In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the experiment it represents. For example, the value in rolling a six-sided die is 3.5. Less roughly, the law of large states that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions approaches infinity. The expected value is known as the expectation, mathematical expectation, EV, mean value, mean. More practically, the value of a discrete random variable is the probability-weighted average of all possible values. In other words, each value the random variable can assume is multiplied by its probability of occurring. The same principle applies to a random variable, except that an integral of the variable with respect to its probability density replaces the sum. The expected value does not exist for random variables having some distributions with large tails, for random variables such as these, the long-tails of the distribution prevent the sum/integral from converging.
The expected value is a key aspect of how one characterizes a probability distribution, by contrast, the variance is a measure of dispersion of the possible values of the random variable around the expected value. The variance itself is defined in terms of two expectations, it is the value of the squared deviation of the variables value from the variables expected value. The expected value plays important roles in a variety of contexts, in regression analysis, one desires a formula in terms of observed data that will give a good estimate of the parameter giving the effect of some explanatory variable upon a dependent variable. The formula will give different estimates using different samples of data, a formula is typically considered good in this context if it is an unbiased estimator—that is, if the expected value of the estimate can be shown to equal the true value of the desired parameter. In decision theory, and in particular in choice under uncertainty, one example of using expected value in reaching optimal decisions is the Gordon–Loeb model of information security investment.
According to the model, one can conclude that the amount a firm spends to protect information should generally be only a fraction of the expected loss. Suppose random variable X can take value x1 with probability p1, value x2 with probability p2, the expectation of this random variable X is defined as E = x 1 p 1 + x 2 p 2 + ⋯ + x k p k. If all outcomes xi are equally likely, the weighted average turns into the simple average and this is intuitive, the expected value of a random variable is the average of all values it can take, thus the expected value is what one expects to happen on average. If the outcomes xi are not equally probable, the simple average must be replaced with the weighted average, the intuition however remains the same, the expected value of X is what one expects to happen on average. Let X represent the outcome of a roll of a fair six-sided die, more specifically, X will be the number of pips showing on the top face of the die after the toss
In probability theory, conditional probability is a measure of the probability of an event given that another event has occurred. For example, the probability that any person has a cough on any given day may be only 5%. But if we know or assume that the person has a cold, the conditional probability of coughing given that you have a cold might be a much higher 75%. The concept of probability is one of the most fundamental. But conditional probabilities can be slippery and require careful interpretation. For example, there need not be a causal or temporal relationship between A and B, P may or may not be equal to P. If P = P, events A and B are said to be independent, in such a case, also, in general, P is not equal to P. For example, if you have cancer you might have a 90% chance of testing positive for cancer. In this case what is being measured is that the if event B having cancer has occurred, you can test positive for cancer but you may have only a 10% chance of actually having cancer because cancer is very rare.
In this case what is being measured is the probability of the event B - having cancer given that the event A - test is positive has occurred, falsely equating the two probabilities causes various errors of reasoning such as the base rate fallacy. Conditional probabilities can be reversed using Bayes theorem. The logic behind this equation is that if the outcomes are restricted to B, Note that this is a definition but not a theoretical result. We just denote the quantity P / P as P and call it the conditional probability of A given B. Further, this multiplication axiom introduces a symmetry with the axiom for mutually exclusive events, P = P + P − P0 If P =0. However, it is possible to define a probability with respect to a σ-algebra of such events. The case where B has zero measure is problematic, see conditional expectation for more information. Conditioning on an event may be generalized to conditioning on a random variable, Let X be a random variable, we assume for the sake of presentation that X is discrete, that is, X takes on only finitely many values x.
