Question

A fair die is cast at random three independent times. Let the random variable \(X_{i}\) be equal to the number of spots that appear on the \(i\) th trial, \(i=1,2,3\). Let the random variable \(Y\) be equal to \(\max \left(X_{i}\right) .\) Find the cdf and the pmf of \(Y\). Hint: \(P(Y \leq y)=P\left(X_{i} \leq y, i=1,2,3\right)\).

Short answer

The Cumulative Distribution Function (CDF) of Y is given by \(P(Y \leq y) = (y/6)^3\), and the Probability Mass Function (PMF) of Y is given by \(P(Y = y) = ((y/6)^3 - ((y - 1)/6)^3)\), for y in {1,2,3,4,5,6}.

Step-by-step solution

Understand the probability of a single trial

Firstly, note that the die is fair, meaning each face (number from 1 to 6) has an equal probability of 1/6 of occurring in each throw. This probability remains constant for each throw, as the throws are independent of one another.

Formulate CDF of Y

The cumulative distribution function (CDF) of a random variable is the probability that the variable takes a value less than or equal to a certain value. From the hint, the CDF of Y can be computed by \(P(Y \leq y) = P(X_i \leq y, i=1,2,3)\). As the 3 throws are independent, this can be written as the cube of the probability of a single trial. Therefore, \(P(Y \leq y) = [P(X_i \leq y)]^3 = (y/6)^3\), for y in {1,2,3,4,5,6}. This is because there are y outcomes less than or equal to y, and each has probability 1/6.

Formulate PMF of Y

The probability mass function (PMF) gives the probability of each possible outcome of the random variable, and it is the derivative of the cumulative distribution function. Deriving our CDF thus gives the PMF of Y: \(P(Y = y) = P(Y \leq y) - P(Y \leq y - 1) = ((y/6)^3 - ((y - 1)/6)^3)\), for y in {1,2,3,4,5,6}.

Key concepts

Probability Mass Function

The Probability Mass Function (PMF) is a fundamental concept in probability theory when dealing with discrete random variables such as the roll of a die. The PMF, denoted as P(X = x), gives us the probability that a discrete random variable X will take on the exact value x. In the given exercise, we're dealing with a fair six-sided die, so each outcome (1 through 6) has an equal probability of 1/6.

When we derive the PMF of the random variable Y, which represents the maximum value observed in three independent die throws, we calculate the difference between the probabilities of Y being less than or equal to a specific value y and Y being less than or equal to y - 1. The PMF informs us how probable each possible result of Y actually is. Understanding how to derive and utilize the PMF is crucial for solving probability-related questions and for interpreting the distribution of discrete outcomes in a range of fields, from games of chance to statistical sampling.

An improved explanation for the PMF could involve emphasizing its role in calculating the likelihood of singular outcomes. This can be visualized in the context of our die example, where the PMF allows us to predict the chances of rolling any one number on a single die throw.

Independent Random Variables

In probability, when we describe random variables as independent, we mean that the outcome of one variable does not influence the outcome of another. This is a core concept because it allows us to compute the joint probability of multiple events happening by simply multiplying their individual probabilities. In our exercise with the fair die, each of the three throws is independent.

To illustrate, if you roll a three on your first try, this has absolutely no bearing on what you roll on your second or third tries. Because of this independence, we can find the CDF of the maximum value Y after three throws just by raising the CDF of a single throw to the power of three. It is this property of independent variables that makes certain types of calculations much simpler to handle. When teaching the concept of independent random variables, it’s helpful to draw connections to real-life scenarios where events occur without affecting each other, such as tossing a coin multiple times or selecting cards from a well-shuffled deck without replacement.

Outcome Probability

Outcome probability refers to the likelihood of a specific event occurring. It is the bread-and-butter of probability theory and forms the basis of calculating more complex probabilities. Each outcome has a probability that ranges between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

In the context of throwing a die, the outcome probability is straightforward – since the die is fair, each of the six faces has an equal probability, which is 1/6. However, when we look at the probability of a compound event, such as the maximum value of three independent die throws, calculating the outcome probability involves understanding both the PMF and the CDF.

For example, considering the variable Y which represents the maximum roll of the die in three throws, we cannot simply assign 1/6 to each outcome. Instead, we need to examine the set of all possible outcomes and their respective probabilities using the rules of CDF and the independence of each throw. It is critical to clearly convey the distinction between the probability of single, simple events and the probability of complex, compound events to facilitate deeper understanding of how different factors can influence outcome probability.