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The mean of a probability distribution is also known as the expected value. It tells us the average outcome we can expect if we were to repeat an experiment multiple times. For any probability distribution, we calculate the mean using the formula: \(\text{Mean} = \, \text{Expected Value} = \, \, \mu = \, \sum [x \, \times \, P(x)]\). In simpler terms, we multiply each value of the random variable \(x\) by its corresponding probability \(P(x)\) and then sum all these products. This gives us a weighted average.
In our example, we had the values 0, 1, 2, and 3 for \(x\) with their respective probabilities. Calculating, we get:
- Mean \(= 0 \, \times \, 0.091 + 1 \, \times \, 0.334 + 2 \, \times \, 0.408 + 3 \, \times \, 0.166 = \mu = 1.648\).
This indicates that on average, 1.648 out of the 3 adults say the most fun way to flirt is in person.

Variance measures the spread of the random variable's possible values. It tells us how much the values deviate from the mean. To find the variance, we use the formula \( \, \, \sigma^2 = \, \, \sum [(x \, - \, \mu)^2 \, \times \, P(x)]\). This involves computing the squared difference between each value and the mean, multiplying each by their respective probabilities, and then summing these products.
For our example, after calculating the squared differences from the mean and respective products, we get:
- \(2.715 \, \times \, 0.091 + 0.419 \, \times \, 0.334 + 0.123 \, \times \, 0.408 + 1.828 \, \times \, 0.166 = \, \sigma^2 = \, 0.74\).
This tells us how much the number of adults who prefer flirting in person deviates from the average.

Standard deviation is a measure of the amount of variation or dispersion of a set of values. It is the square root of the variance and gives us an easy-to-interpret number for how much the values of the random variable tend to deviate from the mean. The formula for standard deviation is \( \, \sigma = \, \, \sqrt{\sigma^2}\).
In our example, the standard deviation is calculated as:
- \( \, \sigma = \, \, \sqrt{0.74} = \, \, \approx 0.86\).
This value tells us that the number of adults who say the most fun way to flirt is in person typically varies by 0.86 from the mean of 1.648.

Expected value is closely related to the mean of a probability distribution. It represents the average outcome if an experiment were repeated a large number of times under the same conditions. We often use expected value to make informed predictions about future outcomes. The formula for expected value is the same as the mean: \( \, E(x) = \, \, \, \mu = \, \, \sum [x \, \times \, P(x)]\).
From the earlier calculations in our example:
- \( E(x) = \, 0 \, \times \, 0.091 + 1 \, \times \, 0.334 + 2 \, \times \, 0.408 + 3 \, \times \, 0.166 = \, 1.648\).
The expected value tells us what we would expect the number of adults who prefer to flirt in person to be, based on the given probabilities.