The conditional probability of A given X is defined as the variable, written P
Maximum likelihood estimation
The method of maximum likelihood corresponds to many well-known estimation methods in statistics. For example, one may be interested in the heights of adult female penguins, MLE would accomplish this by taking the mean and variance as parameters and finding particular parametric values that make the observed results the most probable given the model. In general, for a set of data and underlying statistical model. Maximum likelihood estimation gives a unified approach to estimation, which is well-defined in the case of the normal distribution, maximum-likelihood estimation was recommended and widely popularized by Ronald Fisher between 1912 and 1922. Maximum-likelihood estimation finally transcended heuristic justification in a proof published by Samuel S. Wilks in 1938, the only difficult part of the proof depends on the expected value of the Fisher information matrix, which is provided by a theorem by Fisher. Wilks continued to improve on the generality of the theorem throughout his life, some of the theory behind maximum likelihood estimation was developed for Bayesian statistics.
Reviews of the development of maximum likelihood estimation have been provided by a number of authors, suppose there is a sample x1, x2, …, xn of n independent and identically distributed observations, coming from a distribution with an unknown probability density function f0. It is however surmised that the function f0 belongs to a family of distributions, called the parametric model. The value θ0 is unknown and is referred to as the value of the parameter vector. It is desirable to find an estimator θ ^ which would be as close to the true value θ0 as possible, either or both the observed variables xi and the parameter θ can be vectors. To use the method of maximum likelihood, one first specifies the joint density function for all observations, for an independent and identically distributed sample, this joint density function is f = f × f × ⋯ × f. Note that, denotes a separation between the two categories of arguments, the parameters θ and the observations x 1, …, x n. The hat over ℓ indicates that it is akin to some estimator, indeed, ℓ ^ estimates the expected log-likelihood of a single observation in the model.
The method of maximum likelihood estimates θ0 by finding a value of θ that maximizes ℓ ^ and this method of estimation defines a maximum likelihood estimator of θ0, ⊆, if a maximum exists. An MLE estimate is the same regardless of whether we maximize the likelihood or the log-likelihood function, for many models, a maximum likelihood estimator can be found as an explicit function of the observed data x1. For many other models, however, no solution to the maximization problem is known or available. For some problems, there may be multiple estimates that maximize the likelihood, in the exposition above, it is assumed that the data are independent and identically distributed. In a simpler extension, an allowance can be made for data heterogeneity, put another way, we are now assuming that each observation xi comes from a random variable that has its own distribution function fi
For instance, if the random variable X is used to denote the outcome of a coin toss, the probability distribution of X would take the value 0.5 for X = heads, and 0.5 for X = tails. In more technical terms, the probability distribution is a description of a phenomenon in terms of the probabilities of events. Examples of random phenomena can include the results of an experiment or survey, a probability distribution is defined 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 numbers or a higher-dimensional vector space, or it may be a list of non-numerical values, for example. Probability distributions are divided into two classes. A discrete probability distribution can be encoded by a discrete list of the probabilities of the outcomes, on the other hand, a continuous probability distribution is typically described by probability density functions. The normal distribution represents 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 the set of numbers is called univariate.
Important and commonly encountered univariate probability distributions include the distribution, the hypergeometric distribution. The multivariate normal distribution is a commonly encountered multivariate distribution, to define probability distributions for the simplest cases, one needs to distinguish between discrete and continuous random variables. For example, the probability that an object weighs exactly 500 g is zero. Continuous probability distributions can be described in several ways, the cumulative distribution function is the antiderivative of the probability density function provided that the latter function exists. As probability theory is used in diverse applications, terminology is not uniform. The following terms are used for probability distribution functions, Distribution. Probability distribution, is a table that displays the probabilities of outcomes in a sample. Could be called a frequency distribution table, where all occurrences of outcomes sum to 1. Distribution function, is a form of frequency distribution table.
Probability distribution function, is a form of probability distribution table
The variance has a central role in statistics. It is used in statistics, statistical inference, hypothesis testing, goodness of fit. This makes it a central quantity in numerous such as physics, chemistry, economics. The variance of a random variable X is the value of the squared deviation from the mean of X, μ = E . This definition encompasses random variables that are generated by processes that are discrete, neither, the variance can be thought of as the covariance of a random variable with itself, Var = Cov . The variance is equivalent to the second cumulant of a probability distribution that generates X, the variance is typically designated as Var , σ X2, or simply σ2. On computational floating point arithmetic, this equation should not be used, if a continuous distribution does not have an expected value, as is the case for the Cauchy distribution, it does not have a variance either. Many other distributions for which the value does exist do not have a finite variance because the integral in the variance definition diverges.
