91Ó°ÊÓ

Understanding the probability mass function (PMF) is essential when dealing with discrete random variables, like the one in our exercise, where a number is selected at random from a finite set. A PMF gives us the probability that a discrete random variable is exactly equal to some value. In mathematical terms, if we have a random variable X, the PMF P(X=k) denotes the probability that X takes on the value k.

In this case, the set from which a number is randomly selected is \(\{1, 2, 3, 4\}\). Since each number has an equal chance of being selected, the PMF is uniform with \(P(X=k) = \frac{1}{4}\) for each value of k. This sort of equal probability scenario is common in 'fair' situations, like a fair dice roll or drawing a card from a well-shuffled deck.

Variance is a measure of how much the values of a random variable X spread out from their expected value E(X). It provides an indication of the degree of dispersion or spread in a set of values. A low variance means the values cluster closely around the mean, while a high variance indicates they are spread out over a wider range.

The formula to calculate the variance of a random variable X is \(Var(X) = E(X^2) - (E(X))^2\). This involves computing the expected value of the square of X and subtracting the square of the expected value of X. When we applied this formula to our exercise, we first calculated the expected value to be 2.50. Then, we found \(E(X^2)\), that is, the expected value of the squares of each possible number, and then finally, calculated the variance by subtracting the square of the expected value \(2.50^2\) from \(E(X^2)\), which equaled 1.25 in this instance.

Following the calculation of variance, the standard deviation is the next step. It is the square root of the variance and represents the average amount by which the values in a data set differ from the mean value. The standard deviation is a very useful measure because it is expressed in the same units as the data, which helps in understanding the variability in context.

In mathematical notation, the standard deviation of X is represented as \(\sigma(X)\) and is calculated as \(\sigma(X) = \sqrt{Var(X)}\). When we applied this formula to the results of our exercise, we took the square root of the variance (1.25) that we had previously calculated, resulting in a standard deviation of approximately 1.12. This indicates that on average, each number in the set \(\{1, 2, 3, 4\}\) deviates from the mean (2.50) by about 1.12 units.