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MAST20006 Probability for Statistics / MAST90057 Elements of Probability

Maple Reference Sheet •

All commands and constants must be typed i n th e case (UPPER or lower) that is

•

indicated. Description of a topic can be display ed by typi ng help(topic) .

Arithmetic Operations + * ^

Addition Multiplication exponentiation

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exp(1) e

-infinity

!#

binomial(n,r); a*(b+2*a)+c^2+(a+b)/c; exp(x) log(x), ln(x) abs(x) min(a1,a2,…), max(a1,a2,…) sin(x),cos(x),tan(x),csc(x),… arcsin(x),arccos(x),arctan(x)…

Continuous distributions:

ex log( x), ln( x) |x| minimum/maximum of a sequence a1, a2, … trigonometric functions inverse trigonometric functions

Some useful commands % assign(a,b); or assign(a=b); assume(a>0, b0 and b$2 differentiate f w.r.t x compute the kth derivative of f w.r.t. x evaluate the expression f at x=a numerical evaluation of expression f factorize a multivariate polynomial f b ! a fdx d b ! c ! a f dxdy

int(int(f, x=a..b), y=c..d); limit(f,x=a);

computes a truncated Taylor series (up to the nth order term) of f, expanded about x=a remove the values of a1, a2, etc. call Statistics package

infinity #

labels the expression as f define g as a function of variable x with expression f define h as a function of variables x and y with expression f combinatorial number of n choosing r a +b a (b + 2a ) + c 2 + c

h:=unapply(f,x,y);

! lj =i ! kn =m f

taylor(f,x=a,n);

Discrete distributions:

Mathematical Functions and Expressions f:= expression; g:=unapply(f,x);

sum(sum(f,k=m..n),j=i..l);

Random variables and probability distributions

Some Mathematical Constants Pi

plot the expression f against x on (a,b) apply simplification rules to f Solve the equation f=0 for x. substitute a for x in the expression f ! kn= m f

unassign(‘a1’,‘a2’,…); with(Statistics):

subtraction division factorial

! / !

plot(f,x=a..b); simplify(f); solve(f,x); fsolve(f,x); subs(x=a,f); sum(f,k=m..n);

X:=RandomVariable(Distribution(PDF=f));

create a random variable X with the pdf specified by a separately created function f. X:=RandomVariable(DistributionName(parameters)); create a random variable X having a e.g. X:=RandomVariable(Uniform(a,b)); Maple-built-in distribution specified by its name and parameters. ProbabilityFunction(X,x); PDF(X,x); CDF(X,x); MGF(X,t); Probability({X>a, X a, X ! b) compute the 95th percentile of X compute the “percent” percentile of X E( X ) Var( X ) k E (X ) generate a sample of n numbers from X and save the sample into A. plot a histogram of A plot the density function of X

x! a

piecewise(cond_1,f_1,cond_2,f_2, …,cond_n,f_n, f_otherwise); 1

create a piecewise-continuous function Prepared by Guoqi Qian

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Prepared by Guoqi Qian...