Final answer:
To find the mean and standard deviation of a probability distribution, calculate the expected value by multiplying each outcome by its probability and summing them up. Then calculate the variance by taking the square of the difference between each outcome and the expected value, multiply it by its probability, and sum them up. Finally, the standard deviation is the square root of the variance.
Step-by-step explanation:
To find the mean and standard deviation of the probability distribution, we need to first calculate the expected value and variance. The expected value, or mean, is found by multiplying each possible outcome by its corresponding probability and summing them up. The variance is found by taking the square of the difference between each possible outcome and the expected value, multiplying it by its corresponding probability, and summing them up. Finally, the standard deviation is the square root of the variance.
For example, if we have the following probability distribution for the number of pairs of shoes a student owns:
X: 0 1 2 3
P(X): 0.2 0.3 0.4 0.1
We can calculate the mean as follows:
Expected value = (0 * 0.2) + (1 * 0.3) + (2 * 0.4) + (3 * 0.1) = 0 + 0.3 + 0.8 + 0.3 = 1.4
And we can calculate the variance as follows:
Variance = ((0 - 1.4)^2 * 0.2) + ((1 - 1.4)^2 * 0.3) + ((2 - 1.4)^2 * 0.4) + ((3 - 1.4)^2 * 0.1) = (1.4)^2 * 0.2 + (-0.4)^2 * 0.3 + (0.6)^2 * 0.4 + (1.6)^2 * 0.1 = 0.56 + 0.12 + 0.36 + 0.26 = 1.3
Finally, the standard deviation is the square root of the variance, which is sqrt(1.3) = 1.14.