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### Checking a probability distribution for validity

posted Mar 17, 2014, 4:50 PM by Prof Kiernan
When you are asked if a probability distribution (table) is valid you need to answer 3 questions.

1. Does the sum of P(x) add up to any number other than 1 ?
2. Are there any negative probabilities?
3. Are there any probabilities larger than 1?

If you answer YES to any of the questions above your table is NOT a probability distribution.
If you answer NO to all of the questions above your table is a probability distribution.

Example 1:
 x P(x) 0 0.129 1 0.257 2 0.659 3 0.008 4 -0.053

1. No, the sum of P(x) adds up to 1
0.129+0.257+0.659+0.008+(-0.053) = 1.000
2. Yes, there are negative probabilities.
3. No, there not probabilities larger than 1.
Since we said there are negative probabilities, example 1 is NOT a probability distribution.

Example 2:
 x P(x) -2 0.2 -1 0.2 0 0.2 1 0.2 2 0.2

1. No, the sum of P(x) adds up to 1
0.2+0.2+0.2+0.2+0.2 = 1.000
2. No, there aren't any negative probabilities. The negative numbers are values for x, not the probability of x.
3. No, there aren't any probabilities larger than 1.
Since we said No to all of the questions, example 2 is a probability distribution.

Example 3:
 x P(x) 1 0.200 2 0.200 3 0.200 4 0.200 5 0.199

1. Yes, the sum of P(x) adds up to 0.999
0.2+0.2+0.2+0.2+0.199 = 0.999
2. No, there aren't any negative probabilities. The negative numbers are values for x, not the probability of x.
3. No, there aren't any probabilities larger than 1.
Since we said Yes to the first question, example 3 is NOT a probability distribution.