Probability and Statistics¶
Probability¶
Distributions¶
Each distribution is represented as an entity. For each distribution known to
the system the consistency of parameters is checked. If the parameters for a
distribution are invalid, the functions return Undefined
. For example,
NormalDistribution(a,1)
evaluates to Undefined
, because of
negative variance.

BernoulliDistribution
(p)¶ Bernoulli distribution
 Param p
number, probability of an event in a single trial
A random variable has a Bernoulli distribution with probability
p
if it can be interpreted as an indicator of an event, wherep
is the probability to observe the event in a single trial. Numerical value ofp
must satisfy0 < p < 1
.See also

BinomialDistribution
(p, n)¶ binomial distribution
 Param p
number, probability to observe an event in single trial
 Param n
number of trials
Suppose we repeat a trial
n
times, the probability to observe an event in a single trial isp
and outcomes in all trials are mutually independent. Then the number of trials when the event occurred is distributed according to the binomial distribution. The probability of that isBinomialDistribution(p,n)
. Numerical value ofp
must satisfy0 < p < 1
. Numerical value ofn
must be a positive integer.See also

ChiSquareDistribution
(m)¶

DiscreteUniformDistribution
(a, b)¶

ExponentialDistribution
(l)¶

GeometricDistribution
(p)¶

NormalDistribution
(m, s2)¶

PoissonDistribution
(l)¶

tDistribution
(m)¶ Student’s \(t\) distribution
 Param m
number of degrees of freedom
Functions¶

PDF
(dist, x)¶ probability density function
 Param dist
a distribution type
 Param x
a value of random variable
If
dist
is a discrete distribution, thenPDF()
returns the probability for a random variable with distributiondist
to take a value ofx
. Ifdist
is a continuous distribution, thenPDF
returns the density function at pointx
.See also
Statistics¶

ChiSquareTest
(observed, expected, params)¶ Pearson’s ChiSquare test
 Param observed
list of observed frequencies
 Param expected
list of expected frequencies
 Param params
number of estimated parameters
Chisquared test is intended to find out if our sample was drawn from a given distribution or not. To find this out, one has to calculate observed frequencies into certain intervals and expected ones. To calculate expected frequency the formula \(n_i=n p_i\) must be used, where \(p_i\) is the probability measure of \(i\)th interval, and \(n\) is the total number of observations. If any of the parameters of the distribution were estimated, this number is given as
params
. The function returns a list of three local substitution rules. First of them contains the test statistic, the second contains the value of the parameters, and the last one contains the degrees of freedom. The test statistic is distributed asChiSquareDistribution()
.