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List of probability distributions

From Wikipedia, the free encyclopedia

Many probability distributions that are important in theory or applications have been given specific names.

YouTube Encyclopedic

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  • ✪ Overview of Some Discrete Probability Distributions (Binomial,Geometric,Hypergeometric,Poisson,NegB)
  • ✪ Probability: Types of Distributions
  • ✪ The Shape of Data: Distributions: Crash Course Statistics #7
  • ✪ Constructing a probability distribution for random variable | Khan Academy
  • ✪ 2- Types of Probability Distribution

Transcription

Let's look at a quick overview of some discrete probability distributions and their relationships. I intend this video to be used as a recap after having been introduced to these distributions, but it could possibly be used as an introductory overview. I don't any calculations in this video, nor do I discuss how to calculate the probabilities. I simply discuss how these different distributions arise, and the relationships between them. The Bernoulli distribution is the distribution of the number of successes on a single Bernoulli trial. In a Bernoulli trial we get either a success or a failure. It's like an answer to a yes or no question. A Bernoulli random variable can take on only the values 0 and 1. For example, we can use the Bernoulli distribution to answer questions like: if a single coin is tossed once, what is the probability it comes up heads? Or, if a single adult American is randomly selected, what is the probability they are a heart surgeon? Some other important distributions are built on the notion of independent Bernoulli trials, where we have a series of trials, and each one results in a success or a failure. An important one is the binomial distribution, which is the distribution of the number of successes in n independent Bernoulli trials. For example, with the binomial distribution we can answer a question like: if a coin is tossed 20 times, what is the probability heads comes up exactly 14 times? And since the binomial distribution is the distribution of the number of successes in n independent Bernoulli trials, the Bernoulli distribution is a special case of the binomial distribution with n=1, a single trial. Continuing on with the theme of independent Bernoulli trials, the geometric distribution is the distribution of the number of trials needed to get the first success. For example, with the geometric distribution we can answer a question like: if a coin has repeatedly tossed, what is the probability the first time heads appears occurs on the 8 toss? The negative binomial distribution is a generalization of the geometric distribution. The negative binomial distribution is the distribution of the number of trials needed to get a certain number of successes in repeated independent Bernoulli trials. So the negative binomial distribution can help us answer questions like: if a coin has repeatedly tossed, what is the probability the third time heads appears occurs on the ninth trial? The way the binomial distribution and the negative binomial distribution arise can sound similar, and they can sometimes be confused. They differ in what the random variable is. In the binomial distribution, the number of trials is fixed, and the number of successes is a random variable. For instance, we're tossing a coin a fixed number of times, and the number of heads that comes up is a random variable. In the negative binomial distribution, the number of successes is fixed, and the number of trials required to get that number of successes is the random variable. For instance, we might be tossing a coin until we get heads 4 times. And the number of tosses required to get heads 4 times is the random variable. Now I'll talk about two distributions that are related to the binomial, but aren't based on independent Bernoulli trials. The hypergeometric distribution is similar to the binomial distribution in that we're interested in the number of successes in n trials, but it's different because the trials are not independent. The hypergeometric distribution is the distribution of the number of successes when we are drawing without replacement from a source that contains a certain number of successes and a certain number of failures. For example, we can use the hypergeometric distribution to answer a question like: if 5 cards are drawn without replacement from a well shuffled deck, what is the probability exactly 3 hearts are drawn? It's different from the binomial because the probability of success, the probability of getting a heart, would change from card to card, depending on what happened before. However, if the cards are drawn with replacement, meaning the card was put back in and reshuffled before the next card was drawn, then the trials would be independent and we would use the binomial distribution instead. If we are sampling only a small fraction of objects without replacement from a large population then the trials are still not independent, but that dependency has only a small effect, and the binomial distribution closely approximates the hypergeometric distribution. So there are times when a problem is in its nature a hypergeometric problem, but we use the binomial distribution as an approximation. This can make our life a little bit easier sometimes. Another distribution related to the binomial is the Poisson distribution. But this one's a little harder to explain. The Poisson distribution is the distribution of the number of events in a given time or length, or area, or volume etc., if those events are occurring randomly and independently. There's a bit more to it than that, and I go into this in much greater detail in my Poisson videos. But that's the gist of it. So we might use the Poisson distribution to answer a question like: what is the probability there will be exactly 4 car accidents on a certain university campus in a given week? There is a relationship between the Poisson distribution and the binomial distribution. The Poisson distribution closely approximates the binomial distribution if n, the number of trials, in the binomial, is large and p, the probability of success, is very small. So suppose we have a question like: what is the probability that in a random sample 100,000 births, there is at least one case of progeria? Progeria is an extremely rare disease that causes premature aging, and it occurs in about 1 in every eight million births. This is truly a binomial problem. But we have a binomial problem with a very large n, 100,000, and a very small probability of success, 1 in eight million or so, because progeria such a rare disease. And so this could be very well approximated by the Poisson distribution. I look into all of these concepts discussed in this video in greater detail in the videos for these specific distributions.

Contents

Discrete distributions

With finite support

With infinite support

Continuous distributions

Supported on a bounded interval

Supported on intervals of length 2π – directional distributions

Supported on semi-infinite intervals, usually [0,∞)

Supported on the whole real line

With variable support

Mixed discrete/continuous distributions

Joint distributions

For any set of independent random variables the probability density function of their joint distribution is the product of their individual density functions.

Two or more random variables on the same sample space

Distributions of matrix-valued random variables

Non-numeric distributions

Miscellaneous distributions

See also

This page was last edited on 14 September 2019, at 21:27
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