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In probability theory and statistics, the chi distribution is a continuous probability distribution. It has one parameter k which specifies the number of degrees of freedom. The distribution usually arises when a k-dimensional vector's orthogonal components are independent and each follow a standard normal distribution. The length of the vector will then have a chi distribution. The most familiar example is the Maxwell distribution of (normalized) molecular speeds which is a chi distribution with 3 degrees of freedom. If X is chi-distributed, then X2 will follow the chi-square distribution with the same number of degrees of freedom. Image File history File links Download high resolution version (1300x975, 178 KB) Probability density function for the chi distribution File links The following pages link to this file: Chi distribution ...
Image File history File links Download high resolution version (1300x975, 164 KB) Cumulative distribution function for the chi distribution File links The following pages link to this file: Chi distribution ...
In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...
In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...
In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or...
In probability (and especially gambling), the expected value (or (mathematical) expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds...
In probability theory and statistics, the median is a number that separates the highest half of a sample, a population, or a probability distribution from the lowest half. ...
In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ...
In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ...
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ...
In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ...
Entropy of a Bernoulli trial as a function of success probability. ...
In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith...
Some mathematicians use the phrase characteristic function synonymously with indicator function. The indicator function of a subset A of a set B is the function with domain B, whose value is 1 at each point in A and 0 at each point that is in B but not in A...
Probability theory is the mathematical study of probability. ...
Statistics is a type of data analysis whose practice includes the planning, summarizing, and interpreting of observations of a system possibly followed by predicting or forecasting of future events based on a mathematical model of the system being observed. ...
In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...
The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields, especially in physics and engineering. ...
The Maxwell-Boltzmann distribution is a probability distribution with applications in physics and chemistry. ...
For any positive integer , the chi-square distribution with degrees of freedom is the probability distribution of the random variable where the are independent standard normal variables (zero expected value and unit variance). ...
Properties
The probability density function is where Γ(z) is the Gamma function. The cumulative distribution function is given by: The Gamma function along an interval In mathematics, the Gamma function is a function that extends the concept of factorial to the complex numbers. ...
where P(k,x) is the regularized Gamma function. The moment generating function is given by: In mathematics, the gamma function is defined by a definite integral. ...
In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith...
where M(a,b,z) is Kummer's confluent hypergeometric function. The raw moments are then given by: In mathematics, the confluent hypergeometric function is formed from hypergeometric series. ...
Moment refers to either of two related concepts in mathematics and physics: Moment (physics) Moment (mathematics) See also Moment (magazine), a Jewish general publication. ...
where Γ(z) is the Gamma function. The first few raw moments are: The Gamma function along an interval In mathematics, the Gamma function is a function that extends the concept of factorial to the complex numbers. ...
where the rightmost expressions are derived using the recurrence relationship for the Gamma function: From these expressions we may derive the following relationships:
Mean:
Variance:
Skewness:
Kurtosis excess:
The characteristic function is given by: Some mathematicians use the phrase characteristic function synonymously with indicator function. The indicator function of a subset A of a set B is the function with domain B, whose value is 1 at each point in A and 0 at each point that is in B but not in A...
where again, M(a,b,z) is Kummer's confluent hypergeometric function. The entropy is given by: In mathematics, the confluent hypergeometric function is formed from hypergeometric series. ...
where ψ0(z) is the Polygamma function. In mathematics, the polygamma function of order m is defined as the m+1 th derivative of the logarithm of the gamma function: Here is the digamma function and is the gamma function. ...
Related distributions - If X is chi distributed then X2 is chi-square distributed:
- The Rayleigh distribution with σ = 1 is a chi distribution with two degrees of freedom.
- The Maxwell distribution for normalized molecular speeds is a chi distribution with three degrees of freedom.
- The chi distribution for k = 1 is the half-normal distribution.
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