Show without using the preceding results * that the probability p m ( r , n ) = n − 1 E m ( r , n ) of finding exactly 𝑚 cells empty satisfies p m ( r + 1 , n ) = p m ( r , n ) n − m n + p m + 1 ( r , n ) m + 1 n ( 1 )* The results which I can't use must be E m ( r , n ) = ( n m ) A ( r , n − m ) = ( n m ) ∑ ν = 0 n − m ( − 1 ) ν ( n − m ν ) ( n − m − ν ) r and by association A ( r , n + 1 ) = ∑ k = 1 r ( r k ) A ( r − k , n )By the way, A ( r , n ) is equal to n ! S ( r , n ) where S ( r , n ) are the Stirling numbers of the second kind.

Question

Show without using the preceding results * that the probability       p          m        (  r  ,  n  )  =      n          −      1            E          m        (  r  ,  n  ) of finding exactly 𝑚 cells empty satisfies      p          m        (  r  +  1  ,  n  )  =      p          m        (  r  ,  n  )            n      −      m        n    +      p          m      +      1        (  r  ,  n  )            m      +      1        n        (  1  )* The results which I can&#039;t use must be      E          m        (  r  ,  n  )  =                    (                    n      m                      )              A  (  r  ,  n  −  m  )  =                    (                    n      m                      )                  ∑          ν      =      0              n      −      m        (  −  1      )          ν                          (                            n        −        m            ν                      )              (  n  −  m  −  ν      )          r      and by association  A  (  r  ,  n  +  1  )  =      ∑          k      =      1              r                          (                    r      k                      )              A  (  r  −  k  ,  n  )By the way,   A  (  r  ,  n  ) is equal to   n  !  S  (  r  ,  n  ) where   S  (  r  ,  n  ) are the Stirling numbers of the second kind.

sliceu4i · Accepted Answer

So       p    m    (  r  ,  n  ) is the probability of finding m cells out of n empty when you scatter   r objects randomly among the bins. Now you are trying to find an expression for   m. Think about how you can get there with   r  +  1 objects. You can either have   m cells empty with   r objects and put the new one in an occupied bin, or you can have   m  +  1 cells empty with   r objects and put the new one in an unoccupied bin. This should lead you to the equation you want.

Show without using the preceding results * that the probability pm(r,n)=n^−1Em(r,n) of finding exactly 𝑚 cells empty satisfies pm(r+1,n)=pm(r,n)(n−m)/n+pm+1(r,n)(m+1)/n (1)

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