Which of the following function are probability mass functio

Which of the following function are probability mass functions? For those that are not, find (if possible) a constant a so that ap(w) is a probability mass function.

Solution

a) p(w) = w^2/55 = ( 1+4+9+16+25)/55 = 55/55 = 1   Yes it is PMF

b) p(w) = (1/3)(2/3)^w = (1/3)(2/3)^3+(1/3)(2/3)^4+(1/3)(2/3)^5+(1/3)(2/3)^6+........Its a GP

= (1/3)(2/3)^3 / (1 -2/3)

= 1/3 * 8/27 * 3

=8/27   : Not a PMF , a = 27/8

c) p(w) =1 Not a pmf , a = 1/9

Explanation: It is actually for nine members : (1+1+1+1+1+1+1+1+1) = 9

Hence a =1/9 is chosen to make it as 1.

d) p(w) =1 Not a pmf , a = 1/N

Explanation: It is actually for countably infinite , so let N be the count : (1+1+1+1+1+1+1+1+1+........+1)----> N times = N

Hence a =1/N is chosen to make it as 1.

As per the chegg guidelines I have solved for first four sub parts.

Thank You!