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# 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]:= <<StatisticsContinuousDistributions

In[2]:= <<StatisticsDiscreteDistributions

Among the distributions available are
• Continuous Distributions
• CauchyDistribution[a, b]
• ChiSquareDistribution[n]
• ExponentialDistribution[]
• FRatioDistribution[n1, n2]
• NormalDistribution[, ]
• StudentTDistribution[n]
• UniformDistribution[min, max]
• Discrete Distributions
• BinomialDistribution[n, p]
• GeometricDistribution[p]
• HypergeometricDistribution[n, , ]
• NegativeBinomialDistribution[r, p]
• PoissonDistribution[]
Warning: The parameters of these distributions do not always agree with textbooks. In particular, the in the gamma distribution is the reciprocal of the 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

PDF[dist, x] the p. d. f. f(x). the c. d. f. F(x). the inverse c. d. f. F-1(x). the mean the variance the standard deviation a random variate

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]:= <<StatisticsDescriptiveStatistics

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


Up: Computer Help
Charles Geyer
8/10/1999