Answer to Question #18712 in Statistics and Probability for Mackos

Question

For each of the following, find the constant c so that p(x) satisfies the condition of being a probability density function of a random variable of x:

I. p(x) = c(2/3)^x, x E N

II. p(x) = cx, x E {1,2,3,4,5,6}


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Answer 1

Sum of probabilities should be 1 for both cases.
From this condition we will find normalizing constant

І.

c* Sum_{n=1}^{infinity} (2/3)^n= 1
Sum_{n=1}^{infinity} (2/3)^n = 2
Hence c = 1/( Sum_{n=1}^{infinity} (2/3)^n ) = 1/2

II. There are totally 6 possibilities, hence c = 1/6