Question

which of the following is a valid probability distribution?
probability distribution a
x | p(x)
1 | 0.42
2 | 0.38
3 | 0.13
4 | 0.07
probability distribution b
x | p(x)
1 | 0.27
2 | 0.28
3 | 0.26
4 | 0.27
probability distribution c
x | p(x)
1 | 0.16
2 | 0.39
3 | 0.45

Explanation:

Step1: Check valid distribution rules

A valid probability distribution requires two conditions:

1. All $P(x)$ values are between 0 and 1 (inclusive).
2. The sum of all $P(x)$ values equals 1.

Step2: Evaluate Distribution A

Sum the probabilities:
$0.42 + 0.38 + 0.13 + 0.07 = 1.00$
All $P(x)$ are between 0 and 1.

Step3: Evaluate Distribution B

Sum the probabilities:
$0.27 + 0.28 + 0.26 + 0.27 = 1.08$
The sum exceeds 1, so it is invalid.

Step4: Evaluate Distribution C

The table is incomplete, so we cannot verify the sum of all probabilities, making it invalid to confirm.

Answer:

Probability Distribution A (the first option with x values 1-4 and P(x) values 0.42, 0.38, 0.13, 0.07)