A discrete random variable X takes values x = 1, 2, 3, with probability P(X=x) = k r^x , where k and r are constants and 0 < r < 1 . If the probability that X takes an odd integer value is 3 4 , then the expected value (mean) of X is equal to

Step-by-step solution

For a valid probability distribution, the sum of all probabilities is 1: _ x=1 ^ P(X=x) = 1 k(r + r^2 + r^3 + ) = 1 kr 1-r = 1 k = 1-r r The probability that X takes an odd integer value is given as 3 4 : P(X is odd ) = P(X=1) + P(X=3) + P(X=5) + = 3 4 k(r + r^3 + r^5 + ) = 3 4 This is an infinite geometric progression with common ratio r^2 : kr 1-r^2 = 3 4 Substitute k = 1-r r into the equation: ( 1-r r ) r (1-r)(1+r) = 3 4 1 1+r = 3 4 1+r = 4 3 r = 1 3 Using r = 1 3 , we find k = 1 - 1/3 1/3 = 2 . The expected va