Essays/Bifactorial
Definition
Bifactorial n B m or is the number of ways of picking the marked item in choice m out of n choices with n-1 alternating withdrawals of unmarked items, both without replacement, out of 2n-1 total items.
The value of is the product the mth row of a n x n triangular matrix with unit diagonal, even factorials in the lower half and odd ones in the upper, growing up and to the left from the bottom right corner. Such table describes options in the Generalized Monty Hall problem.
attachment:bitab.png
In general, the value of bifactorial m of n is given explicitly by
By definition, using double factorial, we can write
Fe=: 2&^ * ! Fo=: !@+: % 2&^ * ! Fd=: (Fe@-:)`(Fo@-:@>:)@.(2&|)"0 (Fd@<:@+:@- * <:@[ %&(Fd@+:) -)"0/~ >:i.4 1 0 0 0 1 2 0 0 3 4 8 0 15 18 24 48
Or using the expressions for even and odd factorials directly,
B=: (Fo@- * <:@[ %&Fe -)"0 NB. bifactorial: n B m
Note: as other arithmetic functions, it needs to be scalar, this is why we apply rank 0.
Monty Hall Frequencies
Generalized Monty Hall frequencies in sample spaces by total choices in rows.
B table~ >:i.8 +-+-------------------------------------------------------+ |B| 1 2 3 4 5 6 7 8| +-+-------------------------------------------------------+ |1| 1 0 0 0 0 0 0 0| |2| 1 2 0 0 0 0 0 0| |3| 3 4 8 0 0 0 0 0| |4| 15 18 24 48 0 0 0 0| |5| 105 120 144 192 384 0 0 0| |6| 945 1050 1200 1440 1920 3840 0 0| |7| 10395 11340 12600 14400 17280 23040 46080 0| |8|135135 145530 158760 176400 201600 241920 322560 645120| +-+-------------------------------------------------------+
Properties
The first column and the diagonal contain odd and even double factorials respectively.
(B~ -: Fe@<:) >:i.10
1(B&1 -: Fo@<:) >:i.10
1((B>:)/ -: (B * +:@- % <:@+:@-)/)~ >:i.10
1((B&>:)/ -: (B * 2 * [)"0/)~ >:i.10
1({."1@}. -: +/"1@}:) B/~ >:i.10
1
Writen as a square array with diagonals in rows, bifactorial table has the following form
Bd=: (Fo@[ * Fe@+ % Fe@[)&<:"0 NB. same as (<:@+ B ])"0 Bd table~ >:i.6x NB. cf ([!+)"0/~i.6 +--+-------------------------------------------+ |Bd| 1 2 3 4 5 6| +--+-------------------------------------------+ |1 | 1 2 8 48 384 3840| |2 | 1 4 24 192 1920 23040| |3 | 3 18 144 1440 17280 241920| |4 | 15 120 1200 14400 201600 3225600| |5 |105 1050 12600 176400 2822400 50803200| |6 |945 11340 158760 2540160 45722880 914457600| +--+-------------------------------------------+
is divisible by binomial coefficients
((<:@+ B ])&>: % [!+)"0/~ i.6x 1 2 8 48 384 3840 1 2 8 48 384 3840 3 6 24 144 1152 11520 15 30 120 720 5760 57600 105 210 840 5040 40320 403200 945 1890 7560 45360 362880 3628800
to form the product of odd and even factorials
(([!+)"0/~ * Fo */ Fe) i.6x 1 2 8 48 384 3840 1 4 24 192 1920 23040 3 18 144 1440 17280 241920 15 120 1200 14400 201600 3225600 105 1050 12600 176400 2822400 50803200 945 11340 158760 2540160 45722880 914457600
Thus,
(B&>:/~ -: (!~ * Fo@- * Fe@])"0/~) i.8 1
Bifactorial Sequence
Bifactorial numbers make a naturally ordered monotonous sequence, corresponding to the order of indices .
list ;<@(B >:@i.)"0 >:i.10x 1 1 2 3 4 8 15 18 24 48 105 120 144 192 384 945 1050 1200 1440 1920 3840 10395 11340 12600 14400 17280 23040 46080 135135 145530 158760 176400 201600 241920 322560 645120 2027025 2162160 2328480 2540160 2822400 3225600 3870720 5160960 10321920 34459425 36486450 38918880 41912640 45722880 50803200 58060800 69672960 92897280 185794560
Bifactorial Distribution
Amazingly, the magnitude of each nth sample space (total by nth row) is an odd double factorial too, by value equal to the first cell of the next row.
