# MatheAss - Stochastics

## Statistics

For a master list, the mean (arithmetic mean), the central value (median), the variance and the standard deviation are determined. In addition, the distribution is output as a histogram and as a box plot.

```Data:
9 6 7 7 3 9 10 1 8 7 9 6 9 8 10 5 10 10 9 11 8

Number of data        n = 21
Maximum            max = 11
Minimum              min = 1
Mean                        x = 7,7142857
Median                     c = 8
Variance                 s² = 6,1142857
Standard deviation  s = 2,4727082```

## Regression

With this routine, you can perform a curve adjustment for a series of measurements. You can choose between the following adjustments and, if necessary, move or stretch all points in the x or y direction.

Proportional regression ( y = a·x )

Linear regression ( y = a·x + b )

Polynomial regression n-th order ( y = a0 + ... + an·xn )

Geometric regression ( y = a·xb )

Exponential regression ( y = a·bx )

Logarithmic regression ( y = a + b·ln(x) )

```Polynomial Regression

y =  - 6,9152542
+ 4,7189266·x
- 0,43361582·x2

Coeff.of determin.   = 0,98338318
Correlation coeff.    = 0,99165679
Standard deviation = 0,46028731```

## Logistic Regression(New in version 9.0)

The pogram determines for a series of measurements a curve fit to the logistic function
with the parameters  a1 = ƒ(0)· S , a2 = ƒ(0) , a3 = S - ƒ(0) , and a4 = -k· S  and the saturation limit S .

```Data from:  "hopfenwachstum.csv"

Saturation limit:  6
Dark figure:  1

4,0189
ƒ(x) = ————————————————
0,66981 + 5,3302 · e^(-0,35622·t)

Inflection point W(5,8226/3)

Maximum growth rate ƒ'(xw) = 0,53433

8 Values
Coeff.of determin.   = 0,99383916
Correlation coeff.    = 0,99691482
Standard deviation = 0,16172584```

## Combinatorics

The number of possibilities to select k from n elements is calculated if the order is valued or not and if repetitions are allowed or not.

```n = 49
k = 6

Arrangements without repetit. = 10 068 347 520
Arrangements with repetitions = 13 841 287 201
Combinations without repetit. = 13 983 816
Combinations with repetitions = 25 827 165

Permutations of k :        k! = 720```

## Binomial Distribution

Calculated for a b(k;n;p) distributed random quantity X at fixed n and fixed p

- a rod diagram of the probabilities P(X=k)
- their numerical values in an interval [k-min;k-max]
- the probability P( k-min < = X <= k-max)

```n = 50             p = 0,3

k             P(X=k)         P(0<=X<=k)
¯¯¯¯    ¯¯¯¯¯¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯
8         0,01098914     0,01825335
9         0,02197829     0,04023163
10        0,03861899     0,07885062
11        0,06018544     0,13903606
12        0,08382972     0,22286578
13        0,10501745     0,32788324
14        0,11894834     0,44683157
15        0,12234686     0,56917844
16        0,11470018     0,68387862
17        0,09831444     0,78219306
18        0,07724706     0,85944012
19        0,05575728     0,91519740
20        0,03703876     0,95223616
21        0,02267679     0,97491296
22        0,01281092     0,98772387
¯¯¯¯    ¯¯¯¯¯¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯
P(8<=k<=22)   =          0,98045967```

## Hypergeometric Distribution

Calculations are made for a random variable X distributed h (k; n; m; r) with a fixed n, m and fixed r a bar chart and a table of values for the probabilities P(X=k).

## Normal Distribution

Calculations are carried out for an N(µ, s2) distributed random variable X with a given expected value µ and variance s2 , the density function ƒ(x) and the distribution function Φ(x), i.e. the integral over ƒ(x).

```  μ = 5 ,      σ = .75

x                    ƒ(x)                Φ(x)
¯¯¯¯¯¯¯¯¯¯  ¯¯¯¯¯¯¯¯¯¯  ¯¯¯¯¯¯¯¯¯¯
2                    0,00017844   0,00003167
2,33333333   0,00095649   0,00018859
2,66666666   0,00420802   0,00093192
2,99999999   0,01519465   0,00383038
3,33333332   0,04503153   0,01313415
3,66666665   0,10953585   0,03772017
3,99999998   0,21868009   0,09121120
4,33333331   0,35832381   0,18703139
4,66666664   0,48189843   0,32836063
4,99999997   0,53192304   0,49999998
5,3333333     0,48189845   0,67163934
5,66666663   0,35832383   0,81296859
5,99999996   0,21868012   0,90878878
6,33333329   0,10953586   0,96227982
6,66666662   0,04503154   0,98686585
6,99999995   0,01519465   0,99616962
7,33333328   0,00420802   0,99906808
7,66666661   0,00095649   0,99981141
7,99999994   0,00017844   0,99996833
```