Answer:

Probability distribution and Expected value

A probability distribution is a statistical function that describes all the possible outcomes and likelihoods of a random variable. The expected value of a random variable is the long-run average value of repetitions of the same experiment under the same conditions.

Expected value and Mean

The expected value of a random variable is also known as the mean or the weighted average of all possible values. It represents the center of the distribution and is calculated by multiplying each outcome by its probability and then adding up all the products.

Formula for Expected value

Expected value = ∑ (xi * P(xi))

where xi is the possible value of the random variable and P(xi) is the probability associated with that value.

Median and Mode

The median of a probability distribution is the value that separates the distribution into two equal parts, where half the outcomes are greater than the median and half are less than the median. The mode is the most frequent value in the distribution.

Expected value and Median/Mode

The expected value is not necessarily equal to the median or the mode of the distribution. The median and mode are measures of central tendency that describe the typical or most common value in the distribution, while the expected value is a measure of the long-run average value.

Conclusion

Hence, for a probability distribution, the expected value of x is the mean value of all possible outcomes, and it is not necessarily equal to the median or mode of the distribution.