VISUALIZATION OF DENSITY FUNCTIONS WITH GEOGEBRA · Continuous distributions Statistical functions...

Continuous distributions Statistical functions in GeoGebra Visualization and Curve fitting

VISUALIZATION OF DENSITY FUNCTIONSWITH GEOGEBRA

Csilla Csendes

University of Miskolc, HungaryDepartment of Applied Mathematics

ICAM 2010


Probability density functions

A random variable X has density f , where f is a non-negativeLebesgue-integrable function, if:

P[a ≤ X ≤ b] =

∫ b

af (x) dx .

Hence, if F is the cumulative distribution function of X , then:

F (x) =

∫ x

−∞f (u) du

Intuitively, one can think of f (x)dx as being the probability of Xfalling within the infinitesimal interval [x, x + dx].


Stable distributions


Definition (broad sense)Let X ,X1,X2, ... be iid. random variables. The distribution of Xis stable if it is not concentrated at one point and if for each nthere exist constants an > 0 and bn such that

X1 + X2 + ...+ Xn

an− bn

has the same distribution as X .



Properties

infinite variancenon-known density and distribution functiondescribe with characteristic functionapplications: finance, signal processing, etc.class of possible limit distributions as a solution to thedomain of attraction problem



Parameters

Characterizationcharacteristic exponent or index of stability α ∈ (0,2]

skewness β ∈ [−1,1]

scale γ ≥ 0location δ ∈ R

Characteristic function

φ(u|α, β, γ, δ) = E exp(iuZ ) = exp(−γα[|u|α + iβη(u, α)] + iuδ),

η(u, α) =

{−(signu)tan(πα/2)|u|α, if α 6= 1,(2/π)u ln |u|, if α = 1.



Density functions

α = 1 - Cauchy distributionα = 2 - Normal distribution


Other Continuous Distributions

Exponential distribution

The exponential distribution occurs naturally when describingthe lengths of the inter-arrival times in a homogeneous Poissonprocess.

parameter λ > 0 (rate parameter)An exponential random sample can be generated asln(1− U) where U is uniformly distributed.

Probability density function:

f (x) =

{λe−λx if x ≥ 0,0, otherwise



Gamma distribution

The pdf. of the gamma distribution can be expressed interms of the gamma function parameterized in terms of ashape parameter k and scale parameter θ. Both k and θwill be positive values.sum of k independent exponentially distributed randomvariables, each of which has a mean of θfrequently a probability model for waiting times; forinstance, in life testing, the waiting time until death

Probability density function

f (x ; θ, k) = xk−1 e−x/θ

θk Γ(k)x ≥ 0, k , θ > 0



Chi-square distribution

If X1, ...,Xk are independent, standard normal randomvariables, then the sum of their squares

Q =k∑

i=1

X 2i

is distributed according to the chi-square distribution with kdegrees of freedom.

Probability density function

f (x ; k) =1

2k/2Γ(k/2)xk/2−1e−x/2


Statistical tools

Basics

Mean[list of numbers L]Mode[list of numbers L]Median[list of numbers L]Variance[list of numbers L]CorrelationCoefficient[List of x-Coordinates, List ofy-Coordinates]


Statistical tools

BoxPlot

BoxPlot[yOffset, yScale, List of Raw Data]: Creates abox plot using the given raw data and whose verticalposition in the coordinate system is controlled by variableyOffset and whose height is influenced by factor yScale.BoxPlot[yOffset, yScale, Start Value a, Q1, Median, Q3,End Value b]: Creates a box plot for the given statisticaldata in interval [a, b].


Statistical graphics

Histograms

A histogram is a graphical display of tabular frequencies,shown as adjacent rectangles.A histogram may also be based on relative frequencies. Itthen shows the proportion of cases that fall into each ofseveral categories, with the total area equaling 1.Histograms are used to plot density of data, and often fordensity estimation: estimating the probability densityfunction of the underlying variable. The total area of ahistogram used for probability density is always normalizedto 1.


GeoGebra functions - Syntax

Histogram

Histogram[List of Class Boundaries, List of Heights]:Creates a histogram with bars of the given heights. Theclass boundaries determine the width and position of eachbar of the histogram.Histogram[List of Class Boundaries, List of Raw Data]:Creates a histogram using the raw data. The classboundaries determine the width and position of each bar ofthe histogram and are used to determine how many dataelements lie in each class.


GeoGebra functions - Syntax

BarChart

BarChart[Start Value, End Value, List of Heights]: Creates abar chart over the given interval where the number of bars isdetermined by the length of the list whose elements are theheights of the bars.

BarChart[Start Value a, End Value b, Expression, Variable k,From Number c, To Number d]: Creates a bar chart over thegiven interval [a, b], that calculates the bars heights using theexpression whose variable k runs from number c to number d.

BarChart[Start Value a, End Value b, Expression, Variable k,From Number c, To Number d, Step Width s]: Creates a barchart over the given interval [a, b], that calculates the barsheights using the expression whose variable k runs from numberc to number d using step width s.

BarChart[List of Raw Data, Width of Bars]: Creates a barchart using the given raw data whose bars have the given width.


Histograms

Normal distribution


Histograms

Levy distribution


Histograms

Cauchy distribution


Curve fitting commands in GeoGebra

Commands

FitExp[List of Points] - Calculates the exponentialregression curve.FitLog[List of Points] - Calculates the logarithmicregression curve (i.e. the regression curve of the formy=A+Bln(x)).FitPoly[list of points P, number N] - Calculates theregression polynomial of degree N.FitPow[list of points P] - Calculates the regression curve inthe form y = axb.



Normal distribution, polynomials of 8 and 9 degrees



Normal distribution, polynomials of 10 and 12 degrees



Cauchy distribution, polynomials of 8 and 9 degrees



Cauchy distribution, polynomials of 12 and 13 degrees



Exponential distribution, exponential curve

f (x) = 0.29e−1.06x



THANKS FOR YOUR ATTENTION!

VISUALIZATION OF DENSITY FUNCTIONS WITH GEOGEBRA · Continuous distributions Statistical functions...

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