How to find a probability that sum of geometric variables is less than a numberLet X i , i = 1 , … , n be Geometric i.i.d random variables, which represent the number of fails, with parameter p.Calculate or estimate from above and below: P ( ∑ i = 1 n X i ≤ A ) , A ∈ N .I know that sum of the geometric random variables is the negative binomial, but I would not know all the parameters for the negative binomial r.v.

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How to find a probability that sum of geometric variables is less than a numberLet       X    i    ,  i  =  1  ,  …  ,  n be Geometric i.i.d random variables, which represent the number of fails, with parameter p.Calculate or estimate from above and below:  P  (      ∑          i      =      1        n        X    i    ≤  A  )  ,    A  ∈  N  .I know that sum of the geometric random variables is the negative binomial, but I would not know all the parameters for the negative binomial r.v.

kuglatid4 · Accepted Answer

Step 1You probably mean   A  ∈      N  , i.e. A is a natural number. You should probably use CLT here, as n is large, Xi are iid with   μ  &amp;lt;  ∞  ,      σ    2    &amp;lt;  ∞  P  (      S    n    &amp;lt;  A  )  =  P  (                    S        n            −      n      μ              σ              n              &amp;lt;            A      −      n      μ              σ              n              )  ≈  Φ  (  z  ).Step 2Here   z  =            A      −      n      μ              σ              n            , and   μ  ,  σ are mean and sd of each       X    i  , which you can easily find.

Let X_i, i=1,…,n be Geometric i.i.d random variables, which represent the number of fails, with parameter p.

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