Question

a discrete random variable X takes values 1, 2, 3,...... with probability 
P(X=j)= c(1/2)^j ; j= 1, 2,...
Find c.

Answer 1

the sum of all discrete probability = 1
1 = c[1/2+ 1/2² + 1/2³ + ............]
this is an infinite G.P. expression
sum of infinite G.P. = a/(1-r)
a = 1/2 
r = 1/2 / 1/2² = 1/2
1 = c[1/2 / (1 - 1/2)]
1 = c[1/2 / 1/2]
c = 1 

Answer 2

$P(X = j) = c* \frac{1}{2^n} \\ \\ Sum\ of\ probabilities\ of\ all\ possible\ outcomes\ of \ X\ is\ unity.\\ \\ So, \ Sum\ over\ n from 1\ to\ Infinity = \ \Sigma\ c \frac{1}{2^n} = c * \Sigma \frac{1}{2^n} \\ \\ It\ is\ a\ Geometric\ progression\ with \ a= \frac{1}{2} \ and \ ratio \ \frac{1}{2}. \\ \\ So\ sum = a\ \frac{1-r^n}{1-r} = \frac{1}{2} * \frac{1-\frac{1}{2^n}}{1-\frac{1}{2}} = \frac{1}{2}*\frac{1-0}{\frac{1}{2}},\ \ \ as\ n -> Infinity. \\ \\ = 1. \\ \\ \Sigma\ P(X) = 1 = c$