determine whether the following formula is a probability distribution
P(X)=X/20 for X=1,4,7
AND
P(X)=X+0.1 for X=0.24,0.26,0.15
To determine if a given formula is a probability distribution, we need to make sure that three conditions are satisfied:
1. The probabilities must be non-negative: This means that for each value of X, the probability should be greater than or equal to zero.
2. The probabilities must sum to 1: The sum of all the probabilities for each value of X should be equal to 1.
3. The probabilities should be mutually exclusive: This means that the probability of each value of X should be distinct and unrelated to the probability of other values.
Let's check these conditions for each of the given formulas:
Formula 1: P(X) = X/20 for X = 1, 4, 7
Condition 1: The probabilities are non-negative since X is positive for all values X = 1, 4, 7.
Condition 2: We need to determine if the sum of the probabilities is 1.
P(1) + P(4) + P(7) = (1/20) + (4/20) + (7/20) = 12/20 = 0.6
Since the sum is not equal to 1, the formula does not satisfy this condition and is not a probability distribution.
Let's move on to the second formula:
Formula 2: P(X) = X + 0.1 for X = 0.24, 0.26, 0.15
Condition 1: The probabilities are non-negative since X is positive for all values X = 0.24, 0.26, 0.15.
Condition 2: We need to determine if the sum of the probabilities is 1.
P(0.24) + P(0.26) + P(0.15) = (0.24 + 0.10) + (0.26 + 0.10) + (0.15 + 0.10) = (0.34 + 0.36 + 0.25) = 0.95
Since the sum is not equal to 1, the formula does not satisfy this condition and is not a probability distribution.
Therefore, neither of the given formulas is a probability distribution since they do not satisfy the condition that the sum of probabilities should equal 1.