Download VISUALIZATION OF DENSITY FUNCTIONS WITH GEOGEBRA

Continuous distributions
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VISUALIZATION OF DENSITY FUNCTIONS
WITH GEOGEBRA
Csilla Csendes
University of Miskolc, Hungary
Department of Applied Mathematics
ICAM 2010
Continuous distributions
Statistical functions in GeoGebra
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Probability density functions
A random variable X has density f , where f is a non-negative
Lebesgue-integrable function, if:
Z
P[a ≤ X ≤ b] =
b
f (x) dx.
a
Hence, if F is the cumulative distribution function of X , then:
Z x
F (x) =
f (u) du
−∞
Intuitively, one can think of f (x)dx as being the probability of X
falling within the infinitesimal interval [x, x + dx].
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Stable distributions
Stable distributions
Definition (broad sense)
Let X , X1 , X2 , ... be iid. random variables. The distribution of X
is stable if it is not concentrated at one point and if for each n
there exist constants an > 0 and bn such that
X1 + X2 + ... + Xn
− bn
an
has the same distribution as X .
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Stable distributions
Properties
infinite variance
non-known density and distribution function
describe with characteristic function
applications: finance, signal processing, etc.
class of possible limit distributions as a solution to the
domain of attraction problem
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Stable distributions
Parameters
Characterization
characteristic exponent or index of stability α ∈ (0, 2]
skewness β ∈ [−1, 1]
scale γ ≥ 0
location δ ∈ R
Characteristic function
φ(u|α, β, γ, δ) = E exp(iuZ ) = exp(−γ α [|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.
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Stable distributions
Density functions
α = 1 - Cauchy distribution
α = 2 - Normal distribution
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Other Continuous Distributions
Exponential distribution
The exponential distribution occurs naturally when describing
the lengths of the inter-arrival times in a homogeneous Poisson
process.
parameter λ > 0 (rate parameter)
An exponential random sample can be generated as
ln(1 − U) where U is uniformly distributed.
Probability density function:
λe−λx
f (x) =
0,
if
x ≥ 0,
otherwise
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Other Continuous Distributions
Gamma distribution
The pdf. of the gamma distribution can be expressed in
terms of the gamma function parameterized in terms of a
shape parameter k and scale parameter θ. Both k and θ
will be positive values.
sum of k independent exponentially distributed random
variables, each of which has a mean of θ
frequently a probability model for waiting times; for
instance, in life testing, the waiting time until death
Probability density function
f (x; θ, k ) = x k −1
e−x/θ
θk Γ(k )
x ≥ 0, k , θ > 0
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Other Continuous Distributions
Chi-square distribution
If X1 , ..., Xk are independent, standard normal random
variables, then the sum of their squares
Q=
k
X
Xi2
i=1
is distributed according to the chi-square distribution with k
degrees of freedom.
Probability density function
f (x; k ) =
1
2k /2 Γ(k /2)
x k /2−1 e−x/2
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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 of
y-Coordinates]
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Statistical tools
BoxPlot
BoxPlot[yOffset, yScale, List of Raw Data]: Creates a
box plot using the given raw data and whose vertical
position in the coordinate system is controlled by variable
yOffset 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 statistical
data in interval [a, b].
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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. 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.
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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.
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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.
Continuous distributions
Histograms
Normal distribution
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Continuous distributions
Histograms
Levy distribution
Statistical functions in GeoGebra
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Continuous distributions
Histograms
Cauchy distribution
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Continuous distributions
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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 = ax b .
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Curve fitting commands in GeoGebra
Normal distribution, polynomials of 8 and 9 degrees
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Normal distribution, polynomials of 10 and 12 degrees
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Cauchy distribution, polynomials of 8 and 9 degrees
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Cauchy distribution, polynomials of 12 and 13 degrees
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Exponential distribution, exponential curve
f (x) = 0.29e−1.06x
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Continuous distributions
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THANKS FOR YOUR ATTENTION!
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