Question

If the probability distribution of a random variable X is as follows.

Then, the value of k is _______

• A. 0.1
• B. 0.2
• C. 0.3
• D. 0.4

Hint:

The total sum of probabilities in any probability distribution (discrete or continuous) must always equal 1.
This normalization condition is a fundamental axiom of probability theory.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question requires determining the value of an unknown constant $k$ in a given discrete probability distribution.

Step 2: Key Formula or Approach:
For any valid discrete probability distribution, the sum of all individual probabilities must be equal to 1:
\[ \sum_{i} P(X = x_i) = 1 \]

Step 3: Detailed Explanation:

• Identify the given probability values from the distribution table:
$P(X = 1) = k$
$P(X = 2) = 4k$
$P(X = 3) = 4k$
$P(X = 4) = k$

• Apply the total probability condition:
\[ P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 1 \]
• Substitute the values in terms of $k$ into the equation:
\[ k + 4k + 4k + k = 1 \]
• Combine the terms on the left side:
\[ 10k = 1 \]
• Solve for $k$:
\[ k = \frac{1}{10} = 0.1 \]

Step 4: Final Answer:
The value of $k$ is $0.1$, which corresponds to option (A).