An example is a Pareto distribution whose index k satisfies 1 < k ≤2. e, the normal distribution with parameters μ and σ is a continuous distribution whose probability density function is given by f =12 π σ2 e −22 σ2. In this distribution, E = μ and the variance Var is related with σ via Var = ∫ − ∞ ∞22 π σ2 e −22 σ2 d x = σ2. The role of the distribution in the central limit theorem is in part responsible for the prevalence of the variance in probability. The exponential distribution with parameter λ is a distribution whose support is the semi-infinite interval. Its probability density function is given by f = λ e − λ x, the variance is equal to Var = ∫0 ∞2 λ e − λ x d x = λ −2. So for an exponentially distributed random variable, σ2 = μ2, the Poisson distribution with parameter λ is a discrete distribution for k =0,1,2, …. Its probability mass function is given by p = λ k k, E − λ, and it has expected value μ = λ. The variance is equal to Var = ∑ k =0 ∞ λ k k, E − λ2 = λ, So for a Poisson-distributed random variable, σ2 = μ.
The binomial distribution with n and p is a discrete distribution for k =0,1,2, …, n. Its probability mass function is given by p = p k n − k, the variance is equal to Var = ∑ k =0 n p k n − k 2 = n p
Entropy (information theory)
In information theory, systems are modeled by a transmitter and receiver. The transmitter produces messages that are sent through the channel, the channel modifies the message in some way. The receiver attempts to infer which message was sent, in this context, entropy is the expected value of the information contained in each message. Messages can be modeled by any flow of information, in a more technical sense, there are reasons to define information as the negative of the logarithm of the probability distribution of possible events or messages. The amount of information of every event forms a random variable whose expected value, units of entropy are the shannon, nat, or hartley, depending on the base of the logarithm used to define it, though the shannon is commonly referred to as a bit. The logarithm of the probability distribution is useful as a measure of entropy because it is additive for independent sources, for instance, the entropy of a coin toss is 1 shannon, whereas of m tosses it is m shannons.
Generally, you need log2 bits to represent a variable that can take one of n if n is a power of 2. If these values are equally probable, the entropy is equal to the number of bits, equality between number of bits and shannons holds only while all outcomes are equally probable. If one of the events is more probable than others, observation of event is less informative. Conversely, rarer events provide more information when observed, since observation of less probable events occurs more rarely, the net effect is that the entropy received from non-uniformly distributed data is less than log2. Entropy is zero when one outcome is certain, Shannon entropy quantifies all these considerations exactly when a probability distribution of the source is known. The meaning of the events observed does not matter in the definition of entropy, entropy refers to disorder or uncertainty. Shannon entropy was introduced by Claude E. Shannon in his 1948 paper A Mathematical Theory of Communication, Shannon entropy provides an absolute limit on the best possible average length of lossless encoding or compression of an information source.
Entropy is a measure of unpredictability of the state, or equivalently, to get an intuitive understanding of these terms, consider the example of a political poll. Usually, such polls happen because the outcome of the poll is not already known, consider the case that the same poll is performed a second time shortly after the first poll. Now consider the example of a coin toss, assuming the probability of heads is the same as the probability of tails, the entropy of the coin toss is as high as it could be. Such a coin toss has one shannon of entropy since there are two possible outcomes that occur with probability, and learning the actual outcome contains one shannon of information. Contrarily, a toss with a coin that has two heads and no tails has zero entropy since the coin will always come up heads
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive or negative, or even undefined, the qualitative interpretation of the skew is complicated and unintuitive. Skew must not be thought to refer to the direction the curve appears to be leaning, in fact, positive skew indicates that the tail on the right side is longer or fatter than the left side. In cases where one tail is long but the tail is fat. Further, in multimodal distributions and discrete distributions, skewness is difficult to interpret, the skewness does not determine the relationship of mean and median. In cases where it is necessary, data might be transformed to have a normal distribution, consider the two distributions in the figure just below. Within each graph, the values on the side of the distribution taper differently from the values on the left side. A left-skewed distribution usually appears as a right-leaning curve, positive skew, The right tail is longer, the mass of the distribution is concentrated on the left of the figure.
