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Probability distribution

From Wikipedia, the free encyclopedia

In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. For instance, if the random variable X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 for X = heads, and 0.5 for X = tails (assuming the coin is fair). Examples of random phenomena can include the results of an experiment or survey.

A probability distribution is specified in terms of an underlying sample space, which is the set of all possible outcomes of the random phenomenon being observed. The sample space may be the set of real numbers or a set of vectors, or it may be a list of non-numerical values; for example, the sample space of a coin flip would be {heads, tails} .

Probability distributions are generally divided into two classes. A discrete probability distribution (applicable to the scenarios where the set of possible outcomes is discrete, such as a coin toss or a roll of dice) can be encoded by a discrete list of the probabilities of the outcomes, known as a probability mass function. On the other hand, a continuous probability distribution (applicable to the scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day) is typically described by probability density functions (with the probability of any individual outcome actually being 0). The normal distribution is a commonly encountered continuous probability distribution. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

A probability distribution whose sample space is one-dimensional (for example real numbers, list of labels, ordered labels or binary) is called univariate, while a distribution whose sample space is a vector space of dimension 2 or more is called multivariate. A univariate distribution gives the probabilities of a single random variable taking on various alternative values; a multivariate distribution (a joint probability distribution) gives the probabilities of a random vector – a list of two or more random variables – taking on various combinations of values. Important and commonly encountered univariate probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.

