VISUALIZATION OF DENSITY FUNCTIONS WITH GEOGEBRA
Csilla Csendes
University of Miskolc, Hungary Department of Applied Mathematics
ICAM 2010
Probability density functions
Probability density functions
Random variable
A random variable is defined as a measurable functionX from a probability space(Ω,F,P)to measurable space(X,A).
A probability density function is most commonly associated with continuous univariate distributions.
Probability density functions
Probability density functions
A random variableX has densityf, wheref is a non-negative Lebesgue-integrable function, if:
P[a≤X ≤b] = Z b
a
f(x)dx.
Hence, if F is the cumulative distribution function ofX, then:
F(x) = Z x
−∞
f(u)du
Intuitively, one can think off(x)dx as being the probability ofX falling within the infinitesimal interval [x, x + dx].
Continuous Distributions
Stable Distributions
Normal distribution (α=2) Levy distribution (α=1.5) Cauchy distribution (α=1) Characterization
characteristic exponent or index of stabilityα∈(0,2]
skewnessβ ∈[−1,1]
scaleγ ≥0 locationδ ∈R
Continuous Distributions
Stable density functions
α=1 - Cauchy distribution α=2 - Normal distribution
Continuous Distributions
Exponential distribution
withλ >0 parameter density function:
f(x) =λe−λx if x ≥0, 0 otherwise
An exponential random sample can be generated as ln(1−U) whereUis uniformly distributed.
Histograms
Histograms
A histogram is a graphical display of tabular frequencies, shown as adjacent rectangles.
A histogram may also be based on relative frequencies. It then shows the proportion of cases that fall into each of several categories, with the total area equaling 1.
Histograms are used to plot density of data, and often for density estimation: estimating the probability density function of the underlying variable. The total area of a histogram used for probability density is always normalized to 1.
Histograms
GeoGebra functions - Syntax
Histogram
Histogram[List of Class Boundaries, List of Heights]:
Creates a histogram with bars of the given heights. The class boundaries determine the width and position of each bar of the histogram.
Histogram[List of Class Boundaries, List of Raw Data]:
Creates a histogram using the raw data. The class
boundaries determine the width and position of each bar of the histogram and are used to determine how many data elements lie in each class.
GeoGebra functions - Syntax
BarChart
BarChart[Start Value, End Value, List of Heights]: Creates a bar chart over the given interval where the number of bars is determined by the length of the list whose elements are the heights of the bars.
BarChart[Start Value a, End Value b, Expression, Variable k, From Number c, To Number d]: Creates a bar chart over the given interval [a, b], that calculates the bars heights using the expression 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 bar chart over the given interval [a, b], that calculates the bars heights using the expression whose variable k runs from number c to number d using step width s.
BarChart[List of Raw Data, Width of Bars]: Creates a bar chart 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 exponential regression curve.
FitLog[List of Points] - Calculates the logarithmic regression curve (i.e. the regression curve of the form y=A+Bln(x)).
FitPoly[list of points P, number N] - Calculates the regression polynomial of degree N.
FitPow[list of points P] - Calculates the regression curve in the form y=axb.
Curve fitting commands in GeoGebra
Normal distribution, polynomials of 8 and 9 degrees
Curve fitting commands in GeoGebra
Normal distribution, polynomials of 10 and 12 degrees
Curve fitting commands in GeoGebra
Cauchy distribution, polynomials of 8 and 9 degrees
Curve fitting commands in GeoGebra
Cauchy distribution, polynomials of 12 and 13 degrees
Curve fitting commands in GeoGebra
Exponential distribution, exponential curve
0.29e−1.06x
Curve fitting commands in GeoGebra
THANKS FOR YOUR ATTENTION!
