Determine whether each of the distributions given below represents a probability distribution. Justify your answer.
(A)
x 1 2 3 4
P(x) 1/5 5/12 1/3 1/6
(B)
x 3 6 8
P(x) 0.2 2/5 0.3
(C)
x 20 35 40 50
P(x) 0.3 -0.1 0.5 0.3
Here are some hints:
P(x) must add up to 1. Also, all numerical values for P(x) must be between 0 and 1. These two properties need to be satisfied for these problems to be considered a probability distribution.
I'll let you take it from here. (Here is another hint: only one of these problems fits the criteria.)
To determine whether each of the distributions represent a probability distribution, we need to check two conditions:
1. The sum of all the probabilities should be equal to 1.
2. The probabilities should be non-negative.
Let's analyze each distribution one by one.
(A)
x 1 2 3 4
P(x) 1/5 5/12 1/3 1/6
To check condition 1, we sum up all the probabilities:
1/5 + 5/12 + 1/3 + 1/6 = 12/60 + 25/60 + 20/60 + 10/60
= 47/60
The sum of probabilities is not equal to 1, so this distribution does not represent a probability distribution.
(B)
x 3 6 8
P(x) 0.2 2/5 0.3
To check condition 1, we sum up all the probabilities:
0.2 + 2/5 + 0.3 = 1/5 + 2/5 + 3/10
= 1/5 + 4/10 + 3/10
= 1/5 + 7/10
= 9/10
The sum of probabilities is not equal to 1, so this distribution does not represent a probability distribution.
(C)
x 20 35 40 50
P(x) 0.3 -0.1 0.5 0.3
To check condition 2, we need to ensure that all probabilities are non-negative.
In this case, the probability P(x) = -0.1 is negative. Therefore, this distribution does not represent a probability distribution.
In summary, neither of the distributions (A), (B), and (C) represent a probability distribution.