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Determining the constant in a probability mass function (PMF) involves a few straightforward steps. Your goal is to find a value for the constant, often denoted as \( c \), that makes the function satisfy the properties of a PMF. Given a function, such as \( f(x) = c x \) for specific values of \( x \), the sum of all function values should equal 1. This is because probabilities, as a rule, must always total to 1 when considering all possible outcomes. To solve for \( c \), you set up the equation by summing all the individual probabilities:\[ c imes 1 + c imes 2 + c imes 3 + c imes 4 = 1. \] This sum leads us to a simpler expression: \( 10c = 1 \), because \( 1 + 2 + 3 + 4 = 10 \). Finally, solve for \( c \) by dividing both sides by 10, providing \( c = \frac{1}{10} \). This ensures that the sum of the probabilities equals exactly 1, adhering to the PMF requirements.

In the context of probability mass functions, ensuring that function values are non-negative is crucial. Probability values can never be negative, as they represent the likelihood of an event, which logically cannot be a negative amount. For the function \( f(x) = c x \), this means that every output of \( f(x) \) must be non-negative. As long as \( c \) is greater than or equal to zero, the function \( c x \) satisfies this requirement. If \( c < 0 \), the function would yield negative probabilities, which is invalid within probability theory. Therefore, one of the first checks when determining a PMF is to ensure all function values meet this non-negativity condition, guiding us to confirm \( c \geq 0 \). Thus, when \( c = \frac{1}{10} \), the function values are indeed non-negative, affirming part of the criteria for validating a PMF.

In a probability mass function, the sum of probabilities plays a key role. This rule states that when all individual probabilities are added together, they must sum up to 1. This concept stems from the fundamental principle that one of the possible outcomes must happen.To ensure a PMF behaves correctly, verify the following steps:

• Calculate all individual probabilities, which in the example \( f(x) = c x \) previously determined as \( f(1) = \frac{1}{10} \), \( f(2) = \frac{2}{10} \), \( f(3) = \frac{3}{10} \), and \( f(4) = \frac{4}{10} \).
• Sum these probabilities: \( \frac{1}{10} + \frac{2}{10} + \frac{3}{10} + \frac{4}{10} \).
• Ensure the total equals 1, which confirms the sum of probabilities rule is satisfied.

The computation shows \( \frac{10}{10} = 1 \), which verifies our determination was correct. Thus, with the constant \( c = \frac{1}{10} \), our PMF properly satisfies this critical rule.