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 | 0.05 | 0.12 | 0.15 | 0.18 | 0.23 | 0.27 |
| Distribution 2 | A | B | C | D | E | F |
|---|---|---|---|---|---|---|
| Probability | 0.14 | 0.20 | 0.24 | 0.22 | 0.14 | 0.09 |
| Distribution 3 | A | B | C | D | E | F |
|---|---|---|---|---|---|---|
| Probability | 0.17 | 0.17 | 0.16 | 0.16 | 0.17 | 0.17 |
Let’s start by finding the sums of each distribution.
Here is the sum of Distribution 1: \(0.05+0.12+0.15+0.18+0.23+0.27 = 1.00\)
Here is the sum of Distribution 2: \(0.14+0.20+0.24+0.22+0.14+0.09 = 1.03\)
Here is the sum of Distribution 3: \(0.17+0.17+0.16+0.16+0.17+0.17 = 1.00\)
Distributions 1 and 3 are all positive and add up to 1.0. Therefore, they are both valid distributions.
The probabilities in distribution 2 are all positive. However, they do not add up to 1.0. So, distribution 2 is invalid.