Geometric probability expressed as a geometric seriesSuppose that we want to show that P ( X ≤ n ) = ( 1 − p ) n Where X is a geometric random variable with probability of success p. P ( X &gt; n ) can be written as a geometric series as follows: Σ X = n + 1 ∞ ( 1 − p ) n − 1 = ( 1 − p ) n But how do I proof this? What is a and r in this particular geometric series?

Question

Geometric probability expressed as a geometric seriesSuppose that we want to show that   P  (  X  ≤  n  )  =  (  1  −  p      )          n      Where X is a geometric random variable with probability of success p.  P  (  X  &amp;gt;  n  ) can be written as a geometric series as follows:      Σ          X      =      n      +      1              ∞        (  1  −  p      )          n      −      1        =  (  1  −  p      )          n      But how do I proof this? What is a and r in this particular geometric series?

Alexia Mata · Accepted Answer

Step 1It helps to start from the right place.When X is geometric with success rate p, then   X  =  n is the event of   n  −  1 failures before the first success, so:      P    (  X  =  n  )  =  (  1  −  p      )          n      −      1        pStep 2So:                               P                (        X        &amp;gt;        n        )                            =                  ∑                      k            =            n            +            1                    ∞                          P                (        X        =        k        )                                          =                  ∑                      k            =            n            +            1                    ∞                (        1        −        p                  )                      k            −            1                          p                                          =        p        (        1        −        p                  )                      n                                             ∑                      r            =            0                    ∞                (        1        −        p                  )          r                                                            ⋮

Suppose that we want to show that P(X leq n)=(1-p)^n. <br> Where X is a geometric random variable with probability of success p.

Open question

Answer & Explanation

New Questions in High school geometry