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Transcription

- [Voiceover] Let's define a random variable x as being equal to the number of heads, I'll just write capital H for short, the number of heads from flipping coin, from flipping a fair coin, we're gonna assume it's a fair coin, from flipping coin five times. Five times. Like all random variables this is taking particular outcomes and converting them into numbers. And this random variable, it could take on the value x equals zero, one, two, three, four or five. And I what want to do is figure out what's the probability that this random variable takes on zero, can be one, can be two, can be three, can be four, can be five. To do that, first let's think about how many possible outcomes are there from flipping a fair coin five times. Let's think about this. Let's write possible outcomes. Possible outcomes from five flips. From five flips. These aren't the possible outcomes for the random variable, this is literally the number of possible outcomes from flipping a coin five times. For example, one possible outcome could be tails, heads, tails, heads, tails. Another possible outcome could be heads, heads, heads, tails, tails. That is one of the equally likely outcomes, that's another one of the equally likely outcomes. How many of these are there? For each flip you have two possibilities. Let's write this down. Let me... The first flip, the first flip there's two possibilities, times two for the second flip, times two for the third flip. Actually maybe we'll not use the time notation, you might get confused with the random variable. Two possibilities for the first flip, two possibilities for the second flip, two possibilities for the third flip, two possibilities for the fourth flip, and then two possibilities for the fifth flip, or two to the fifth equally likely possibilities from flipping a coin five times, which is, of course, equal to 32. This is going to be helpful because for each of the values that the random variable can take on, we just have to think about how many of these equally likely possibilities would result in the random variable taking on that value. Let's just delve into it to see what we're actually talking about. I'll do it in this light, let me do it in... I'll start in blue. Let's think about the probability that our random variable x is equal to one. Well actually, let me start with zero. The probability that our random variable x is equal to zero. That would mean that you got no heads out of the five flips. Well there's only one way, one out of the 32 equally likely possibilities, that you get no heads. That's the one where you just get five tails. So this is just going to be, this is going to be equal to one out of the 32 equally likely possibilities. Now, for this case, to think in terms of binomial coefficients, and combinatorics, and all of that, it's much easier to just reason through it, but just so we can think in terms it'll be more useful as we go into higher values for our random variable. This is all buildup for the binomial distribution, so you get a sense of where the name comes from. So let's write it in those terms. This one, this one, this one right over here, one way to think about that in combinatorics is that you had five flips and you're choosing zero of them to be heads. Five flips and you're choosing zero of them to be heads. Let's verify that five choose zero is indeed one. So five choose zero. Write it over here. Five choose zero is equal to five factorial over, over five minus zero factorial. Well actually over zero factorial times five minus zero factorial. Well zero factorial is one, by definition, so this is going to be five factorial, over five factorial, which is going to be equal to one. Once again I like reasoning through it instead of blindly applying a formula, but I just wanted to show you that these two ideas are consistent. Let's keep going. I'm going to do x equals one all the way up to x equals five. If you are inspired, and I encourage you to be inspired, try to fill out the whole thing, what's the probability that x equals one, two, three, four or five. So let's go to the probability that x equals two. Or sorry, that x equals one. The probability that x equals one is going to be equal to... Well how do you get one head? It could be, the first one could be head and then the rest of them are gonna be tails. The second one could be head and then the rest of them are gonna be tails. I could write them all out but you can see that there's five different places to have that one head. So five out of the 32 equally likely outcomes involve one head. Let me write that down. This is going to be equal to five out of 32 equally likely outcomes. Which of course is the same thing, this is going to be the same thing as saying I got five flips, and I'm choosing one of them to be heads. So that over 32. You could verify that five factorial over one factorial times five minus-- Actually let me just do it just so that you don't have to take my word for it. So five choose one is equal to five factorial over one factorial, which is just one, times five minus four-- Sorry, five minus one factorial. Which is equal to five factorial over four factorial, which is just going to be equal to five. All right, we're making good progress. Now in purple let's think about the probability that our random variable x is equal to two. Well this is going to be equal to, and now I'll actually resort to the combinatorics. You have five flips and you're choosing two of them to be heads. Over 32 equally likely possibilities. This is the number of possibilities that result in two heads. Two of the five flips have chosen to be heads, I guess you can think of it that way, by the random gods, or whatever you want to say. This is the fraction of the 32 equally likely possibilities, so this is the probability that x equals two. What's this going to be? I'll do it right over here. And actually no reason for me to have to keep switching colors. So five choose two is going to be equal to five factorial over two factorial times five minus two factorial. Five minus two factorial. So this is five factorial over two factorial times three factorial. And this is going to be equal to five times four times three times two, I could write times one but that doesn't really do anything for us. Then two factorial's just going to be two. Then the three factorial is three times two. I could write times one, but once again doesn't do anything for us. That cancels with that. Four divided by two is two. Five times two is 10. So this is equal to 10. This right over here is equal to 10/32. 10/32. And obviously we could simplify this fraction, but I like to leave it this way because we're now thinking everything is in terms of 32nds. There's a 1/32 chance x equals zero, 5/32 chance that x equals one and a 10/32 chance that x equals two. Let's keep on going. I'll go in orange. What is the probability that our random variable x is equal to three? Well this is going to be five, out of the five flips we're going to need to choose three of them to be heads to figure out which of the possibilities involve exactly three heads. And this is over 32 equally likely possibilities. And this is going to be equal to, five choose three is equal to five factorial over three factorial times five minus three factorial. Let me just write it down. Five minus three factorial, which is equal to five factorial over three factorial times two factorial. That's exactly what we had up here and we just swapped three and the two, so this also is going to be equal to 10. So this is also going to be equal to 10/32. All right, two more to go. And I think you're going to start seeing a little bit of a symmetry here. One, five, 10, 10, let's keep going. Let's keep going, and I haven't used white yet. Maybe I'll use white. The probability that our random variable x is equal to four. Well, out of our five flips we want to select four of them to be heads, or out of the five-- We're obviously not actively selecting. One way to think of it, we want to figure out the possibilities that involve out of the five flips, four of them are chosen to be heads, or four of them are heads. And this is over 32 equally likely possibilities. So five choose four is equal to five factorial over four factorial times five minus four factorial which is equal to, well that's just going to be five factorial, this is going to be one factorial right over here. That doesn't change the value, you just multiply one factorial times four factorial, so it's five factorial over four factorial, which is equal to five. So once again this is 5/32. And you could have reasoned through this because if you're saying you want five heads, that means you have one tail. There's five different places you could put that one tail. There are five possibilities with one tail. Five of the 32 equally likely. And then, and you could probably guess what we're gonna get for x equals five because having five heads means you have zero tails, and there's only gonna be one possibility out of the 32 with zero tails, where you have all heads. Let's write that down. The probability, the probability that our random variable x is equal to five. So we have all five heads. You could say this is five and we're choosing five of them to be heads. Out of the 32 equally likely possibilities. Well five choose five, that's going to be... Let me just write it here since I've done it for all of the other ones. Five choose five is five factorial over five factorial times five minus five factorial. Well this right over here is zero factorial, which is equal to one, so this whole thing simplifies to one. This is going to be one out of-- 1/32. So you see the symmetry. 1/32, 1/32. 5/32, 5/32; 10/32, 10/32. And that makes sense because the probability of getting five heads is the same as the probability of getting zero tails, and the probability of getting zero tails should be the same as the probability of getting zero heads. I'll leave you there for this video. In the next video we'll graphically represent this and we'll see the probability distribution for this random variable.

