Determine whether each of the following is a valid probability distribution.
a) X: 0, 2, 2, 3
P(x): .1, .3, .4, .1
b) X: 5, -6, 10, 0
P(x): .01, .01, .01, .97
c) X: 14, 12, -7, 13
P(x): .25, .46, .04, .2
If these are all the scores in the distribution, the sum of their probabilities = 1.
To determine whether each of the following is a valid probability distribution, we need to check if the probabilities sum up to 1.
a) X: 0, 2, 2, 3
P(x): .1, .3, .4, .1
To check the sum of probabilities, we calculate:
.1 + .3 + .4 + .1 = 1
Since the sum of probabilities is equal to 1, this is a valid probability distribution.
b) X: 5, -6, 10, 0
P(x): .01, .01, .01, .97
To check the sum of probabilities, we calculate:
.01 + .01 + .01 + .97 = 1.00
Since the sum of probabilities is equal to 1, this is a valid probability distribution.
c) X: 14, 12, -7, 13
P(x): .25, .46, .04, .2
To check the sum of probabilities, we calculate:
.25 + .46 + .04 + .2 = 0.95
Since the sum of probabilities is not equal to 1 (it is 0.95), this is not a valid probability distribution.
To determine whether each of the given probability distributions is valid, we need to make sure that three conditions are met:
1) The sum of all the probabilities is equal to 1.
2) Each probability is between 0 and 1 (inclusive).
3) The probabilities for each value in the distribution are non-negative.
Let's now analyze each distribution:
a) X: 0, 2, 2, 3
P(x): 0.1, 0.3, 0.4, 0.1
To check the first condition, we sum up all the probabilities:
0.1 + 0.3 + 0.4 + 0.1 = 0.9
Since the sum is not equal to 1, this distribution does not satisfy the first condition and is not a valid probability distribution.
b) X: 5, -6, 10, 0
P(x): 0.01, 0.01, 0.01, 0.97
For the first condition:
0.01 + 0.01 + 0.01 + 0.97 = 1
The sum of all probabilities is equal to 1, so it satisfies the first condition.
Now for the second and third conditions, we need to check that all probabilities are between 0 and 1 and non-negative. Here, all probabilities are between 0 and 1, and none of them are negative. Therefore, this distribution satisfies all the conditions and is a valid probability distribution.
c) X: 14, 12, -7, 13
P(x): 0.25, 0.46, 0.04, 0.2
For the first condition:
0.25 + 0.46 + 0.04 + 0.2 = 0.95
The sum of all probabilities is not equal to 1, so it does not satisfy the first condition and is not a valid probability distribution.
In summary:
a) Not a valid probability distribution.
b) Valid probability distribution.
c) Not a valid probability distribution.