91Ó°ÊÓ

Discrete probability distributions are one of the fundamental concepts in probability and statistics. Essentially, they are functions that assign probabilities to discrete outcomes, such as integers. Imagine you are rolling a die; the possible outcomes (1, 2, 3, 4, 5, and 6) are all discrete.

In the context of this exercise, we see that the function given is \( f(x) = cx \) where \( x \) can take values 1, 2, 3, and 4. This is a classic example of a discrete probability distribution. Each different \( x \) value has an associated probability value, which is determined by the constant \( c \).

Key Characteristics of Discrete Probability Distributions:

• The set of possible outcomes must be countable.
• Each outcome is assigned a probability value.
• The sum of all probabilities must equal 1.

These aspects ensure that the function \( f(x) \) is a valid probability mass function, allowing us to determine the constant \( c \) through further analysis.

Determining the constant \( c \) in a probability mass function is a crucial step to ensure that the function is valid and useful for statistical analysis. For the function \( f(x) = cx \), the determination of \( c \) involves the use of the probability mass function condition.

To find the correct value of \( c \), we substitute each possible \( x \) value (1, 2, 3, 4) into the function and sum up the probabilities. This requires using the equation\[\sum_{i=1}^{4} f(x_i) = 1\]By substituting the function values into this equation as shown in the original solution, we perform the following steps:

• Calculate \( f(1) = c \cdot 1 \)
• Calculate \( f(2) = c \cdot 2 \)
• Calculate \( f(3) = c \cdot 3 \)
• Calculate \( f(4) = c \cdot 4 \)

The sum of these probabilities must equal 1, which gives us a straightforward equation to solve: \( 10c = 1 \). Solving for \( c \) gives \( c = \frac{1}{10} \).

This determined constant ensures that the sum of probabilities adheres to the rules of a valid probability mass function.

To verify and validate a probability mass function, it must adhere to certain probability conditions. These conditions are the guiding principles that maintain the integrity of the distribution.

The primary condition is that the sum of all probabilities must equal 1. This ensures that all possible outcomes of the random variable are accounted for. It's like making sure that all pieces of a pie add up to the whole pie. For our function \( f(x) = cx \), substituting the calculated \( c \) for each value of \( x \) yields:

• \( f(1) = \frac{1}{10} \)
• \( f(2) = \frac{2}{10} \)
• \( f(3) = \frac{3}{10} \)
• \( f(4) = \frac{4}{10} \)

When these probabilities are added together, we validate that they indeed sum to 1:
\[\frac{1}{10} + \frac{2}{10} + \frac{3}{10} + \frac{4}{10} = 1\]
Another important condition is that each probability must be between 0 and 1, inclusive. This confirms that none of the probabilities are negative or exceed certainty. By confirming these conditions, we can ascertain that our probability mass function is both valid and reliable.