Final answer:
The variance of a probability distribution measures how spread out the values of the random variable are. To find the variance, calculate the deviation of each value from the mean, square the deviation, multiply it by its probability, and add up all the products. Then, take the square root to find the standard deviation.
Step-by-step explanation:
The variance of a probability distribution is a measure of how spread out the values of the random variable are. To find the variance of a discrete probability distribution, you need to calculate the deviation of each value from the mean, square the deviation, multiply it by its probability, and add up all the products. Then, to find the standard deviation, simply take the square root of the variance.
For example, suppose you have a discrete probability distribution with values 1, 2, 3, and 4, each with a probability of 0.25. The mean is (1*0.25 + 2*0.25 + 3*0.25 + 4*0.25) = 2.5. The deviations from the mean are (-1.5, -0.5, 0.5, and 1.5), squared they are (2.25, 0.25, 0.25, and 2.25). Multiplying each squared deviation by their corresponding probability gives (0.5625, 0.0625, 0.0625, and 0.5625), and summing them up results in a variance of 1.25. Taking the square root of the variance gives a standard deviation of 1.118.