Here are three distributions. One of them is invalid. Determine which is the invalid distribution and what can be done to fix it.
| Distribution 1 | A | B | C | D | E | F |
|---|---|---|---|---|---|---|
| Probability | 0.24 | 0.28 | 0.24 | 0.19 | -0.11 | 0.16 |
| Distribution 2 | A | B | C | D | E | F |
|---|---|---|---|---|---|---|
| Probability | 0.04 | 0.09 | 0.13 | 0.16 | 0.26 | 0.32 |
| Distribution 3 | A | B | C | D | E | F |
|---|---|---|---|---|---|---|
| Probability | 0.13 | 0.17 | 0.24 | 0.21 | 0.16 | 0.09 |
Let’s start by finding the sums of each distribution.
Here is the sum of Distribution 1: \(0.24+0.28+0.24+0.19+(-0.11)+0.16=1.00\)
Here is the sum of Distribution 2: \(0.04+0.09+0.13+0.16+0.26+0.32 = 1.00\)
Here is the sum of Distribution 3: \(0.13+0.17+0.24+0.21+0.16+0.09 = 1.00\)
The probabilities in all three distributions add up to 1.0. So, they all pass that test.
However, \(P(E) = -0.11\) in Distribution 1. All probabilities need to be between 0 and 1, so distribution 1 is invalid.