Getting Started with Mathematica

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.

Getting In and Out

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.

Integrals

Indefinite Integrals

In[1]:= Integrate[x^2 Exp[- x], x]

                    2
        -2 - 2 x - x
Out[1]= -------------
              x
             E

Definite Integrals

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]

Differentiation

Ordinary and Partial Derivatives

In[5]:= D[1 / (1 - t / lambda)^alpha, t]

                     t    -1 - alpha
        alpha (1 - ------)
                   lambda
Out[5]= ----------------------------
                   lambda

Higher-Order Derivatives

In[6]:= D[1 / (1 - t / lambda)^alpha, {t, 2}]

                                    t    -2 - alpha
          (-1 - alpha) alpha (1 - ------)
                                  lambda
Out[6]= -(-----------------------------------------)
                                 2
                           lambda

Defining Functions

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
        E

defines 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]= lambda

calculate the mean and variance.

Probability Distributions

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

• Continuous Distributions
  • BetaDistribution[$\alpha,$ $\beta$]
  • CauchyDistribution[a, b]
  • ChiSquareDistribution[n]
  • ExponentialDistribution[$\lambda$]
  • FRatioDistribution[n₁, n₂]
  • GammaDistribution[$\alpha$, $\lambda$]
  • NormalDistribution[$\mu$, $\sigma$]
  • StudentTDistribution[n]
  • UniformDistribution[min, max]
• Discrete Distributions
  • BinomialDistribution[n, p]
  • GeometricDistribution[p]
  • HypergeometricDistribution[n, $n_{\textit{succ}}$, $n_{\textit{tot}}$]
  • NegativeBinomialDistribution[r, p]
  • PoissonDistribution[$\mu$]

Warning: The parameters of these distributions do not always agree with textbooks. In particular, the $\lambda$ in the gamma distribution is the reciprocal of the $\lambda$ in some textbooks.

In[3]:= PDF[GammaDistribution[alpha, lambda], x]

                    -1 + alpha
                   x
Out[3]= ----------------------------------
         x/lambda       alpha
        E         lambda      Gamma[alpha]

Operations on these distributions include

Examples

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