#OK to post homework 

#Zhizhang Deng, 12/02/2020, Assignment 17a



1. 

    Year 2000: 

        Total electorial vote is 538. 



        Outcome is 271 - 267



        coeff(GFv(USEC(), x), x, 271) = 16965465344318 ways



    Year 2004: 

        Total electorial vote is 537 where Minnesota only voted 9 votes when it has 10 votes. 



        Outcome is 286 - 251



        data := USEC(); 

        [3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 13, 14, 15, 16, 16, 18, 20, 20, 29, 29, 38, 55]



        # now we make a 10 to 9 due to Minnesota 

        data := [3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 12, 13, 14, 15, 16, 16, 18, 20, 20, 29, 29, 38, 55]



        coeff(GFv(data, x), x, 286) = 16161969590617 ways 



    Year 2008: 

        Total electorial vote is 538



        Outcome is 365 - 173



        coeff(GFv(USEC(), x), x, 365) = 3182416524832 ways 



    Year 2012:

        Total electorial vote is 538



        Outcome is 332 - 206



        coeff(GFv(USEC(), x), x, 332) = 8628577597686



    Year 2016: 

        Total electorial vote is 531



        Outcome is 227 - 304 - 7



        Assuming this 7 votes for others is added to 227, making a outcome as 234 - 304



        coeff(GFv(USEC(), x), x, 234) = 13873406885786 ways



2. 

    (i). 

        eqn := GFvp(USEC(), 1/2, x)

        evalf(add(coeff(eqn, x, n), n = 270 .. 538)) = .4962304641

    (ii). 

        From 2000 - 2016, lists of winner votes is: 

        lst := [271, 286, 365, 332, 304]



        seq(evalf(add(coeff(eqn, x, n), n = lst[i] .. 538)), i = 1 .. 5)



        = .4886926154, .3774150266, 0.2906376147e-1, .1138587085, .2558478272



3. 

    Observed value: 

    res := [seq(SimuCount(USEC(), 3/10, 2000, 4), i = 1 .. 4)] = [[163.8520000, 47.02447338, .2816623650, 2.935646384], .4170000000, [161.7525000, 46.79671190, .2920772324, 2.918609751], .4115000000, [162.9730000, 46.83705020, .2810178755, 2.901708903], .4085000000, [162.2635000, 46.23926976, .2913412550, 2.765843725], .4205000000]



    Theoretical value: 

    evalf(StatAnal(GFvp(USEC(), 3/10, x), x, 4)) = [161.4000000, 46.65683230, .2695069204, 2.844567676]



    Expectation, standard-deviation, 3rd, 4th momentum are very close to each other and are close to the theoretical value. As for the probability of consistency, it seems like it's around 41%. And the four runes are very close to each other on this one as well.