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

For a discrete random variable, the sum of all possible probabilities must equal 1.
Given $P(X=k) = c(2k+1)$ for $k \in \{0, 1, 2, 3\}$.
This means $\sum_{k=0}^{3} P(X=k) = 1$.

For $k=0$, $2(0)+1 = 1$.
For $k=1$, $2(1)+1 = 3$.
For $k=2$, $2(2)+1 = 5$.
For $k=3$, $2(3)+1 = 7$.

• A. $c$ is determined by $c = \frac{1}{4}$ because the random variable has 4 possible values. This is incorrect. Our calculation shows $c = \frac{1}{16}$.
• 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 $c = \frac{1}{\sum_{k=0}^{3} (2k+1)}$ ensuring the total probability equals 1. This statement correctly describes the method to find $c$. As calculated in Step 3, $\sum_{k=0}^{3} (2k+1) = 16$, so $c = \frac{1}{16}$. This matches our result.
• D. $c$ equals $\frac{1}{7}$ 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.