Given P(X=k) = c(2k+1) for k = 0, 1, 2, 3, what is the value of the constant c?

Step 1: Understand the properties of a discrete probability distribution. For a discrete random variable, the sum of all possible probabilities must equal 1. Given P(X=k) = c(2k+1) for k \0, 1, 2, 3\. This means _k=0^3 P(X=k) = 1. Step 2: Calculate the sum of the terms 2k+1 for k=0, 1, 2, 3. For k=0, 2(0)+1 = 1. For k=1, 2(1)+1 = 3. For k=2, 2(2)+1 = 5. For k=3, 2(3)+1 = 7. Step 3: Set up the equation for the sum of probabilities. P(X=0) + P(X=1) + P(X=2) + P(X=3) = 1 c(1) + c(3) + c(5) + c(7) = 1 c(1+3+5+7) = 1 c(16) = 1 c = (1)/(16) Step 4: Evaluate the given options. A. c is determined by c = (1)/(4) because the random variable has 4 possible values. This is incorrect. Our calculation shows c = (1)/(16). B. c can be any positive constant because the probabilities are all proportional and do not need to sum to 1. This is incorrect. Probabilities for a valid distribution must sum to 1. C. c must satisfy c = (1)/(_k=0)^3 (2k+1) ensuring the total probability equals 1. This statement correctly describes the method to find c. As calculated in Step 3, _k=0^3 (2k+1) = 16, so c = (1)/(16). This matches our result. D. c equals (1)/(7) because the smallest positive normalization constant is always the reciprocal of the largest coefficient, here 7. This is incorrect. c is the reciprocal of the sum of all coefficients, not just the largest one. The correct statement is C. C. c must satisfy c = (1)/(_k=0)^3 (2k+1) ensuring the total probability equals 1. Last free one today — make it count tomorrow, or type /upgrade for unlimited.