Question

A random variable \( X \) has the following probability distribution table: 

\( X \)	0	1	2
\( P(X) \)	\( k \)	\( 2k \)	\( 3k \)

Find the exact value of the unknown parameter \( k \).

• A. \( \frac{1}{6} \)
• B. \( \frac{1}{3} \)
• C. \( 1 \)
• D. \( \frac{1}{5} \)

Hint:

Always use the total probability condition (\( \sum P(X) = 1 \)) as your first step to clear out unknowns like \( k \) or \( a \) before attempting to calculate further metrics like the Mean (\( \mu \)) or Variance (\( \sigma^2 \)).

Answer 1

The Correct Option is A

Concept: For any discrete random variable to have a mathematically valid probability distribution, it must follow two foundational rules:

• Every individual probability value must be non-negative: \( P(X_i) \ge 0 \)
• The sum of all individual probabilities across the entire sample space must equal exactly 1: \[ \sum_{i=1}^{n} P(X_i) = 1 \]

Step 1: Sum all the entries in the probability row.
From the provided distribution table, extract and add the individual probability terms: \[ \sum P(X) = P(0) + P(1) + P(2) = k + 2k + 3k \] Combine the like terms together: \[ \sum P(X) = 6k \]

Step 2: Equate this total sum to 1 to solve for the parameter \( k \).
Applying the total probability axiom rule: \[ 6k = 1 \quad \Rightarrow \quad k = \frac{1}{6} \]