Beta distribution
Encyclopedia : B : BE : BET : Beta distribution
(\alpha,\beta)}\!]| cdf =[I_x(\alpha,\beta)\!]| mean =[\frac\!]| median =| mode =[\frac\!] for [\alpha>1, \beta>1]| variance =[\frac\!]| skewness =[\frac}}]| kurtosis =see text| entropy =| mgf =[1 +\sum_^ \left( \prod_^ \frac \right) \frac]| char =[_1F_1(\alpha; \alpha+\beta; i\,t)\!]| }} In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]:
- [ f(x;\alpha,\beta) = \frac(\alpha,\beta)} x^(1-x)^]
The beta function is a normalization constant to ensure that the integral of the pdf is unity:
- [ f(x;\alpha,\beta) = \frac(1-x)^} (1-u)^\, du} \!]
- :[= \frac\, x^(1-x)^\!]
- :[= \frac(\alpha,\beta)}\, x^(1-x)^\!]
[B(i,j)] with integer values of i and j is the distribution of the j-th highest of a sample of [i+j-1] independent random variables uniformly distributed between 0 and 1. The cumulative probability from 0 to x is thus the probability that the j-th highest value is less than x, in other words, it is the probability that at least i of the random variables are less than x, a probability given by summing over the binomial distribution with its p parameter set to x. This shows the intimate connection between the beta distribution and the binomial distribution.
The special case of the beta distribution when α = 1 and β = 1 is the standard uniform distribution.
The expected value and variance of a beta random variable X with parameters α and β are given by the formulae:
- [ \operatorname(X) = \frac ]
- [ \operatorname(X) = \frac]
- [6\,\frac\!]
Parameter estimation
When the expected value and variance of a beta random variable X are given, the parameters α and β are calculated by the formulae:- [\alpha=\operatorname(X)\left( \frac(X) (1 - \operatorname(X))}(X)} - 1\right),]
- [\beta = (1-\operatorname(X))\left( \frac(X) (1 - \operatorname(X))}(X)} - 1\right).]
For any two numbers u, v such that 0 < u < 1 and 0 < v < u(1 − u) there is a beta distribution having expected value E(X) = u and variance var(X) = v.
Cumulative distribution function
The cumulative distribution function is
- [F(x;\alpha,\beta) = \frac_x(\alpha,\beta)}(\alpha,\beta)} = I_x(\alpha,\beta) \!]
- [ I_x(\alpha,\beta) = \sum_^ x^\alpha (1-x)^ ]
Shapes
The beta function can take on different shapes depending on the values of the two parameters:- [\alpha < 1,\ \beta < 1] is U-shaped (red plot)
- [\alpha < 1,\ \beta \geq 1] or [\alpha = 1,\ \beta > 1] is strictly decreasing (blue plot)
- * [\alpha = 1,\ \beta > 2] is strictly convex
- * [\alpha = 1,\ \beta = 2] is a straight line
- * [\alpha = 1,\ 1 < \beta < 2] is strictly concave
- [\alpha = 1,\ \beta = 1] is the uniform distribution
- [\alpha = 1,\ \beta < 1] or [\alpha > 1,\ \beta \leq 1] is strictly increasing (green plot)
- *[\alpha > 2,\ \beta = 1] is strictly convex
- *[\alpha = 2,\ \beta = 1] is a straight line
- *[1 < \alpha < 2,\ \beta = 1] is strictly concave
- [\alpha > 1,\ \beta > 1] is unimodal (purple & black plots)
Related distributions
- The connection with the binomial distribution has been mentioned above.
- [X \sim \mathrm(0,1)] (that is, it follows a uniform distribution) if [X \sim \mathrm(\alpha = 1, \beta = 1)].
- If [X \sim \mathrm(\alpha, \theta)] and [Y \sim \mathrm(\beta, \theta)] are independent gamma variates, then [X/(X+Y) \sim \mathrm(\alpha,\beta)].
- If [X \sim \mathrm(\alpha,\beta)] is a beta variate and [Y \sim \mathrm(2\beta,2\alpha)] is an independent F variate, then [\Pr[X leq alpha/(alpha+x,beta)] = \Pr[Y > x]] for all [x>0].
- The Dirichlet distribution is the multivariate generalization of the beta distribution.
- The Kumaraswamy distribution resembles the beta distribution.
Applications
Beta distributions are used extensively in Bayesian statistics, since beta distributions provide a family of conjugate prior distributions for binomial distributions.The Beta distribution can be used to model events which are constrained to take place within an interval defined by a minimum and maximum value. For this reason, the Beta distribution - along with the triangular distribution - is used extensively in PERT, CPM and other project management / control systems to describe the time to completion of a task.
The c.d.f of the Beta distribution is used as a convenient way of obtaining the sum over a set of binomial outcomes.
External links
- [Beta Distribution], wolfram.com
- [Beta Distribution - Overview and Example], xycoon.com
- [Beta Distribution], brighton-webs.co.uk
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