I have equation for truncated increasing geometric distribution given as p r = ( 1 − α ) α C W 1 − α C W . α − r with r = 1 , . . . . . , C W. α is parameter we can set between 0 and 1.Here probability of picking r = 1 is very low and r = C W is very high. How can i derive an equation from this equation that should give me values generated between r = 1 and CW with given distribution.i should get values like this if r = 1 to 10. C W = 10 10 , 10 , 9 , 9 , 6 , 1 , 6 , 10 , 10 , 3 , 4(following geometric distribution from equation provided)

Question

I have equation for truncated increasing geometric distribution given as   p  r  =            (      1      −      α      )              α                  C          W                            1      −              α                  C          W                      .      α          −      r       with   r  =  1  ,  .  .  .  .  .  ,  C  W.   α is parameter we can set between 0 and 1.Here probability of picking   r  =  1 is very low and   r  =  C  W is very high. How can i derive an equation from this equation that should give me values generated between   r  =  1 and CW with given distribution.i should get values like this if   r  =  1 to 10.   C  W  =  10     10  ,  10  ,  9  ,  9  ,  6  ,  1  ,  6  ,  10  ,  10  ,  3  ,  4(following geometric distribution from equation provided)

Mckenna Friedman · Accepted Answer

Step 1The standard way to generate a geometric variable with probability mass function   P  (  n  )  =  (  1  −  p  )      p    n   for   n  ∈            N        0   from a variable u uniformly distributed over [0,1] is   n  =      ⌊                  log        ⁡        u                    log        ⁡        p              ⌋      .Step 2To restrict to   0  ≤  n  &amp;lt;  C  W, we need to transform u to   [      u    0    ,  1  ] such that             log      ⁡              u        0                    log      ⁡      p        =  C  W. Thus       u    0    =      p          C      W      , so you can use   u  →  1  −  u      (    1    −          p              C        W              )  , and the formula for generating n becomes  n  =      ⌊                  log        ⁡                  (          1          −          u                      (            1            −                          p                              C                W                                      )                    )                            log        ⁡        p              ⌋      .To map this to your case, take   r  =  C  W  −  n and   p  =  α

I have equation for truncated increasing geometric distribution given as pr=((1-alpha)alpha^{CW})/(1-alpha^{CW}). alpha^{-r} with r=1,.....,CW. alpha is parameter we can set between 0 and 1.

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