MaplePrimes Questions

I found this strange result:
When the central moment of order 2 and the variance are assessed on a sample of a random variable, they don't return the same value.

Let S be the sample, N its size and M its empirical mean.  The difference comes from the fact that 

  • Variance(S)               = add( (S[n]-M)^2, n=1..N) / (N-1)
  • CentralMoment(S, 2) = add( (S[n]-M)^2, n=1..N) / N

By definition the variance of a random variable X is its 2nd order central moment.
This definition should also apply when these statistics are calculated on a sample of X.

It seems to me this is a mistake.
 

restart:

with(Statistics):

N := 100:

# Example 1

U := RandomVariable(BetaDistribution(3, 2)):
CentralMoment(U, 2, numeric);
Variance(U, numeric);

print():
S := Sample(U, N):
CentralMoment(S, 2);
Variance(S);

0.4000000000e-1

 

0.4000000000e-1

 

 

HFloat(0.05053851854005207)

 

HFloat(0.05104900862631522)

(1)

# Example 2

V := RandomVariable(Normal(3, 2)):
CentralMoment(V, 2, numeric);
Variance(V, numeric);

print():

S := Sample(V, N):
CentralMoment(S, 2);
Variance(S)

4.

 

4.

 

 

HFloat(3.7514118336684517)

 

HFloat(3.7893048824933824)

(2)

# Examples revisited

S := Sample(U, N):
CentralMoment(S, 2);
Variance(S)*(N-1)/N;
print():
S := Sample(V, N):
CentralMoment(S, 2);
Variance(S)*(N-1)/N
 

HFloat(0.03346262243902275)

 

HFloat(0.03346262243902278)

 

 

HFloat(4.159851396769612)

 

HFloat(4.159851396769612)

(3)

 


 

Download StrangerThings.mw

Hello. Let's say I have a set A1 in which there are nested sets {5,6},{8,9}. How can I get A2 from A1? Expand nested sets so that all elements are in the same set

The eval command doesn't work on random variables.

For instance:

with(Statistics):
X := RandomVariable(Normal(mu, sigma)):
Y := eval(X, mu=1);
Mean(Y); # returns mu

I once found a Maple command whose syntax is the same as eval's and acts as eval
Y := COMMAND(X, mu=1)
Mean(Y); # returns 1

Could someone remind me of its name?

Thanks in advance

How can i plot a probability function such as cos(x-y)*cos(y-z)*cos^3(x-2z)=0.6 where

x=0..5, y=0..x, z=0..y.

please guide me.

Hi, 

When you declare a random variable X  by writting X := Statistics:-RandomVariable(...) Maple associates to the name X an "object" dubbed _R.
My question is very simple: is it possible to recover the definition of X from the "object" _R?

To illustrate, suppose I've written
> X := Statistics:-RandomVariable(Normal(0, 1)) ;
       _R
I would like to extract the information "_R represents a random variable Normal(0, 1)" from _R

Thanks in advance

 

Dear Maple users,

I am progressing, but one last hitch, see below. A want the invariants of the PDE below. However, the final expression is too general to be useful. I would like to insert specific values for F1(R), F2(R), F3(R), and F4(R):

F3=F4=0; and F2=1, and F3(R)=aR, where a is a constant. R is one of my independent variables (the Reynolds number). 

 

I would like to do this at the step where the Infinitiesimals are generated by

infinies := Infinitesimals(PDE)

For example, the first entry would then be, infinies:=[ _xi[y](y, R, l, u) = y, ... ]. Then Invariants should give me much simplified expressions which I need.  How can i do this?

Thanks

Nadeem

 

with(PDEtools)

declare(u(y, R))

` u`(y, R)*`will now be displayed as`*u

(1)

declare(l(y, R))

` l`(y, R)*`will now be displayed as`*l

(2)

L := diff_table(l(y, R))

table( [(  ) = l(y, R) ] )

(3)

NULL``

U := diff_table(u(y, R))

table( [(  ) = u(y, R) ] )

(4)

DepVars := ([l, u])(y, R)

[l(y, R), u(y, R)]

(5)

PDE := U[y, y]+2*l(y, R)^2*U[y]*U[y, y]+2*l(y, R)*L[y]*U[y]^2+1/R = 0

diff(diff(u(y, R), y), y)+2*l(y, R)^2*(diff(u(y, R), y))*(diff(diff(u(y, R), y), y))+2*l(y, R)*(diff(l(y, R), y))*(diff(u(y, R), y))^2+1/R = 0

(6)

infinies := Infinitesimals(PDE)

