How come if X is a Geometric random variable, then P(X>x)=q^x?

undergoe8m

undergoe8m

Answered question

2022-11-09

How come if X is a Geometric random variable, then P ( X > x ) = q x ?

Answer & Explanation

erlent00s

erlent00s

Beginner2022-11-10Added 15 answers

Step 1
Let X Geometric(p) then we know the CDF of X is F ( k ) = 1 ( 1 p ) k = 1 q k
So then what does 1 F ( k ) = P ( X > k ) mean then? It means at least k-many trials have failed.
Step 2
Symbolically we have P ( X > k ) = 1 P ( X k ) = 1 [ 1 ( 1 p ) k ] = 1 [ 1 q k ] = q k
This is for the definition of Geometric distribution supported on k { 1 , 2 , 3 , 4 , } = N { 0 }

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