Find the mean of the distribution shown.
x 2 5 7
P(x) 0.28 0.42 0.30
What is the mean and the standard deviation of this distribution?
To find the mean of a distribution, we multiply each value by its corresponding probability, and then sum up the results.
In this case, we have the following values and corresponding probabilities:
x: 2, 5, 7
P(x): 0.28, 0.42, 0.30
To find the mean, we multiply each value by its probability:
(2 * 0.28) + (5 * 0.42) + (7 * 0.30) = 0.56 + 2.10 + 2.10 = 4.76
Therefore, the mean of this distribution is 4.76.
To calculate the standard deviation, we first need to find the variance. The variance is the average of the squared differences between each value and the mean. To do this, we follow these steps:
1. Subtract the mean from each value in the distribution:
For x = 2: 2 - 4.76 = -2.76
For x = 5: 5 - 4.76 = 0.24
For x = 7: 7 - 4.76 = 2.24
2. Square each result:
For x = 2: (-2.76)² = 7.6176
For x = 5: (0.24)² = 0.0576
For x = 7: (2.24)² = 5.0176
3. Multiply each squared result by its corresponding probability:
For x = 2: 7.6176 * 0.28 = 2.13171
For x = 5: 0.0576 * 0.42 = 0.02419
For x = 7: 5.0176 * 0.30 = 1.50528
4. Sum up the squared results:
2.13171 + 0.02419 + 1.50528 = 3.66118
Therefore, the variance of this distribution is 3.66118.
The standard deviation is the square root of the variance. Taking the square root of 3.66118, we get approximately 1.913.
Therefore, the standard deviation of this distribution is approximately 1.913.