Random Variables
A numerical value is often assigned to a possible outcome of a random phenomenon.
For example, a researcher might count how many people are in a random sample that have blood type A, or how many times a coin has to be tossed before observing three successive heads, or how much weight a person loses on a certain diet program.
A random variable is obtained when a numerical value is assigned to each outcome in the sample space for some experiment.
Discrete Random Variables A discrete random variable is a variable that can only take a countable number of values, and so a discrete random variable is often a count. A continuous random variable can take any value within a given interval.
Examples of discrete random variables are the sum rolled on two dice, the amount won on a hand of Blackjack, or the number of phone calls received by an operator in an hour.
Examples of continuous random variables are the amount of gas purchased by a customer at a gas station, tomorrow's weather temperature, or the length of the operator's next phone call.
The distribution between discrete random variables and continuous random variables is important to know, since the probability properties of these two types of random variables are different.
The probability properties of a discrete random variable are based upon the assignment of a probability pᵢ to each possible value xᵢ that can be taken by the random variable X. These probabilities together are known as the probability distribution of the random variable X.
The function P(X = xᵢ) = pᵢ is a probability distribution for some discrete random variable X if the following is true:
The distribution of a random variable X is usually written in tabular form, but it can also be represented graphically.
Example The following is the distribution for the sum of the numbers on two fair dice:
Example The following is the distribution for the maximum of the numbers on two fair dice:
Example Suppose someone is betting $10 on a game of Blackjack in which six decks of cards are being used, and they are using the optimal strategy. Let X be the amount of money they win on a given hand. The following is its distribution:
Expected Value The expected value, or expectation of a discrete random variable X, denoted as E(X), is the long-run average value of X that would appear after a very large number of observations. The expected value is often denoted as μ.
When the population is finite, the expected value is the population mean.
Therefore, to compute the expected value of X, the possible values of X and their probabilities are multiplied. If these probabilities are known, then the population size is irrelevant.
The population mean, or expected value, is a weighted mean of the possible values of X, where each weight is the corresponding probabilities of those values.
The following is the formula for the expected value of X:
Example From the previous example, if X was the amount of money someone wins on a given hand of Blackjack when they bet $10, then E(X) = (0.4909)(-10) + (0.0848)(0) + (0.3768)(10) + (0.0475)(15) = -0.4285.
Variance The variance of a random variable X, denoted as Var(X), is defined as the expected squared deviation from the population mean.
The following is the formula for the variance:
The standard deviation of a random discrete variable is the positive square root of the variance, where Var(X) is also denoted as σ² and so the standard deviation is denoted as σ.
Example If X is the amount won on a hand of Blackjack when betting $10, then Var(X) ≈ 97.27. The standard deviation is then σ = √(Var(X)) = √(97.27) ≈ 9.86.
Consider the experiment in which two dice are tossed. a) Let the random variable X be defined as the sum of the numbers showing on the two dice. What values can X take? b) Let X be the maximum of the numbers showing on two dice. What values can X take?
a)
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12
b)
1, 2, 3, 4, 5, and 6
When playing a game of BlackJack against the dealer at a casino, there are four possible outcomes: S = {lose, push, win, blackjack}. If the player bets $10, they lose their bet if they lose, they keep their bet if there is a push (tie), they win a $10 profit if they win, or they win a $15 profit if they get blackjack. If X was the amount of money won on one hand on Blackjack, what values of X are there?
-10, 0, 10, 15
Consider a university with 15,000 students, and let X be the number of courses for which a randomly selected student is registered. The probability distribution of X is the following:
What is the expected value of X?
E(X) = 1p(1) + 2p(2) + 3p(3) + 4p(4) + 5p(5) = 4.06
If X is the sum of the numbers showing when rolling two fair dice with an expected value of 7, what is the variance of X? What is the standard deviation of X?
Var(X) ≈ 5.833
σ = √(5.833) ≈ 2.415
If X is the maximum of the values showing when two fair dice are rolled with an expected value of 4.47, what is the variance of X? What is the standard deviation of X?
Var(X) ≈ 1.971
σ √(97.27) ≈ 1.404
