Look at the relative-frequency table below of the probability distribution for the frequency with which customers buy items, with discrete random variable X = "number of items purchased by a customer."x P(X = x)1 .022 .173 .294 .275 .156 .077 .03What is the standard deviation?

To calculate the standard deviation of a probability distribution, we first need to calculate the mean (expected value), then use this to calculate the variance, and finally take the square root of the variance to get the standard deviation.

Step 1: Calculate the mean (expected value) The mean (μ) of a probability distribution is calculated by multiplying each outcome by its probability and then summing these products.

μ = Σ [x * P(X = x)] = (1*.02) + (2*.17) + (3*.29) + (4*.27) + (5*.15) + (6*.07) + (7*.03) = .02 + .34 + .87 + 1.08 + .75 + .42 + .21 = 3.69

Step 2: Calculate the variance The variance (σ^2) is calculated by multiplying the square of the difference between each outcome and the mean by the probability of the outcome, and then summing these products.

σ^2 = Σ [(x - μ)^2 * P(X = x)] = [(1-3.69)^2 * .02] + [(2-3.69)^2 * .17] + [(3-3.69)^2 * .29] + [(4-3.69)^2 * .27] + [(5-3.69)^2 * .15] + [(6-3.69)^2 * .07] + [(7-3.69)^2 * .03] = [.0721 * .02] + [.4624 * .17] + [.4761 * .29] + [.0961 * .27] + [.1764 * .15] + [.5376 * .07] + [1.1025 * .03] = .001442 + .078608 + .138069 + .025946 + .026496 + .037632 + .033075 = 0.341268

Step 3: Calculate the standard deviation The standard deviation (σ) is the square root of the variance.

σ = √σ^2 = √0.341268 = 0.584

So, the standard deviation of the probability distribution is approximately 0.584.