Charles J. Geyer (charlie@stat.umn.edu )
Mathematica is far too complex for this page to do more than hint at the possibilities. We just provide a few examples to indicate the possibilities.
To start Mathematica in command line mode, type math at a UNIX prompt. To start Mathematica in notebook mode, type Mathematica at a UNIX prompt. To leave Mathematica type Quit at a Mathematica prompt.
In[1]:= Integrate[x^2 Exp[- x], x] 2 -2 - 2 x - x Out[1]= ------------- x E
In[2]:= Integrate[x^2 Exp[- x], {x, 0, Infinity}] Out[2]= 2 In[3]:= Integrate[x^k Exp[- lambda x], {x, 0, Infinity}] -1 - k Out[3]= lambda Gamma[1 + k]
In[5]:= D[1 / (1 - t / lambda)^alpha, t] t -1 - alpha alpha (1 - ------) lambda Out[5]= ---------------------------- lambda
In[6]:= D[1 / (1 - t / lambda)^alpha, {t, 2}] t -2 - alpha (-1 - alpha) alpha (1 - ------) lambda Out[6]= -(-----------------------------------------) 2 lambda
To define a function of one or more variables, you indicate the free variables in the function definition by placing an underscore after the variable name, for example
In[1]:= f[x_] = lambda Exp[ - lambda x] lambda Out[1]= --------- lambda x Edefines the exponential density function. Then
In[2]:= mu = Integrate[ x f[x], {x, 0, Infinity} ] 1 Out[2]= ------ lambda In[3]:= sigma2 = Integrate[ (x - mu)^2 f[x], {x, 0, Infinity} ] -2 Out[3]= lambdacalculate the mean and variance.
For these you have to load a package, either or both of
In[1]:= <<Statistics`ContinuousDistributions` In[2]:= <<Statistics`DiscreteDistributions`Among the distributions available are
In[3]:= PDF[GammaDistribution[alpha, lambda], x] -1 + alpha x Out[3]= ---------------------------------- x/lambda alpha E lambda Gamma[alpha]
Operations on these distributions include
PDF[dist, x] | the p. d. f. f(x). |
---|---|
CDF[dist, x] | the c. d. f. F(x). |
Quantile[dist, p] | the inverse c. d. f. F-1(x). |
Mean[dist] | the mean |
Variance[dist] | the variance |
StandardDeviation[dist] | the standard deviation |
Random[dist] | a random variate |
In[3]:= dist = NormalDistribution[2, 3] Out[3]= NormalDistribution[2, 3] In[4]:= CDF[dist, -2] Out[4]= 0.0912112 In[5]:= Quantile[dist, 0.091] Out[5]= -2.003866860100579 In[6]:= Mean[dist] Out[6]= 2 In[7]:= StandardDeviation[dist] Out[7]= 3 In[8]:= Random[dist] Out[8]= -1.092274190123777 In[9]:= Random[dist] Out[9]= 0.871083809725889 In[10]:= Table[Random[dist], {5}] Out[10]= {4.712827641845193, 5.556764945655314, -0.089456940162683, > 2.668718852551182, 1.822779619000173}There are also descriptive statistics packages that calculate sample moments.
In[1]:= <<Statistics`DescriptiveStatistics` In[2]:= data = {2.3, 4.5, 1.03, 17.6} Out[2]= {2.3, 4.5, 1.03, 17.6} In[3]:= Mean[data] Out[3]= 6.3575 In[4]:= StandardDeviation[data] Out[4]= 7.63085