Given the following relative-frequency table for the probability distribution of X, what are the expected value (𝜇) and the standard deviation (𝜎)?x P(X = x)5 .1010 .1415 .2920 .47

To find the expected value (mean) and the standard deviation of a probability distribution, we need to follow these steps:

Step 1: Calculate the Expected Value (Mean) The expected value (mean) of a random variable is calculated by multiplying each possible outcome by its probability, and then summing all these values.

So, for the given table, the expected value (𝜇) would be calculated as follows:

𝜇 = Σ [x * P(X = x)] = (5 * .10) + (10 * .14) + (15 * .29) + (20 * .47) = 0.5 + 1.4 + 4.35 + 9.4 = 15.65

Step 2: Calculate the Variance The variance is calculated by taking each difference from the mean, squaring it, multiplying it by the corresponding probability, and then summing these values.

Variance = Σ [(x - 𝜇)² * P(X = x)] = [(5 - 15.65)² * .10] + [(10 - 15.65)² * .14] + [(15 - 15.65)² * .29] + [(20 - 15.65)² * .47] = [112.0225 * .10] + [31.6225 * .14] + [0.4225 * .29] + [18.8225 * .47] = 11.20225 + 4.42715 + 0.122525 + 8.846575 = 24.5985

Step 3: Calculate the Standard Deviation The standard deviation is the square root of the variance.

𝜎 = √Variance = √24.5985 = 4.96 (rounded to two decimal places)

So, the expected value (𝜇) is 15.65 and the standard deviation (𝜎) is 4.96.