Question

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

Then the ratio between \( k \) and \( P(X \lt 3) \) is:

• A. \( 1:1 \)
• B. \( 1:2 \)
• C. \( 1:3 \)
• D. \( 3:1 \)

Hint:

You don't always need the actual value of \( k \) to find ratios. Since both terms depend linearly on \( k \), it cancels out during division.

Answer 1

The Correct Option is C

Concept: The sum of all probabilities in a discrete probability distribution must equal 1.

• Total Probability: \( \sum P(X=x) = 1 \).

• Cumulative Probability: \( P(X \lt a) = \sum_{x \lt a} P(X=x) \).

Step 1: Find the value of the constant \( k \).
Summing all probabilities: \[ k + 2k + 3k + 4k = 1 \] \[ 10k = 1 \implies k = 0.1 \]

Step 2: Calculate the probability \( P(X \lt 3) \).
The condition \( X \lt 3 \) includes the values \( X=1 \) and \( X=2 \). \[ P(X \lt 3) = P(X=1) + P(X=2) \] \[ P(X \lt 3) = k + 2k = 3k \]

Step 3: Determine the ratio.
We need the ratio \( k : P(X \lt 3) \). \[ \text{Ratio} = \frac{k}{3k} = \frac{1}{3} \] In ratio form, this is written as \( 1:3 \). Final Answer: (C)