An Introduction to
Statistics with Klong

Probability Distributions

The Klong nstat module contains a set of functions implementing some selected discrete and continuous probability distributions. For instance, n.pdf is the probability density function (PDF) of the normal distribution. Probability functions can be plotted using the interactive plotter interface (the result is the scaling of the y-axis; see fig.4 for the plot):

    v.aplot(n.pdf(;0;4);-6;6)
[0.0 0.2 0.02]

To display the graph in a more familiar frame with a y-axis ranging from zero to one, the y-axis is adjusted using the set.y function, specifying the origin, the limit, and the scaling step width of the axis. The vp function re-plots the most recently plotted graph, using the new parameters. Fig.5 shows the adjusted plot.

    set.y(0;1;0.1)
    vp()

Fig.4: larger. Klong
Fig.5: larger. Klong

While the interactive plotter interface is useful for quickly plotting graphs of data sets and functions, there are sometimes good reasons for using the nplot functions directly. For example, they can

The nplot functions always write Postscript code to the output channel; they never display the generated panels. Hence they are more suitable for batch processing than interactive use. Here are the nplot instructions for plotting normal distribution curves with different parameters in different line styles:

    grid([-6 4 1];[0 1 0.1])
    plot(n.pdf(;0;1))
    setline(1);plot(n.pdf(;-2;4))
    setline(2);plot(n.pdf(;2;0.25))
    draw()

The output of the program can then be viewed with a Postscript viewer or converted to an image file format. The panel generated by the above program can be found in fig.6. A commented version of the above program is linked below the figure.


Fig.6: larger. Klong

The signature of the n.pdf function is n.pdf(x;μ;σ2), where x is the score whose probability is to be computed, μ is the mean and σ2 the variance of the distribution. There are some other functions dealing with the normal distribution. They all have a n. prefix attached to their names. These functions include the probability density function, the cumulative distribution function (CDF), the mean and variance, and the skewness:

    n.pdf(0;0;4)
0.19947114020071634
    n.cdf(3;1;4)
0.841344680841397269
    n.mu(5;1)
5
    n.var(0;7)
7
    n.skew(0;1)
0

These functions are also available for all other probability distributions supported by nstat. Discrete probability distributions, however, have a probability mass function (PMF) instead of a density function. The continuous distributions also include a quantile function (QF), which is the inverse function of the CDF. Here is a summary of the supported distributions and the signatures of the corresponding functions:

Distribution Prefix PMF/PDF CDF μ σ2 Skew QF
Frequency * f. pmf(x;F) cdf(x;F) mu(F) var(F) -/- -/-
Uniform
(discrete)
u. pmf(x;a;b) cdf(x;a;b) mu(a;b) var(a;b) skew(a;b) -/-
Geometric geo. pmf(x;p) cdf(x;p) mu(p) var(p) skew(p) -/-
Binomial b. pmf(x;n;p) cdf(x;n;p) mu(n;p) var(n;p) skew(n;p) -/-
Hypergeometric hyp. pmf(x;n,p,N) cdf(x;n,p,N) mu(n;p;N) var(n;p;N) skew(n;p;N) -/-
Poisson poi. pmf(x;λ) cdf(x;λ) mu(λ) var(λ) skew(λ) -/-
Normal n. pdf(x;μ;σ2) cdf(x;μ;σ2) mu(μ;σ2) var(μ;σ2) skew(μ;σ2) qf(p;μ;σ2)
Standard
Normal
ndf(x) cdf(x)
phi(x)
0 1 0 qf(p)
Lognormal ln. pdf(x;μ;σ2) cdf(x;μ;σ2) mu(μ;σ2) var(μ;σ2) skew(μ;σ2) qf(p;μ;σ2)
χ2 x2. pdf(ν;x) cdf(ν;x) mu(ν) var(ν) skew(ν) qf(ν;p)
Student's t t. pdf(ν;x) cdf(ν;x) mu(ν) var(ν) skew(ν) qf(ν;p)

* The F parameter in the frequency distribution functions indicates a frequency distribution of the form [[value frequency] ...].

The interactive plotter contains functions for highlighting areas under the graphs of probability functions. They can be used, for example, to illustrate confidence intervals and critical regions. For instance, a 90% confidence interval under a t-distribution curve with ν=3 degrees of freedom can be plotted as follows using the vplot interface:

    set.x(-3;3;1)
    set.y(0;1;0.1)
    t::t.qf(3;0.95)
    set.fill(-t;t;0)
    v.plot(t.pdf(3;))

The output will look similar to that to fig.7. Fig.8 shows an α=0.1 critical region in a χ2 distribution with ν=3. For Klong code using the nplot interface, see the programs linked in under the figures.


Fig.7: larger. Klong
Fig.8: larger. Klong

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