Probability density function

Question

p(Y=y) = p(1-p)^(y-1)

(Given y=1,2,3,.... and 0<=p<=1 )

verify the above is a probability density function

Answer 1

have to show that the:

1. probabilities are non negative, and

2. that they add up to 1 (across the values of y)

1. P(Y=y)=p(1-p)^(y-1) >= 0 since p>0 and (1-p)>0 .The latter follows since 0<p<1

2.ΣP(Y=y) = Σ p*(1-p)^(y-1) = p Σ (1-p)^(y-1)

= p * (1/(1-(1-p))) [sum of infinite geometric series= first term/(1-common ratio)]

=p * (1/p)

=1