Step 1: Understanding the Concept:
A probability distribution defines the likelihood of all possible outcomes for a discrete random variable \( X \).
For any distribution to be mathematically valid, it must follow the fundamental axioms of probability.
The most important rule for solving this type of problem is the Summation Rule: the sum of the probabilities of all possible mutually exclusive events in a sample space must equal exactly 1.
This signifies that one of the events listed in the table is guaranteed to happen.
Additionally, every individual probability \( P(X) \) must be a non-negative value between 0 and 1.
Step 2: Key Formula or Approach:
The condition for a valid probability mass function is:
\[ \sum_{i=1}^{n} P(X_i) = 1 \]
We will sum the expressions for \( P(X) \) from the table and set them equal to unity.
Step 3: Detailed Explanation:
From the given table, the probabilities associated with each value of \( X \) are:
- For \( X = 0 \), \( P(0) = k \)
- For \( X = 1 \), \( P(1) = 2k \)
- For \( X = 2 \), \( P(2) = 3k \)
Applying the total probability axiom:
\[ P(0) + P(1) + P(2) = 1 \]
Substitute the algebraic terms into the equation:
\[ k + 2k + 3k = 1 \]
Combine the like terms on the left side of the equation:
\[ (1 + 2 + 3)k = 1 \]
\[ 6k = 1 \]
To find the value of the parameter \( k \), divide both sides of the equation by 6:
\[ k = \frac{1}{6} \]
We can double-check this result by substituting \( k = 1/6 \) back into the table.
The probabilities become \( 1/6, 2/6, 3/6 \). Since all these values are positive and their sum is \( 6/6 = 1 \), our solution is correct.
Step 4: Final Answer:
The value of the unknown parameter \( k \) is \( 1/6 \).