Lgraczyk: Created page with "{| align="right" | TOC |} == Exercise == The following probability density function is given: File:Lab04b_funkcja.png Please do: * calculate constant c in the way the p..."

2022-03-21T12:23:39Z

Created page with "{| align="right" | __TOC__ |} == Exercise == The following probability density function is given: File:Lab04b_funkcja.png Please do: * calculate constant c in the way the p..."

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== Exercise ==
The following probability density function is given:
[[File:Lab04b_funkcja.png]]

Please do:
* calculate constant c in the way the probability distribution is normalized
* randomly select from the probability distribution a pair of numbers (x,y) and then fill with the a histogram of the probability distribution f(x,y) (0.5pkt)
* normalize the histogram of the probability distribution (0.5pkt)
* calculate and draw the histogram of the cumulative distribution F(x,y) (1pkt)
* calculate and draw the histograms of the marginal distributions g(x) i h(y) (1pkt)
* calculate:
** expected values: E(X), E(Y) (0.5pkt)
** standard deviations sigma(X), sigma(Y) (0.5pkt)
** covariance cov(X,Y) (0.5pkt)
** correlation coefficient rho(X,Y) (0.5pkt)

== Attention ==
* As minimum and maximum on x and y axed in all objects we set 0 and PI/2
* For work with histograms we used objects <code>TH1D</code> and <code>TH2D</code>. Some examples can be found in: [ftp://root.cern.ch/root/doc/3Histograms.pdf Histograms]
* The histogram of the probability density we create by random selection of pseudorandom numbers from the probability density functions (we create a TF2 object like previous lab and we implement a loop until a certain number, for each iteration we use '''GetRandom2''' to select those 2 numbers and then we use a function called '''Fill'' to fill the density histogram)
* Cumulative distribution is calculated '''numerically''' (two '''for''' loops, and we integrate the density iterating over x and y, by '''SetBinContent''' we set the specific valued of the histogram)
* Marginal distributions have their own methods for histograms ('''hint:''' they are called projections)
* For parameters (means, standard deviations, covariances, correlation factors) - there are specific methods

== Result ==

Plots:
[[File:lab04_KADD2016.png]]

Rotated distribution:
[[File:lab04b_KADD2016.png]]

Output:
E(X)=0.990827
E(Y)=0.990535
sigma(X)=0.377467
sigma(Y)=0.377583
cov(X,Y)=-0.0137694
rho(X,Y)=-0.09661

KADD 2022 Laboratorium 4 EN - Revision history

Lgraczyk: Created page with "{| align="right" | __TOC__ |} == Exercise == The following probability density function is given: File:Lab04b_funkcja.png Please do: * calculate constant c in the way the p..."

Lgraczyk: Created page with "{| align="right" | TOC |} == Exercise == The following probability density function is given: File:Lab04b_funkcja.png Please do: * calculate constant c in the way the p..."