determine the constant K such that the function can serve as

determine the constant K such that the function can serve as the probability distribution of a random variable with the given range

f(x) = k (3/5)^x ; x=1,2,3,----

Solution

f(x) = k (3/5)^x

x = 1, 2, 3....

sum of all probability is equal to 1

f(1) + f(2) + f(3) + .... =1

k(3/5) + k(3/5)^2 + k(3/5)^3 +... =1

k*3/5 ( 1 + 3/5 + (3/5)^2 + .... ) =1

Sum of GP ( 1 + 3/5 + (3/5)^2 + .... ) is equal to (1/(1-3/5))

k*3/5 *(1/(1-3/5)) = 1

k*(3/5)*(5/2) = 1

k = 2/3.....Answer

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