Prepared by Guoqi Qian 1
MAST20006 Probability for Statistics / MAST90057 Elements of Probability

Maple Reference Sheet

• All commands and constants must be typed in the case (UPPER or lower) that is
indicated.
• Description of a topic can be displayed by typing help(topic).

Arithmetic Operations

+ Addition ! subtraction
* Multiplication / division
^ exponentiation ! factorial

Some Mathematical Constants

Pi " exp(1) e -infinity !# infinity #

Mathematical Functions and Expressions

f:= expression; labels the expression as f
g:=unapply(f,x); define g as a function of variable x with
expression f
h:=unapply(f,x,y); define h as a function of variables x and y with
expression f
binomial(n,r); combinatorial number of n choosing r
a*(b+2*a)+c^2+(a+b)/c;
c
ba
caba
+
+++
2)2(
exp(x) x
e
log(x), ln(x) )ln(),log( xx
abs(x) || x
min(a1,a2,…), max(a1,a2,…) minimum/maximum of a sequence a1, a2, …
sin(x),cos(x),tan(x),csc(x),… trigonometric functions
arcsin(x),arccos(x),arctan(x)… inverse trigonometric functions

Some useful commands

% refer to the immediate past display
assign(a,b); or assign(a=b); assign name or function a an expression b
assume(a>0, b<=2); assume a>0 and b$2
diff(f,x); differentiate f w.r.t x
diff(f,x$k); compute the kth derivative of f w.r.t. x
eval(f,x=a); evaluate the expression f at x=a
evalf(f); numerical evaluation of expression f
factor(f); factorize a multivariate polynomial f
int(f,x=a..b);
!
b
a fdx
int(int(f, x=a..b), y=c..d);
! !
d
c
b
a dxdyf
limit(f,x=a); f
ax!
lim
piecewise(cond_1,f_1,cond_2,f_2,
…,cond_n,f_n, f_otherwise);
create a piecewise-continuous function
Prepared by Guoqi Qian 2
plot(f,x=a..b); plot the expression f against x on (a,b)
simplify(f); apply simplification rules to f
solve(f,x); fsolve(f,x); Solve the equation f=0 for x.
subs(x=a,f); substitute a for x in the expression f
sum(f,k=m..n); ! =
n
mk
f
sum(sum(f,k=m..n),j=i..l); ! != =
l
ij
n
mk
f
taylor(f,x=a,n); computes a truncated Taylor series (up to the
nth order term) of f, expanded about x=a
unassign(‘a1’,‘a2’,…); remove the values of a1, a2, etc.
with(Statistics): call Statistics package

Random variables and probability distributions

Discrete distributions: Binomial(n,p), Geometric(p),
Hypergeometric(N1+N2, N1, n),
NegativeBinomial(r,p),
Poisson(lambda).
Continuous distributions: Uniform(a,b), Exponential(theta),
Gamma(theta,alpha), ChiSquare(df),
Normal(mu,sigma).

X:=RandomVariable(Distribution(PDF=f)); create a random
variable X with the
pdf specified by a
separately created
function f.
X:=RandomVariable(DistributionName(parameters));

e.g. X:=RandomVariable(Uniform(a,b));
create a random
variable X having a
Maple-built-in
distribution specified
by its name and
parameters.

ProbabilityFunction(X,x); pmf of a discrete random variable X, )( xXP ==
PDF(X,x); pdf of a continuous random variable X, )(xf X=
CDF(X,x); cdf of a continuous random variable X, )(xFX=
MGF(X,t);
mgf of a random variable X, ][)( tXeEtM ==
Probability({X>a, X<=b}) ),( bXaXP !>
Percentile(X, 95);
Percentile(X, percent);
compute the 95
th
percentile of X
compute the “percent” percentile of X
Mean(X); )(XE
Variance(X); )(Var X
Moment(X,k);
)( kXE
A:=Sample(X,n); generate a sample of n numbers from X and save
the sample into A.
Histogram(A, range=a..b); plot a histogram of A
DensityPlot(X,range=a..b); plot the density function of X


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