[_xi[y](y, R, l, u) = _F2(R)*y+_F3(R), _xi[R](y, R, l, u) = _F1(R), _eta[l](y, R, l, u) = (1/2)*l*(-R*_F2(R)+_F1(R))/R, _eta[u](y, R, l, u) = (2*R*_F2(R)-_F1(R))*u/R+_F4(R)]

(7)

InfinitesimalGenerator(infinies, DepVars, prolongation = 1)

proc (f) options operator, arrow; add(xi[x[j]]*(diff(f, x[j])), j = 1 .. 2)+add(eta[u[m]]*(diff(f, u[m]))+eta[u[m], [y]]*(diff(f, u[m][y]))+eta[u[m], [R]]*(diff(f, u[m][R])), m = 1 .. 2) end proc

(8)

Phi := Invariants(infinies, DepVars)

l*exp(-(1/2)*(Int((-R*_F2(R)+_F1(R))/(R*_F1(R)), R))), -(Int(_F3(R)*exp(-(Int(_F2(R)/_F1(R), R)))/_F1(R), R))+y*exp(-(Int(_F2(R)/_F1(R), R))), u*exp(Int((-2*R*_F2(R)+_F1(R))/(R*_F1(R)), R))-(Int(_F4(R)*exp(Int((-2*R*_F2(R)+_F1(R))/(R*_F1(R)), R))/_F1(R), R)), u[y]*exp(Int((-R*_F2(R)+_F1(R))/(R*_F1(R)), R)), l[y]*exp(-(1/2)*(Int((-3*R*_F2(R)+_F1(R))/(R*_F1(R)), R))), (1/2)*Intat(-exp((1/2)*(Int((2*(diff(_F1(_j), _j))*_j+_j*_F2(_j)-_F1(_j))/(_j*_F1(_j)), _j)))*(-2*(diff(_F2(_j), _j))*_j^2*l[y]*(y*exp(-(Int(_F2(R)/_F1(R), R)))-(Int(_F3(R)*exp(-(Int(_F2(R)/_F1(R), R)))/_F1(R), R))+Int(_F3(_j)*exp(-(Int(_F2(_j)/_F1(_j), _j)))/_F1(_j), _j))*exp(Int(_F2(_j)/_F1(_j), _j)+(1/2)*(Int((-3*_j*_F2(_j)+_F1(_j))/(_j*_F1(_j)), _j))-(1/2)*(Int((-3*R*_F2(R)+_F1(R))/(R*_F1(R)), R)))+l*(-(diff(_F2(_j), _j))*_j^2+(diff(_F1(_j), _j))*_j-_F1(_j))*exp((1/2)*(Int((-_j*_F2(_j)+_F1(_j))/(_j*_F1(_j)), _j))-(1/2)*(Int((-R*_F2(R)+_F1(R))/(R*_F1(R)), R)))-2*(diff(_F3(_j), _j))*l[y]*_j^2*exp((1/2)*(Int((-3*_j*_F2(_j)+_F1(_j))/(_j*_F1(_j)), _j))-(1/2)*(Int((-3*R*_F2(R)+_F1(R))/(R*_F1(R)), R))))/(_j^2*_F1(_j)), _j = R)+l[R]*exp((1/2)*(Int((2*(diff(_F1(R), R))*R+R*_F2(R)-_F1(R))/(R*_F1(R)), R))), u[R]*exp(Int(((diff(_F1(R), R))*R-2*R*_F2(R)+_F1(R))/(R*_F1(R)), R))+Intat(-exp(Int(((diff(_F1(_k), _k))*_k-2*_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k)-(Int((-_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k))-(Int((-2*_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k)))*(-(diff(_F2(_k), _k))*_k^2*u[y]*(y*exp(-(Int(_F2(R)/_F1(R), R)))-(Int(_F3(R)*exp(-(Int(_F2(R)/_F1(R), R)))/_F1(R), R))+Int(_F3(_k)*exp(-(Int(_F2(_k)/_F1(_k), _k)))/_F1(_k), _k))*exp(Int(_F2(_k)/_F1(_k), _k)+Int((-2*_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k)+Int((-R*_F2(R)+_F1(R))/(R*_F1(R)), R))+(diff(_F4(_k), _k))*exp(Int((-_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k)+Int((-2*_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k))*_k^2-_k^2*u[y]*(diff(_F3(_k), _k))*exp(Int((-2*_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k)+Int((-R*_F2(R)+_F1(R))/(R*_F1(R)), R))+exp(Int((-_k*_F2(_k)+_F1(_k))/(_k*_F1(_k)), _k))*(u*exp(Int((-2*R*_F2(R)+_F1(R))/(R*_F1(R)), R))-(Int(_F4(R)*exp(Int((-2*R*_F2(R)+_F1(R))/(R*_F1(R)), R))/_F1(R), R))+Int(_F4(_k)*exp(-(Int((2*_k*_F2(_k)-_F1(_k))/(_k*_F1(_k)), _k)))/_F1(_k), _k))*(2*(diff(_F2(_k), _k))*_k^2-(diff(_F1(_k), _k))*_k+_F1(_k)))/(_k^2*_F1(_k)), _k = R)

In short, I want to create a density plot but with points. So I do a pointplot and I want to color it according to the value of a third function. 

