Memoryless Property of Exponential Distribution

The exponential distribution has an interesting memoryless property.

The probability distribution of the remaining time until the event occurs is always the same regardless of the time that passed.

There is no memory in the process. The history is not relevant. The time to next arrival is not influenced by when the last event or arrival occurred.

This property is unique to the exponential and the geometric distributions.

Learn about this in detail in lesson 44. I promise you will never forget it.

Lesson 44 – Keep waiting: The memoryless property of exponential distribution

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Exponential distributions: Waiting Time

Your every day “waiting for” experiences are exponential distributions. I explain how in lesson 43. You will find at least one thing in common.

You will also learn how to derive the probability functions of exponential distribution from first principles.

It is a fun and educational read. Trust me.

Lesson 43 – Wait time: The language of exponential distribution

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Beta Distribution and Uniform Distribution

The function I asked you to solve last week is a Beta distribution function. It is defined in the 0 to 1 range. The beta distribution is a bounded distribution. The function is 0 everywhere else.

In lesson 42, we learn the family of Beta distribution and how it relates to the uniform distribution. We solve the function from last week which leads us to the basics of Beta distribution.

Learn more here. There is also a cool animation to make things clear.

Lesson 42 – Bounded: The language of Beta distribution

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The basics of continuous probability distributions

The characteristic feature in all the discrete distributions is that the random variable X is discrete. The possible outcomes are distinct numbers, which is why we called them discrete probability distributions.

Have you asked yourself, “what if the random variable X is continuous?” What is the probability that X can take any particular value x on the real number line which has infinite possibilities?

For a continuous random variable, the number of possible outcomes is infinite, hence,

P(X = x) = 0.

For continuous random variables, the probability is defined in an interval between two values. It is computed using continuous probability distribution functions.

Learn more about these fundamentals in Lesson 41.

Lesson 41 – Struck by a smooth function

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R coding for Geometric, Negative Binomial and Poisson distributions

Today’s temperature in New York is below 30F — a cold November day.

  1. Do you want to know what the probability of a cold November day is?
  2. Do you want to know what the return period of such an event is?
  3. Do you want to know how many such events happened in the last five years?

Get yourself some warm tea. Let the room heater crackle. We are diving into rest of the discrete distributions in R. The lesson with complete code is here. Happy coding.

Lesson 40 – Discrete distributions in R: Part II

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Learn discrete distributions and how to create Animations in R

Today’s lesson includes a journey through Bernoulli trials and Binomial distribution in R.

I use data from New York City’s parking violations. Since we are learning discrete probability distributions, the violation tickets data can serve as a neat example.

We also learn how to create GIFs in R. We first save the plots as “.png” files and then combine them into a GIF using the “animation” and “magick”packages.

The lesson with complete code is here. Happy coding.

Lesson 39 – Discrete distributions in R: Part I

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What is Hypergeometric Distribution?

If there are R Pepsi cans in a total of N cans (N-R Cokes) and we are asked to identify them correctly, in our choice selection of R Pepsi, we can get k = 0, 1, 2, … R Pepsi. The number of correct guesses and the probability of correctly selecting k Pepsi cans is Hypergeometric distribution.

Hypergeometric distribution is typically used in quality control analysis for estimating the probability of defective items out of a selected lot.

The Pepsi-Coke marketing analysis is another example application. Companies can analyze the preferences of one product to other among a subset of customers in their region.

Learn more about Hypergeometric distribution and how to derive the probability from the ground up in lesson 38 of our data analysis classroom.

Lesson 38 – Correct guesses: The language of Hypergeometric distribution

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Poisson distribution

If we assume events are independent (the occurrence of one event does not affect the probability that a second event will occur), then the counts per unit interval can be assumed a random variable that follows a probability distribution. Counts, i.e., the number of times an event occurs in an interval follows a Poisson distribution.

Poisson distribution has one control parameter. It is the rate of occurrence; the average number of events per unit interval.

In lesson 36, we learn the fundamentals of Poisson distribution. You will also meet Able and Mumble two of my friends.

Lesson 36 – Counts: The language of Poisson distribution