Contents

Introduction

The probability mass function (pmf) p(S) specifies the probability distribution for the sum S of counts from two dice.  For example, the figure shows that p(11) = 2/36 = 1/18.  The pmf allows the computation of probabilities of events such as P(S > 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.
The probability mass function (pmf) p(S) specifies the probability distribution for the sum S of counts from two dice. For example, the figure shows that p(11) = 2/36 = 1/18. The pmf allows the computation of probabilities of events such as P(S > 9) = 1/12 + 1/18 + 1/36 = 1/6, and all other probabilities in the distribution.

To define probability distributions for the simplest cases, it is necessary to distinguish between discrete and continuous random variables. In the discrete case, it is sufficient to specify a probability mass function assigning a probability to each possible outcome: for example, when throwing a fair dice, each of the six values 1 to 6 has the probability 1/6. The probability of an event is then defined to be the sum of the probabilities of the outcomes that satisfy the event; for example, the probability of the event "the dice rolls an even value" is

In contrast, when a random variable takes values from a continuum then typically, any individual outcome has probability zero and only events that include infinitely many outcomes, such as intervals, can have positive probability. For example, the probability that a given object weighs exactly 500 g is zero, because the probability of measuring exactly 500 g tends to zero as the accuracy of our measuring instruments increases. Nevertheless, in quality control one might demand that the probability of a "500 g" package containing between 490 g and 510 g should be no less than 98%, and this demand is less sensitive to the accuracy of measurement instruments.

Continuous probability distributions can be described in several ways. The probability density function describes the infinitesimal probability of any given value, and the probability that the outcome lies in a given interval can be computed by integrating the probability density function over that interval. On the other hand, the cumulative distribution function describes the probability that the random variable is no larger than a given value; the probability that the outcome lies in a given interval can be computed by taking the difference between the values of the cumulative distribution function at the endpoints of the interval. The cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists.

The probability density function (pdf) of the normal distribution, also called Gaussian or "bell curve", the most important continuous random distribution.  As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.
The probability density function (pdf) of the normal distribution, also called Gaussian or "bell curve", the most important continuous random distribution. As notated on the figure, the probabilities of intervals of values correspond to the area under the curve.

Terminology

As probability theory is used in quite diverse applications, terminology is not uniform and sometimes confusing. The following terms are used for non-cumulative probability distribution functions:

  • Frequency distribution: A frequency distribution is a table that displays the frequency of various outcomes in a sample.
  • Relative frequency distribution: A frequency distribution where each value has been divided (normalized) by a number of outcomes in a sample i.e. sample size.
  • Probability distribution: Sometimes used as an alias for Relative frequency distribution but most books use it as a limit to which Relative frequency distribution tends when sample size tends to population size. It's a general term to indicate the way the total probability of 1 is distributed over all various possible outcomes (i.e. over entire population). It may for instance refer to a table that displays the probabilities of various outcomes in a finite population or to the probability density of an uncountably infinite population.
  • Cumulative distribution function: is a general functional form to describe a probability distribution.
  • Probability distribution function: somewhat ambiguous term sometimes referring to a functional form of probability distribution table. Could be called a "normalized frequency distribution function", where area under the graph equals to 1.
  • Probability mass, Probability mass function, p.m.f., Discrete probability distribution function: for discrete random variables.
  • Categorical distribution: for discrete random variables with a finite set of values.
  • Probability density, Probability density function, p.d.f., Continuous probability distribution function: most often reserved for continuous random variables.

The following terms are somewhat ambiguous as they can refer to non-cumulative or cumulative distributions, depending on authors' preferences:

  • Probability distribution function: continuous or discrete, non-cumulative or cumulative.
  • Probability function: even more ambiguous, can mean any of the above or other things.

