Question

which of the following represents a valid probability distribution? probability distrbution a x p(x) 1 -0.14 2 0.6 3 0.25 4 0.29 probability distrbution b x p(x) 1 0 2 0.45 3 0.16 4 0.39 probability distribution c x p(x) 1 0.45 2 1.23 3 -0.87 4 0.19

Explanation:

Step1: Recall Probability Distribution Rules

A valid probability distribution must have: 1) Each probability \( P(x) \) between 0 and 1 (inclusive), and 2) The sum of all \( P(x) \) equal to 1.

Step2: Check Distribution A

• \( P(1) = -0.14 \) (negative, invalid as probabilities can't be negative). So A is invalid.

Step3: Check Distribution B

• \( P(1)=0 \), \( P(2)=0.45 \), \( P(3)=0.16 \), \( P(4)=0.39 \).
• Check each \( P(x) \): All between 0 and 1.
• Sum: \( 0 + 0.45 + 0.16 + 0.39 = 1 \). Valid.

Step4: Check Distribution C

• \( P(2)=1.23 \) (greater than 1), \( P(3)=-0.87 \) (negative). Invalid.

Answer:

Probability Distribution B (with X=1: P(x)=0; X=2: P(x)=0.45; X=3: P(x)=0.16; X=4: P(x)=0.39)