A right-skewed distribution usually appears as a left-leaning curve, Skewness in a data series may sometimes be observed not only graphically but by simple inspection of the values. For instance, consider the sequence, whose values are evenly distributed around a central value of 50. If the distribution is symmetric, the mean is equal to the median, if, in addition, the distribution is unimodal, the mean = median = mode. This is the case of a coin toss or the series 1,2,3,4, however, that the converse is not true in general, i. e. zero skewness does not imply that the mean is equal to the median. Paul T. von Hippel points out, Many textbooks, teach a rule of thumb stating that the mean is right of the median under right skew and this rule fails with surprising frequency. It can fail in multimodal distributions, or in distributions where one tail is long, most commonly, the rule fails in discrete distributions where the areas to the left and right of the median are not equal. Such distributions not only contradict the textbook relationship between mean and skew, they contradict the textbook interpretation of the median.
It is sometimes referred to as Pearsons moment coefficient of skewness, or simply the moment coefficient of skewness, the last equality expresses skewness in terms of the ratio of the third cumulant κ3 to the 1. 5th power of the second cumulant κ2. This is analogous to the definition of kurtosis as the fourth cumulant normalized by the square of the second cumulant, the skewness is sometimes denoted Skew. Starting from a standard cumulant expansion around a distribution, one can show that skewness =6 /standard deviation + O
Other Bayesians prefer to parametrize the inverse gamma distribution differently, as a scaled inverse chi-squared distribution. The inverse gamma distributions probability density function is defined over the support x >0 f = β α Γ x − α −1 exp with shape parameter α, here Γ denotes the gamma function. Many math packages allow you to compute Q, the gamma function. K α in the expression of the function is the modified Bessel function of the 2nd kind. For α >0 and β >0, E = ln − ψ and E = α β, gamma distribution inverse-chi-squared distribution normal distribution V. Witkovsky Computing the distribution of a linear combination of inverted gamma variables, Kybernetika 37, 79-90
There is not necessarily an underlying ordering of these outcomes, but numerical labels are often attached for convenience in describing the distribution. Note that the K-dimensional categorical distribution is the most general distribution over a K-way event, the parameters specifying the probabilities of each possible outcome are constrained only by the fact that each must be in the range 0 to 1, and all must sum to 1. On the other hand, the distribution is a special case of the multinomial distribution. Occasionally, the distribution is termed the discrete distribution. However, this refers not to one particular family of distributions. However, conflating the categorical and multinomial distributions can lead to problems, both forms have very similar-looking probability mass functions, which both make reference to multinomial-style counts of nodes in a category. However, the multinomial-style PMF has a factor, a multinomial coefficient. Confusing the two can easily lead to results in settings where this extra factor is not constant with respect to the distributions of interest.
The factor is constant in the complete conditionals used in Gibbs sampling. A categorical distribution is a probability distribution whose sample space is the set of k individually identified items. It is the generalization of the Bernoulli distribution for a random variable. In one formulation of the distribution, the space is taken to be a finite sequence of integers. The exact integers used as labels are unimportant, they might be or or any other set of values. In the following descriptions, we use for convenience, although this disagrees with the convention for the Bernoulli distribution, which uses. In this case, the probability function f is, f = p i. Another formulation that appears more complex but facilitates mathematical manipulations is as follows, using the Iverson bracket, f = ∏ i =1 k p i, there are various advantages of this formulation, e. g. It is easier to write out the function of a set of independent identically distributed categorical variables. It connects the categorical distribution with the related multinomial distribution and it shows why the Dirichlet distribution is the conjugate prior of the categorical distribution, and allows the posterior distribution of the parameters to be calculated
In contrast, the binomial distribution describes the probability of k successes in n draws with replacement. In statistics, the hypergeometric test uses the hypergeometric distribution to calculate the significance of having drawn a specific k successes from the aforementioned population. The test is used to identify which sub-populations are over- or under-represented in a sample. This test has a range of applications. For example, a group could use the test to understand their customer base by testing a set of known customers for over-representation of various demographic subgroups. The following conditions characterize the distribution, The result of each draw can be classified into one of two mutually exclusive categories. The probability of a success changes on each draw, as each draw decreases the population, the pmf is positive when max ≤ k ≤ min. The pmf satisfies the recurrence relation P = P with P =, as one would expect, the probabilities sum up to 1, ∑0 ≤ k ≤ n =1 This is essentially Vandermondes identity from combinatorics.
