Final answer:
The expected value of a discrete random variable probability distribution is the sum of all possible values times their probabilities, reflecting the long-term average of many trials in an experiment. It is calculated using E(X) = μ = Σ xP(x), not to be mistaken with the mode or median, and it is different from the standard deviation which measures outcome variability.
Step-by-step explanation:
The expected value (or mean) of a discrete random variable probability distribution is the long-term average of the outcomes after many trials of a statistical experiment. Specifically, the expected value is defined as the sum of all possible values multiplied by their corresponding probabilities, which is option (b) from your question. This is not to be confused with the mode, which represents the most frequently occurring value, or the median, which is the middle value when all possible values are ordered from smallest to largest.
To calculate the expected value, represented by the notation E(X) or mean μ, you multiply each outcome value (x) by its probability (P(x)) and sum the products. For instance, if you have a random variable X that represents the number of heads in three coin tosses, and you repeat this experiment many times, the expected value of X would be the average number of heads you would expect to get.
The standard deviation of a probability distribution is a measure of how spread out the possible outcomes are from the mean. It is important to note that the standard deviation is not necessarily equal to the mean, instead, it provides insight into the variability of the outcomes around the mean.