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The expected value, often referred to as the mean, is a foundational concept in probability and statistics. It represents the average outcome you would anticipate if you could repeat an experiment or process an infinite number of times. Imagine rolling a die. Each face has an equal probability of 1/6, so the expected value of this die roll is the average of all possible outcomes, taking into account these probabilities.
For a discrete random variable, the expected value is found using the formula:

• \( E(X) = \sum{[x_i \times P(x_i)]} \)

This means you multiply each possible value \( x_i \) by its probability \( P(x_i) \), and then sum those products.
In the context of our original exercise, the random variable has an expected value of 10. This signifies that, as a weighted average of its outcomes, the mean or expected long-term value when considering all possible outcomes and their probabilities is 10.

Standard deviation is a statistical measure that illustrates the spread or variability of a set of values. A low standard deviation means that values tend to be close to the mean, while a high standard deviation indicates that they are spread out over a wider range.
For a finite random variable, you'll calculate the standard deviation by first determining the variance, which is the expected value of the squared deviations from the mean:

• \( \text{Variance} = E[(X - E(X))^2] = \sum{[(x_i - E(X))^2 \times P(x_i)]} \)
• \( \text{Standard Deviation} = \sqrt{\text{Variance}} \)

In the exercise presented, the standard deviation is 0. This implies there is absolutely no variability; all outcomes are the same as the expected value. Therefore, if you calculate the variance, it results in zero—showing no dispersion of the values around the mean.

A finite random variable is one that has a limited set of distinct outcomes it can take. Since real-world scenarios often have finite outcomes, understanding this concept is crucial.
Such a random variable could represent, for example, the result of drawing a card from a deck, the number of heads in a series of coin flips, or the specific case in our exercise that results in only one value, 10.

• The probability distribution for this kind of random variable is a complete list of possible outcomes along with their likelihoods.
• For instance, if only one outcome consistently occurs, as in the exercise where \( P(X = 10) = 1 \), it indicates no other values occur.

With a standard deviation of 0, the finite random variable can only equal the expected value, as all probabilities consolidate into a single, certain outcome. This ensures all realizations of the random variable equate to that single specific outcome.