Also note the following identity holds, = and this follows from the symmetry of the problem, but it can be shown by expressing the binomial coefficients in terms of factorials and rearranging the latter. The classical application of the distribution is sampling without replacement. Think of an urn with two types of marbles, red ones and green ones, define drawing a green marble as a success and drawing a red marble as a failure. If the variable N describes the number of all marbles in the urn and K describes the number of green marbles, in this example, X is the random variable whose outcome is k, the number of green marbles actually drawn in the experiment. This situation is illustrated by the following table, Now. Standing next to the urn, you close your eyes and draw 10 marbles without replacement, what is the probability that exactly 4 of the 10 are green. This problem is summarized by the following table, The probability of drawing exactly k green marbles can be calculated by the formula P = f =.
Hence, in this example calculate P = f = =5 ⋅814506010272278170 =0.003964583 …, intuitively we would expect it to be even more unlikely for all 5 marbles to be green. P = f = =1 ⋅122175910272278170 =0.0001189375 …, As expected, in Holdem Poker players make the best hand they can combining the two cards in their hand with the 5 cards eventually turned up on the table. The deck has 52 and there are 13 of each suit, for this example assume a player has 2 clubs in the hand and there are 3 cards showing on the table,2 of which are clubs
Probability density function
In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. The probability density function is everywhere, and its integral over the entire space is equal to one. The terms probability distribution function and probability function have sometimes used to denote the probability density function. However, this use is not standard among probabilists and statisticians, further confusion of terminology exists because density function has been used for what is here called the probability mass function. In general though, the PMF is used in the context of random variables. Suppose a species of bacteria typically lives 4 to 6 hours, what is the probability that a bacterium lives exactly 5 hours. A lot of bacteria live for approximately 5 hours, but there is no chance that any given bacterium dies at exactly 5.0000000000, instead we might ask, What is the probability that the bacterium dies between 5 hours and 5.01 hours.
Lets say the answer is 0.02, What is the probability that the bacterium dies between 5 hours and 5.001 hours. The answer is probably around 0.002, since this is 1/10th of the previous interval, the probability that the bacterium dies between 5 hours and 5.0001 hours is probably about 0.0002, and so on. In these three examples, the ratio / is approximately constant, and equal to 2 per hour, for example, there is 0.02 probability of dying in the 0. 01-hour interval between 5 and 5.01 hours, and =2 hour−1. This quantity 2 hour−1 is called the probability density for dying at around 5 hours, therefore, in response to the question What is the probability that the bacterium dies at 5 hours. A literally correct but unhelpful answer is 0, but an answer can be written as dt. This is the probability that the bacterium dies within a window of time around 5 hours. For example, the probability that it lives longer than 5 hours, there is a probability density function f with f =2 hour−1. The integral of f over any window of time is the probability that the dies in that window.
A probability density function is most commonly associated with absolutely continuous univariate distributions, a random variable X has density fX, where fX is a non-negative Lebesgue-integrable function, if, Pr = ∫ a b f X d x. That is, f is any function with the property that. In the continuous univariate case above, the measure is the Lebesgue measure
The Delaporte distribution is a discrete probability distribution that has received attention in actuarial science. It can be defined using the convolution of a binomial distribution with a Poisson distribution. The skewness of the Delaporte distribution is, λ + α β32 The excess kurtosis of the distribution is, λ +3 λ2 + α β2 Murat, M. Szynal, on moments of counting distributions satisfying the kth-order recursion and their compound distributions