Question

Let \(X\) have the \(\operatorname{pmf} p(x)=1 / 3, x=-1,0,1\). Find the pmf of \(Y=X^{2}\).

Short answer

The pmf of \(Y=X^{2}\) is \(P(Y = y) = \frac{2}{3}\) for \(y = 1\), \(P(Y = y) = \frac{1}{3}\) for \(y = 0\), and \(P(Y = y) = 0\) for any other value of \(y\).

Step-by-step solution

Identify the Possible Outcomes

The first step is to identify the possible outcomes of \(Y\). Since \(Y = X^2\), and \(X\) can take the values -1, 0, and 1, the possible values of \(Y\) are 1, 0, and 1, respectively.

Compute the pmf of \(Y\)

The second step is to compute the pmf of \(Y\). The probability that \(Y = y\) is the sum of the probabilities that \(X = x\), for every \(x\) such that \(x^2 = y\). If \(y = 0\), then \(P(Y = 0) = P(X = 0) = \frac{1}{3}\). If \(y = 1\), then \(P(Y = 1) = P(X = -1) + P(X = 1) = \frac{1}{3} + \frac{1}{3} = \frac{2}{3}\).

Write the pmf of \(Y\)

The final step is to write down the pmf of \(Y\). The pmf of \(Y\) is then \(P(Y = y) = \frac{2}{3}\), for \(y = 1\), and \(P(Y = y) = \frac{1}{3}\), for \(y = 0\). For any other value of \(y\), \(P(Y = y) = 0\).

Key concepts

Discrete Random Variables

In probability theory, a discrete random variable is a type of variable that can take on a countable number of distinct values. Think of it as having a handful of marbles, each with a number on it. These variables are the cornerstone of many probability calculations and are especially important when we talk about events with specific outcomes, such as rolling a die or flipping a coin.

Understanding discrete random variables is crucial to grasping the broader concepts of probability because they lay the groundwork for defining probability distributions, which tell us how likely different outcomes are. In the context of our exercise, the random variable X is discrete since it can only take on the values -1, 0, and 1. Notice how there are a finite number of possibilities? That's a telltale sign that we're dealing with a discrete random variable.

PMF Calculation

The probability mass function (pmf) is a function that gives us the probability that a discrete random variable is exactly equal to some value. Think of it as a recipe that tells us how to mix different probabilities. The pmf is essential to efficiently communicating the probabilities of the outcomes of a discrete random variable.

To calculate the pmf of a new variable like Y in our exercise, you have to map the probabilities from the original variable X. Since Y = X^2, some outcomes of X correspond to the same outcome of Y, which is why they must be combined (as seen with Y = 1 from X = -1 and X = 1). This step often involves some addition, as probabilities from different outcomes contributing to the same final result are summed. This calculation step is an excellent exercise in seeing how different variable transformations affect the probabilities of outcomes.

Probability Theory

At its core, probability theory is the branch of mathematics concerned with analyzing random phenomena and uncertainty. It allows us to make sense of the world's intrinsic randomness, providing a mathematical framework to quantify the likelihood of events. From coin flips to stock market fluctuations, probability theory gives us the tools to understand and predict outcomes.

Crucial to this is the concept of a probability distribution — a description of how probabilities are spread over the possible values of a random variable. The probability mass function we calculated for variable Y exemplifies such a distribution specific to discrete random variables. Probability theory not only encompasses the calculations of these probabilities but also underpins important principles such as the law of large numbers and the central limit theorem, which have profound implications in statistics and the real world.