Which of the following statements correctly verifies that the function $p\left(x\right)=\frac{1}{4}$ for $x=1$, $x=2$, $x=3$, $x=4$ and $p\left(x\right)=0$ otherwise is a probability mass function?

• A. The function is a probability mass function because the probabilities can be any real numbers.
• B. The function is not a probability mass function because the sum of the probabilities is $4$.
• C. The function is not a probability mass function because some probabilities are negative.
• D. The function is a probability mass function because $\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=1$ and all probabilities are non-negative.

Answer: (D)

Verified step by step guidance

1. Recall the two main conditions for a function \( p(x) \) to be a probability mass function (pmf): (1) each probability \( p(x) \) must be non-negative, i.e., \( p(x) \geq 0 \) for all \( x \), and (2) the sum of all probabilities over the possible values of \( x \) must equal 1, i.e., \( \sum_x p(x) = 1 \).
2. Identify the values of \( x \) for which \( p(x) \) is defined: \( x = 1, 2, 3, 4 \), and for these values, \( p(x) = \frac{1}{4} \). For all other values of \( x \), \( p(x) = 0 \).
3. Check the non-negativity condition by verifying that \( p(x) = \frac{1}{4} \) is non-negative for each \( x = 1, 2, 3, 4 \), and \( p(x) = 0 \) otherwise, so all probabilities are \( \geq 0 \).
4. Calculate the sum of the probabilities over all possible values of \( x \): \[ \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = 1. \] This satisfies the total probability condition.
5. Conclude that since all probabilities are non-negative and their sum equals 1, the function \( p(x) \) satisfies the conditions of a probability mass function.