Deriving the Mean of a Geometric Distribution

 

Let X  ~ Geometric( p ) .  

 

Then   Geo_probability_f .

 

 

Lemma 1:          Geo_Lemma1

 

            Proof: 

                        Lemma1_proof

 

 

Lemma 2:       Lemma2_Geo

 

 This is a generalization, easy to prove by induction, of the
sum rule  for differentiation;      

  sum_rule_dx .

 

Now,

           Derivation_firstpart.

 

And,

            Derivation_2nd_part   .

 

Hence,

            Derivation_conclusion