Basic terms

  • Mode: for a discrete random variable, the value with highest probability (the location at which the probability mass function has its peak); for a continuous random variable, a location at which the probability density function has a local peak.
  • Support: the smallest closed set whose complement has probability zero.
  • Head: the range of values where the pmf or pdf is relatively high.
  • Tail: the complement of the head within the support; the large set of values where the pmf or pdf is relatively low.
  • Expected value or mean: the weighted average of the possible values, using their probabilities as their weights; or the continuous analog thereof.
  • Median: the value such that the set of values less than the median, and the set greater than the median, each have probabilities no greater than one-half.
  • Variance: the second moment of the pmf or pdf about the mean; an important measure of the dispersion of the distribution.
  • Standard deviation: the square root of the variance, and hence another measure of dispersion.
  • Symmetry: a property of some distributions in which the portion of the distribution to the left of a specific value is a mirror image of the portion to its right.
  • Skewness: a measure of the extent to which a pmf or pdf "leans" to one side of its mean. The third standardized moment of the distribution.
  • Kurtosis: a measure of the "fatness" of the tails of a pmf or pdf. The fourth standardized moment of the distribution.

Cumulative distribution function

Because a probability distribution P on the real line is determined by the probability of a scalar random variable X being in a half-open interval (−∞, x], the probability distribution is completely characterized by its cumulative distribution function:

Discrete probability distribution

The probability mass function of a discrete probability distribution. The probabilities of the singletons {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.
The probability mass function of a discrete probability distribution. The probabilities of the singletons {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero.
The cdf of a discrete probability distribution, ...
The cdf of a discrete probability distribution, ...
... of a continuous probability distribution, ...
... of a continuous probability distribution, ...
... of a distribution which has both a continuous part and a discrete part.
... of a distribution which has both a continuous part and a discrete part.

A discrete probability distribution is a probability distribution characterized by a probability mass function. Thus, the distribution of a random variable X is discrete, and X is called a discrete random variable, if

as u runs through the set of all possible values of X. A discrete random variable can assume only a finite or countably infinite number of values.[1] For the number of potential values to be countably infinite, even though their probabilities sum to 1, the probabilities have to decline to zero fast enough. For example, if for n = 1, 2, ..., the sum of probabilities would be 1/2 + 1/4 + 1/8 + ... = 1.

Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution. Additionally, the discrete uniform distribution is commonly used in computer programs that make equal-probability random selections between a number of choices.

When a sample (a set of observations) is drawn from a larger population, the sample points have an empirical distribution that is discrete and that provides information about the population distribution.

Measure theoretic formulation

A measurable function between a probability space and a measurable space is called a discrete random variable provided that its image is a countable set. In this case measurability of means that the pre-images of singleton sets are measurable, i.e., for all . The latter requirement induces a probability mass function via . Since the pre-images of disjoint sets are disjoint,

This recovers the definition given above.

Cumulative distribution function

Equivalently to the above, a discrete random variable can be defined as a random variable whose cumulative distribution function (cdf) increases only by jump discontinuities—that is, its cdf increases only where it "jumps" to a higher value, and is constant between those jumps. The points where jumps occur are precisely the values which the random variable may take.

Delta-function representation

Consequently, a discrete probability distribution is often represented as a generalized probability density function involving Dirac delta functions, which substantially unifies the treatment of continuous and discrete distributions. This is especially useful when dealing with probability distributions involving both a continuous and a discrete part.

Indicator-function representation

For a discrete random variable X, let u0, u1, ... be the values it can take with non-zero probability. Denote

These are disjoint sets, and for such sets

It follows that the probability that X takes any value except for u0, u1, ... is zero, and thus one can write X as

except on a set of probability zero, where is the indicator function of A. This may serve as an alternative definition of discrete random variables.

Continuous probability distribution

A continuous probability distribution is a probability distribution that has a cumulative distribution function that is continuous. Most often they are generated by having a probability density function. Mathematicians call distributions with probability density functions absolutely continuous, since their cumulative distribution function is absolutely continuous with respect to the Lebesgue measure λ. If the distribution of X is continuous, then X is called a continuous random variable. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.

Formally, if X is a continuous random variable, then it has a probability density function ƒ(x), and therefore its probability of falling into a given interval, say [a, b] is given by the integral

In particular, the probability for X to take any single value a (that is aXa) is zero, because an integral with coinciding upper and lower limits is always equal to zero.

The definition states that a continuous probability distribution must possess a density, or equivalently, its cumulative distribution function be absolutely continuous. This requirement is stronger than simple continuity of the cumulative distribution function, and there is a special class of distributions, singular distributions, which are neither continuous nor discrete nor a mixture of those. An example is given by the Cantor distribution. Such singular distributions however are never encountered in practice.

