#C9.txt, Maple Code for Lecture 9, of Dr.Z.'s ExpMath class, Spring 2021
Help9:=proc(): print(`RRV(n,K), Ave(X), Mom(X,k) , SA(X,K), Pgf(X,x) `): end:

#RRV(n,K): A random random variable on {1, ... n}
#where the values are from 0 to K (uniformly)
RRV:=proc(n,K) local i,ra:
ra:=rand(0..K):
[ seq(ra(),i=1..n)]:
end:


#Ave(X): inputs a RV X (given as list of values where the value of
#Mr. or Ms. i is L[i]) outputs the average (alias expectation, alias mean)
Ave:=proc(X) local i:add(X[i],i=1..nops(X))/nops(X):end:

#Mom(X,k): inputs a RV X (given as list of values where the value of
#Mr. or Ms. i is L[i]) outputs the k-th moment about the mean
Mom:=proc(X,k) local i,mu:
mu:=Ave(X):add((X[i]-mu)^k ,i=1..nops(X))/nops(X):
end:

#SA(X,K): inputs a discrete r.v. X (given as a list of numbers) and
#a pos. integer K outputs a list of length K whose
#(i) first entry is the average
#(ii) second entry variance
#The remaining entries are the scaled-third moment (about the mean) aka skeweness
#scaled fourth-moment (about the mean) aka kurtosis and so on
#up to the K-th moment
SA:=proc(X,K) local i,M,mu,sig2:
if not (type(X,list) and type(K,integer) and K>=2) then
 print(`Bad input`):
 RETURN(FAIL):
fi:
mu:=Ave(X,1):
sig2:=Mom(X,2):
M:=[mu,sig2]:

if sig2=0 then
 RETURN(M):
fi:

[op(M), seq(evalf(Mom(X,i)/sig2^(i/2)),i=3..K)]:

end:

#Pgf(X,x): inputs a RV X (given as list of values where the value of
#Mr. or Ms. i is L[i]) outputs the prob. generating function of X
#using the variable
Pgf:=proc(X,x) local i:add(x^X[i],i=1..nops(X))/nops(X):end:

#f:=Sum(x^a[i], i=1..n);
#diff(f,x)= subs(x=1,Sum(a[i]*x^(i-1),i=1..n))/n;
#SAc(f,x,K): The SA(X,x,K) of the rv X such whose Pgf is f of x
SAc:=proc(f,x,K) local f1:
if subs(x=1,f)=0 then
 RETURN(0):
fi:

f1:=f/subs(x=1,f):

mu:=subs(x=1,x*diff(f1,x)):
#TO BE CONTINUED AS HOMEWORK
end: