restart;

with(Student[Statistics]):

A := BinomialRandomVariable(5, 1/3):

Probability({A>=2, A<=4});

$\frac{130}{243}$

Mean(A);

$\frac{5}{3}$

Correlation(A, A^2);

$\frac{11}{1370}\sqrt{10}\sqrt{1370}$

M := Matrix([[1,2,4,1],[2,4,1,2],[6,5,2,7],[1,2,0,9]]);

Variance(M);

L := [1, 2, 3, 1, 2, 3, 1, 2, 2, 2, 6, 2, 3, 4, 5, 2, 4]:

Mode(L);

$\left\{2\right\}$

Percentile(L, 30);

$2$

B := NormalRandomVariable(1,2):

Probability(B<x, inert);

${\&int;}_{-\mathrm{\&infin;}}^x\frac{1}{4}\frac{\sqrt{2}{\&ExponentialE;}^{-\frac{1}{8}{\left(\mathrm{\_t2}-1\right)}^2}}{\sqrt{\mathrm{\&pi;}}}\&DifferentialD;\mathrm{\_t2}$

CumulativeDistributionFunction(B, 3, output=plot);

X := [6, 3, 2, 1, 9, 1, 2, 3, 3, 2]:

StandardDeviation(X);

$\frac{1}{15}\sqrt{1390}$

ChiSquareSuitableModelTest(X, PoissonRandomVariable(3));

Chi-Square Test for Suitable Probability Model

----------------------------------------------
Null Hypothesis:
Sample was drawn from specified probability distribution
Alt. Hypothesis:
Sample was not drawn from specified probability distribution
Bins:                    5
Degrees of freedom:      4
Distribution:            ChiSquare(4)
Computed statistic:      2.20801
Computed pvalue:         0.697563
Critical value:          9.48772903678116
Result: [Accepted]
There is no statistical evidence against the null hypothesis

$\left[\mathrm{hypothesis}\&equals;\mathrm{true}\&comma;\mathrm{criticalvalue}\&equals;9.48772903678116\&comma;\mathrm{distribution}\&equals;\mathrm{ChiSquare}\left(4\right)\&comma;\mathrm{pvalue}\&equals;0.697562873819393\&comma;\mathrm{statistic}\&equals;2.20801128786324\right]$

ExploreRV(NormalRandomVariable(mu, sigma));

Distribution1 := BinomialRandomVariable(7,1/2):

Mean(Distribution1);

$\frac{7}{2}$

StandardDeviation(Distribution1);

$\frac{1}{2}\sqrt{7}$

StandardDeviation(Distribution1, numeric);

$1.322875656$

ProbabilityFunction(Distribution1,x,output=plot);

CDF(Distribution1, 3, output = plot);

Probability(Distribution1 <= 4, inert);

$\sum _{\mathrm{\_t3}\&equals;0}^4\left(\&lcub;\begin{array}{cc}0& \mathrm{\_t3}<0\\ \mathrm{binomial}\left(7\&comma;\mathrm{\_t3}\right){\left(\frac{1}{2}\right)}^{\mathrm{\_t3}}{\left(\frac{1}{2}\right)}^{7-\mathrm{\_t3}}& \mathrm{otherwise}\end{array}\right)$

Sample1 := Sample(ExponentialRandomVariable(5), 1000);

InterquartileRange(Sample1);

$5.21348341554859$

Median(Sample1);

$3.06138615867682$

Sample(ExponentialRandomVariable(5), 1000, output = plot);

ChiSquareSuitableModelTest(Sample1, ExponentialRandomVariable(5));

Chi-Square Test for Suitable Probability Model

----------------------------------------------
Null Hypothesis:
Sample was drawn from specified probability distribution
Alt. Hypothesis:
Sample was not drawn from specified probability distribution
Bins:                    32
Degrees of freedom:      31
Distribution:            ChiSquare(31)
Computed statistic:      33.472
Computed pvalue:         0.348188
Critical value:          44.9853428040743
Result: [Accepted]
There is no statistical evidence against the null hypothesis

$\left[\mathrm{hypothesis}\&equals;\mathrm{true}\&comma;\mathrm{criticalvalue}\&equals;44.9853428040742\&comma;\mathrm{distribution}\&equals;\mathrm{ChiSquare}\left(31\right)\&comma;\mathrm{pvalue}\&equals;0.348188188879230\&comma;\mathrm{statistic}\&equals;33.47200000\right]$

Matrix1 := Matrix(5, 3, {(1, 1) = 1, (1, 2) = 2, (1, 3) = 3, (2, 1) = 2, (2, 2) = Pi, (2, 3) = 5, (3, 1) = 9, (3, 2) = 7, (3, 3) = 3, (4, 1) = 5, (4, 2) = 5, (4, 3) = 2, (5, 1) = 2, (5, 2) = 8, (5, 3) = 10}):

Mean(Matrix1);

Value, Graph := InterquartileRange(Matrix1, output = both):

Value;

Graph;