Note on terminology: some authors use the term "continuous distribution" to denote the distribution with continuous cumulative distribution function. Thus, their definition includes both the (absolutely) continuous and singular distributions.

By one convention, a probability distribution is called continuous if its cumulative distribution function is continuous and, therefore, the probability measure of singletons for all .

Another convention reserves the term continuous probability distribution for absolutely continuous distributions. These distributions can be characterized by a probability density function: a non-negative Lebesgue integrable function defined on the real numbers such that

Discrete distributions and some continuous distributions (like the Cantor distribution) do not admit such a density.

Some properties

  • The probability distribution of the sum of two independent random variables is the convolution of each of their distributions.
  • Probability distributions are not a vector space—they are not closed under linear combinations, as these do not preserve non-negativity or total integral 1—but they are closed under convex combination, thus forming a convex subset of the space of functions (or measures).

Kolmogorov definition

In the measure-theoretic formalization of probability theory, a random variable is defined as a measurable function X from a probability space to measurable space . A probability distribution of X is the pushforward measure X*P  of X , which is a probability measure on satisfying X*P = PX −1.[2]

Random number generation

A frequent problem in statistical simulations (the Monte Carlo method) is the generation of pseudo-random numbers that are distributed in a given way. Most algorithms are based on a pseudorandom number generator that produces numbers X that are uniformly distributed in the half-open interval [0,1). These random variates X are then transformed via some algorithm to create a new random variate having the required probability distribution.

Applications

The concept of the probability distribution and the random variables which they describe underlies the mathematical discipline of probability theory, and the science of statistics. There is spread or variability in almost any value that can be measured in a population (e.g. height of people, durability of a metal, sales growth, traffic flow, etc.); almost all measurements are made with some intrinsic error; in physics many processes are described probabilistically, from the kinetic properties of gases to the quantum mechanical description of fundamental particles. For these and many other reasons, simple numbers are often inadequate for describing a quantity, while probability distributions are often more appropriate.

Several more specific example of an application:

Common probability distributions

The following is a list of some of the most common probability distributions, grouped by the type of process that they are related to. For a more complete list, see list of probability distributions, which groups by the nature of the outcome being considered (discrete, continuous, multivariate, etc.)

Note also that all of the univariate distributions below are singly peaked; that is, it is assumed that the values cluster around a single point. In practice, actually observed quantities may cluster around multiple values. Such quantities can be modeled using a mixture distribution.

Related to real-valued quantities that grow linearly (e.g. errors, offsets)

  • Normal distribution (Gaussian distribution), for a single such quantity; the most common continuous distribution

Related to positive real-valued quantities that grow exponentially (e.g. prices, incomes, populations)

Related to real-valued quantities that are assumed to be uniformly distributed over a (possibly unknown) region

Related to Bernoulli trials (yes/no events, with a given probability)

Related to categorical outcomes (events with K possible outcomes, with a given probability for each outcome)

Related to events in a Poisson process (events that occur independently with a given rate)

Related to the absolute values of vectors with normally distributed components

  • Rayleigh distribution, for the distribution of vector magnitudes with Gaussian distributed orthogonal components. Rayleigh distributions are found in RF signals with Gaussian real and imaginary components.
  • Rice distribution, a generalization of the Rayleigh distributions for where there is a stationary background signal component. Found in Rician fading of radio signals due to multipath propagation and in MR images with noise corruption on non-zero NMR signals.

Related to normally distributed quantities operated with sum of squares (for hypothesis testing)

Useful as conjugate prior distributions in Bayesian inference

See also

References

  1. ^ 1941-, Çınlar, E. (Erhan) (2011). Probability and stochastics. New York: Springer. p. 51. ISBN 9780387878591. OCLC 710149819.
  2. ^ W., Stroock, Daniel (1999). Probability theory : an analytic view (Rev. ed.). Cambridge [England]: Cambridge University Press. p. 11. ISBN 978-0521663496. OCLC 43953136.
  3. ^ Chen, P.; Chen, Z.; Bak-Jensen, B. (April 2008). "Probabilistic load flow: A review". 2008 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies. pp. 1586–1591. doi:10.1109/drpt.2008.4523658. ISBN 978-7-900714-13-8.

External links

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