For N B ( r , p ) where r is the number of successes and the p is the probability of success. Also we have: f ( x ) = ( r + x − 1 r − 1 ) p r q x Shown that f ( x ) = ( r + x − 1 ) q x f ( x − 1 ). So: f ( x ) &gt; f ( x − 1 ) ⟺ ( r − 1 ) q p &gt; x f ( x ) &lt; f ( x − 1 ) ⟺ ( r − 1 ) q p &lt; xAnd after a long calculation, I've found that the mode is the integer part of ( r − 1 ) q p . Where I did wrong?

Question

For   N  B  (  r  ,  p  ) where   r is the number of successes and the   p is the probability of success. Also we have:  f  (  x  )  =                (                      r        +        x        −        1                    r        −        1                        )            p    r        q      x  Shown that   f  (  x  )  =            (      r      +      x      −      1      )      q        x    f  (  x  −  1  ). So:  f  (  x  )  &amp;gt;  f  (  x  −  1  )    ⟺              (      r      −      1      )      q        p    &amp;gt;  x  f  (  x  )  &amp;lt;  f  (  x  −  1  )    ⟺              (      r      −      1      )      q        p    &amp;lt;  xAnd after a long calculation, I&#039;ve found that the mode is the integer part of             (      r      −      1      )      q        p  . Where I did wrong?

Marlee Guerra · Accepted Answer

There are two different definitions of Negative Binomial distributions in common use. It depends on what you are counting (and what is the terminator).- When   X is the count of failures before success number   r, with success rate   p and   q  =  1  −  p:      p    X    (  k  )  =                          (                              k          +          r          −          1                k                          )                  q    k        p    r          The mode of       X     is:         ⌊                            q          (          r          −          1          )                p              ⌋             1          r      &amp;gt;      1      - When   Y is the count of successes before failure number   r, with success rate   p and   q  =  1  −  p:      p    Y    (  k  )  =                          (                              k          +          r          −          1                k                          )                  q    r        p    k          The mode of       Y     is:         ⌊                            p          (          r          −          1          )                q              ⌋             1          r      &amp;gt;      1

For N B ( r , p ) where r is the number of successes and the p is the probab

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