+/"1 B/~ >:i.7 1 3 15 105 945 10395 135135
So the probability of picking the marked item in choice m of total n choices in uniform Monty Hall sample space produces Bifactorial Distribution with density function , the corresponding Monty Hall densities.
Pmh=: (B % >:@[ B 1:)"0 NB. probability P(m|n) Pmhf=: ]&.(0j3&":)@Pmh NB. formatted Pmhf table~ >:i.8 +----+-----------------------------------------------+ |Pmhf| 1 2 3 4 5 6 7 8| +----+-----------------------------------------------+ |1 | 1 0 0 0 0 0 0 0| |2 |0.333 0.667 0 0 0 0 0 0| |3 | 0.2 0.267 0.533 0 0 0 0 0| |4 |0.143 0.171 0.229 0.457 0 0 0 0| |5 |0.111 0.127 0.152 0.203 0.406 0 0 0| |6 |0.091 0.101 0.115 0.139 0.185 0.369 0 0| |7 |0.077 0.084 0.093 0.107 0.128 0.17 0.341 0| |8 |0.067 0.072 0.078 0.087 0.099 0.119 0.159 0.318| +----+-----------------------------------------------+
In fractional form
x:@Pmh table~ >:i.8 +------+-----------------------------------------------------------------+ |x:@Pmh| 1 2 3 4 5 6 7 8| +------+-----------------------------------------------------------------+ |1 | 1 0 0 0 0 0 0 0| |2 | 1r3 2r3 0 0 0 0 0 0| |3 | 1r5 4r15 8r15 0 0 0 0 0| |4 | 1r7 6r35 8r35 16r35 0 0 0 0| |5 | 1r9 8r63 16r105 64r315 128r315 0 0 0| |6 |1r11 10r99 80r693 32r231 128r693 256r693 0 0| |7 |1r13 12r143 40r429 320r3003 128r1001 512r3003 1024r3003 0| |8 |1r15 14r195 56r715 112r1287 128r1287 256r2145 1024r6435 2048r6435| +------+-----------------------------------------------------------------+
Razed densities for the first n=1..20
<@(Pmh >:@i.)"0 >:i.4 +-+-----------------+---------------------+-----------------------------------+ |1|0.333333 0.666667|0.2 0.266667 0.533333|0.142857 0.171429 0.228571 0.457143| +-+-----------------+---------------------+-----------------------------------+ ;<@(Pmh >:@i.)"0 >:i.4 1 0.333333 0.666667 0.2 0.266667 0.533333 0.142857 0.171429 0.228571 0.457143 load 'plot' plot ;<@(Pmh >:@i.)"0 >:i.20
attachment:freqtab1.png
and the envelopes corresponding to first and last choices of the families for n=20,50,100,200.
plot |:({.,{:)@(Pmh >:@i.)"0 >:i.20x
attachment:freqtab2.png
And the log of frequencies looks like this
plot ^.;<@(B >:@i.)"0 >:i.10
attachment:freqtab3.png
See Also
- Double Factorial
- Generalized Monty Hall
- OEIS:A122774 Triangle of bifactorial numbers, n B m = (2(n-m)-1)!! (2(n-1))!! / (2(n-m))!!, read by rows
- OEIS:A000165 Double factorial numbers: (2n)!! = 2^n*n! (even)
- OEIS:A001147 Double factorial numbers: (2n-1)!! = 1.3.5....(2n-1) (odd)
- OEIS:A006882 Double factorials n!!: a(n)=n*a(n-2)
- OEIS:A007318 Pascal's triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!), 0<=k<=n
- MathWorld:DoubleFactorial, Mathworld
- MathWorld:GeometricDistribution, Mathworld
- MathWorld:BinomialDistribution, Mathworld
- Virtual Laboratories in Probability and Statistics