Probability and Statistics

(directly go to documentation on : BernoulliDistribution, BinomialDistribution, tDistribution, PDF, ChiSquareTest. )

# 30. Probability and Statistics

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 | Bernoulli distribution
BinomialDistribution | binomial distribution
tDistribution | Student's t distribution
PDF | probability density function
ChiSquareTest | Pearson's ChiSquare test

##

30.1 Probability

###

BernoulliDistribution -- Bernoulli distribution

##### Standard library

##### Calling format:

    BernoulliDistribution(p)

##### Parameters:

`p` -- number, probability of an event in a single trial

##### Description:

A random variable has a Bernoulli distribution with probability `p` if it can be interpreted as an indicator of an event, where `p` is the probability to observe the event in a single trial.

Numerical value of `p` must satisfy 0<p<1.

##### See also:

BinomialDistribution .

###

BinomialDistribution -- binomial distribution

##### Standard library

##### Calling format:

    BinomialDistribution(p,n)

##### Parameters:

`p` -- number, probability to observe an event in single trial

`n` -- number of trials

##### Description:

Suppose we repeat a trial `n` times, the probability to observe an event in a single trial is `p` 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 is `BinomialDistribution``(p,n)`.

Numerical value of `p` must satisfy 0<p<1. Numerical value of `n` must be a positive integer.

##### See also:

BernoulliDistribution .

###

tDistribution -- Student's t distribution

##### Standard library

##### Calling format:

    {tDistribution}(m)

##### Parameters:

`m` -- integer, number of degrees of freedom

##### Description:

Let Y and Z be independent random variables, Y have the NormalDistribution(0,1), `Z` have ChiSquareDistribution(m). Then Y/Sqrt(Z/m) has tDistribution(m).

Numerical value of `m` must be positive integer.

###

PDF -- probability density function

##### Standard library

##### Calling format:

    PDF(dist,x)

##### Parameters:

`dist` -- a distribution type

`x` -- a value of random variable

##### Description:

If `dist` is a discrete distribution, then `PDF` returns the probability for a random variable with distribution `dist` to take a value of `x`. If `dist` is a continuous distribution, then `PDF` returns the density function at point x.

##### See also:

CDF .

##

30.2 Statistics

###

ChiSquareTest -- Pearson's ChiSquare test

##### Standard library

##### Calling format:

    ChiSquareTest(observed,expected)
    ChiSquareTest(observed,expected,params)

##### Parameters:

`observed` -- list of observed frequencies

`expected` -- list of expected frequencies

`params` -- number of estimated parameters

##### Description:

`ChiSquareTest` 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 as ChiSquareDistribution.