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How do you check if a function is a valid CDF and what are the properties of a CDF?

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A cumulative distribution function (CDF) is a function that gives the probability that a random variable is less than or equal to a given value. CDFs are useful for describing the behavior and characteristics of random phenomena, such as the outcomes of experiments, the distribution of data, and the reliability of systems. In this article, you will learn how to find the CDF of a discrete or continuous random variable, how to check if a function is a valid CDF, and what are the properties of a CDF.

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Tanmayan Pande

Software Engineer at Google Cloud
Lokesh Parab

Data Scientist | Team Leader | Prompt Engineer | OpenAI | Llama 3 | Langchain | PowerBI | Tableau | Looker | AWS | GCP…

1 Finding the CDF of a discrete random variable

A discrete random variable is one that can take only a finite or countable number of values, such as the number of heads in a coin toss, the number of students in a class, or the color of a card. To find the CDF of a discrete random variable, you need to know its probability mass function (PMF), which is a function that gives the probability of each possible value. The CDF is then obtained by summing up the PMF values from the lowest value to the given value. For example, if X is the number of heads in two coin tosses, its PMF is:

P(X = 0) = 0.25

P(X = 1) = 0.5

P(X = 2) = 0.25

The CDF of X is then:

F(X <= 0) = 0.25

F(X <= 1) = 0.25 + 0.5 = 0.75

F(X <= 2) = 0.25 + 0.5 + 0.25 = 1

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Tanmayan Pande

Software Engineer at Google Cloud
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In fact, one can say that the CDF is a more basic property of a probability distribution, with a PMF being the derivative of the CDF. The CDF is defined from first principles in set theory, while the PDF has no such definition (in the case of the continuous probability distribution, think of this as the ratio between the cardinalities of subsets and a larger universal set)

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2 Finding the CDF of a continuous random variable

A continuous random variable is one that can take any value in a given interval, such as the height of a person, the weight of a fruit, or the time of arrival of a bus. To find the CDF of a continuous random variable, you need to know its probability density function (PDF), which is a function that gives the relative likelihood of each possible value. The CDF is then obtained by integrating the PDF from the lowest value to the given value. For example, if Y is the height of a person in centimeters, and its PDF is:

f(Y) = 0.01 for 150 <= Y <= 200

f(Y) = 0 for Y < 150 or Y > 200

The CDF of Y is then:

F(Y <= y) = 0 for y < 150

F(Y <= y) = 0.01 * (y - 150) for 150 <= y <= 200

F(Y <= y) = 1 for y > 200

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Lokesh Parab

Data Scientist | Team Leader | Prompt Engineer | OpenAI | Llama 3 | Langchain | PowerBI | Tableau | Looker | AWS | GCP | Azure
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In the case of Discrete Random Variable For example : Let X be the possible outcomes from tossed Dice. i.e X --> { 1, 2, 3, 4, 5, 6 } & P(X) = 1/6 So CDF in Discrete Random Variable will be F(x) = P{ X =< x } For e.g. x = 3 F(3) = P{ X =< 3 ) = P(1) + P(2) + P(3) = 3/6 In the case of Continous Random Variable Instead of Summation, we have to Integrate with probability density function F(x) = P{ X =< x } = ∫ f(x) dx where f(x) is probability density function and x range from ?∞ to ∞

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3 Checking if a function is a valid CDF

In order for a function to be considered a Cumulative Distribution Function (CDF) of a random variable, it must meet certain conditions. The function must be non-decreasing, right-continuous, and have limits of 0 and 1 at the extremes. If the function does not satisfy these conditions, it is not a valid CDF. As an example, the function G(x) = 0.5 * sin(x) + 0.5 does not meet any of these conditions and is therefore not a valid CDF.

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4 Properties of a CDF

A CDF is a useful tool for analyzing and manipulating random variables and their probabilities. It is equal to the probability of the event X <= x for any value x. Additionally, the probability of any interval (a, b] for a random variable X is equal to the difference between the CDF values at b and a, that is P(a < X <= b) = F(b) - F(a). Moreover, the probability of a complement event X > x is equal to one minus the CDF value at x, that is P(X > x) = 1 - F(x). Furthermore, if X and Y are independent, then F(X + Y) = F(X) * F(Y), where * denotes the convolution operator. Lastly, if Z = g(X), then F(Z) = F(X(g^-1(Z))). Understanding and using CDFs can help you model random phenomena in various fields of science, engineering, and statistics. You can also use CDFs to generate random samples from a desired distribution by applying the inverse transformation method.

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How do you check if a function is a valid CDF and what are the properties of a CDF?

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1 Finding the CDF of a discrete random variable

2 Finding the CDF of a continuous random variable

3 Checking if a function is a valid CDF

4 Properties of a CDF

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