Question

Which of the following can be the probability distribution of a random variable ?

• A.

  \(X\)	1	2	3
  \(P(X)\)	-0.5	0.5	0.1

• B.

  \(X\)	1	2	3	\(4\)	\(5\)
  \(P(X)\)	\(0.1\)	\(0.4\)	\(0.05\)	\(-0.2\)	\(0.2\)

• C.

  \(X\)	\(1\)	\(2\)	\(3\)	\(5\)
  \(P(X)\)	\(0.2\)	\(0.3\)	\(0.2\)	\(0.2\)

• D.

  \(X\)	\(0\)	\(1\)	\(2\)
  \(P(X)\)	\(0.4\)	\(0.2\)	\(0.4\)

Answer 1

The Correct Option is D

To determine which option can be the probability distribution of a random variable, we need to check two main conditions that a probability distribution must satisfy:

1. All probabilities \( P(X) \) must be non-negative.
2. The sum of all probabilities must equal 1.

Let's evaluate each given option:

1. \(X\)	1	2	3
   \(P(X)\)	-0.5	0.5	0.1

   The probability -0.5 is negative, which violates the first condition.

2. \(X\)	1	2	3	4	5
   \(P(X)\)	0.1	0.4	0.05	-0.2	0.2

   The probability -0.2 is negative, which violates the first condition.

3. \(X\)	1	2	3	5
   \(P(X)\)	0.2	0.3	0.2	0.2

   The sum of the probabilities is \(0.2 + 0.3 + 0.2 + 0.2 = 0.9\), which is not equal to 1.

4. \(X\)	0	1	2
   \(P(X)\)	0.4	0.2	0.4

   All probabilities are non-negative, and the sum \(0.4 + 0.2 + 0.4 = 1.0\) satisfies the second condition.

Based on these evaluations, option 4 is the only valid probability distribution for a random variable as it satisfies both conditions required for a probability distribution.