So let's say I do:

>x1:=[seq(x,x=1..2,0.02)];

>y1:=[seq(y^3,y=1..2,0.02)];

>z1:=[seq(x1[i]/y1[i],i=1..51)];

And I want a plot like:

>pointplot([x1,y1])

but with a color corresponding to the gradient of z1.

What I can do is something like:

>pointplot([x1,y1], colorscheme=["valuesplit", z1, [0..0.5="Black",0.5..1="RoyalBlue"]])

but that's not as pretty as it could be. So is there a way to do something like colorscheme=["zgradient", z1]? This doesn't work because zgradient expects its first argument to be colors and then you can add markers, but they apply to the function you plot, not to a third function.

Any ideas?

colorscheme=["zgradient",["Orange","Red","NavyBlue"]]

I'm working towards creating a way to visualise real polynomial ideals! (or at least the solutions of the polynomials in the ideals) this code creates a plot showing the solutions to all the polynomials in the ideal generated by P1 and P2 (these are specified in the code)

with(plots);
P1 := x^2+2*y^2-3;
solve(P1, y);
Plot1 := plot([%], x = -2 .. 2);

P2 := -2*x^2+2*x*y+3*y^2+x-4;
solve(%, y);
Plot2 := plot([%], x = -4 .. 2);

P2*a+P1;
solve(%, y);
seq(plot([%], x = -4 .. 2), a = 0 .. 10, .1);
display(%, Plot1, Plot2)




This is because when you multiply two polynomials their set of solution curves is just the union of the sets of curves associated with the previous polynomials.

For the next step I'd like to create a graph of the solutions associated with an ideal with three generators. To stop this from being excessively messy I'd like to do it with the RGB value of the colour of a curve is determined by  a and b where the formula for a generic polynomial that we are solving and graphing is given by:

P1+a*P2+b*P3;

where P3 is given by

P3 := x*y-3

I've tried various ways to use cury to make this work (my intuition is cury is the right function to use here)  but got no where. Any ideas how to procede?

I wanted to remove entry from a list that contain y=y or x=x in it. Here is an example

f:= (x-1)*y^4/(x^2*(2*y^2-1));
S:=[singular(f)]

Where I wanted to remove those entries highlighted above to obtain

This is below how I ended up doing it. I'd like to ask if there is a better or more elegent way. I had to use map, since could not get remove() to work on the original list in one shot. 

foo:= z->remove(has,z,{y = y,x = x});
map(foo,[singular(f)])

Which gives the output above.

Is there a better way to do this? I always learn when I find how to do something better.

Maple 2019.1

 

 

Why does the iscont( ) function declare that the square root function is continous over Riscont_error.mw
 

iscont(sqrt(x), x = -infinity .. infinity)

true

(1)

``


 

Download iscont_error.mw

 

Dear Users!

Hoped you will be fine. I want to define a block matrix A for any value of M like this way

A = [A[0], A[1], A[2], ..., A[M]]

A[i] = [A[i,0], A[i,1], A[i,2], ..., A[i,M]]

and

A[i,j] = Transpose([a[i,j,0], a[i,j,1], a[i,j,2], ..., a[i,j,M]]);

Kindly help me in this matter. Thanks in advance.

A research paper published in 1929 claimed that this integral was solved in 1896.

int(exp(I*m*omega + I*b*cos(omega) ),omega=0..2*Pi)           (m integer,  b positive constant)

but it defeats Maple (and other).  Can anybody suggest a way to solve this integral with Maple?

If f is a procedure e.g.  f := x -> x^2;  then f[a](b)  evaluates to f(b)  (= b^2).
Do you know where is this feature documented?
(I have used this in the past, but I need now a reference and I cannot find it).

 

Hello. There is a complex function U11[n]. Let's say I take its derivative and want to extract the real part. That is all right. Now let's say I want to get the derivative at a point and extract the real part. I can't. Are there any solutions?


HELP.mw

 

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