A little problem with this Probability distribution P { X = n } = ( 1 − p ) n − 1 pIn solving a probability exercise, they give me a random variable X with a geometric distribution of parameter p 0 &lt; p &lt; 1 such that probability function of X is P { X = n } = ( 1 − p ) n − 1 p for n ≥ 1. And they use some property that I don't know and they get this: P { X &gt; n } = ( 1 − p ) n Could someone explain how they do it?

Question

A little problem with this Probability distribution  P  {  X  =  n  }  =            (      1      −      p      )              n      −      1        pIn solving a probability exercise, they give me a random variable X with a geometric distribution of parameter   p        0  &amp;lt;  p  &amp;lt;  1 such that probability function of X is   P  {  X  =  n  }  =            (      1      −      p      )              n      −      1        p for   n  ≥  1. And they use some property that I don&#039;t know and they get this:  P  {  X  &amp;gt;  n  }  =            (      1      −      p      )              n      Could someone explain how they do it?

Tiberlaue · Accepted Answer

Explanation:      P    [  X  &amp;gt;  n  ]  =      ∑          x      =      n      +      1              ∞        p  (  1  −  p      )          x      −      1        =  p  (  1  −  p      )    n        ∑          y      =      0              ∞        (  1  −  p      )    y    =  p  (  1  −  p      )    n    ⋅      1          1      −      (      1      −      p      )        =  (  1  −  p      )    n

A little problem with this Probability distribution P{X=n}=(1-p)^{n-1}p

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Answer & Explanation

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