F(x)$ ranges between 0 and 1.

Limits:

• CDF approaches 0 as $x$ approaches negative infinity

• CDF approaches 1 as $x$ approaches positive infinity

Applications

The Cumulative Distribution Function (CDF) is widely used in various disciplines, including statistics, finance, economics, and others. Some are listed below:

• Statistics: In statistics, the CDF is an essential idea. It facilitates data analysis, modeling, and inference by calculating probabilities associated with different values of a random variable.

• Finance: The CDF estimates the likelihood of various financial events like stock prices exceeding certain thresholds. This information helps investors and financial analysts make informed decisions.

Conclusion

The cumulative distribution function (CDF) is a helpful tool for determining the probabilities of random variables. It determines the probabilities associated with various experiment outcomes.