Understanding the Square of a Standard Normal Variate

A standard normal variate, often denoted by \(Z\), is a random variable that follows the standard normal distribution. This distribution has a mean of 0 and a variance of 1. In mathematical notation, we write \(Z \sim N(0, 1)\).

The question asks about the distribution of the square of such a variable, which is \(Z^2\).

Distribution of \(Z^2\)

In probability and statistics, there is a well-known result concerning the distribution of the square of a standard normal variate. This result is fundamental in the construction of several important statistical tests and distributions.

• If \(Z\) is a single random variable following the standard normal distribution \(N(0, 1)\), then the random variable \(Y = Z^2\) follows a chi-square distribution.
• The degrees of freedom for the chi-square distribution are determined by the number of independent standard normal variates that are squared and summed. Since we are only squaring a single standard normal variate \(Z\), the resulting chi-square distribution has 1 degree of freedom.

Thus, the distribution of the square of a standard normal variate is a chi-square distribution with 1 degree of freedom, denoted as \(\chi^2(1)\).

Evaluating the Options

Let's briefly look at the other options provided to understand why they are not the correct distribution for the square of a standard normal variate:

• Beta distribution: This distribution is typically used to model probabilities or proportions, defined on the interval [0, 1]. The square of a standard normal variate \(Z^2\) can take any non-negative value (\([0, \infty)\)), not just values between 0 and 1.
• Cauchy distribution: The Cauchy distribution is known for having heavy tails and undefined moments (like mean and variance). The square of a standard normal variate has a well-defined mean (equal to its degrees of freedom, which is 1) and variance (equal to 2 times its degrees of freedom, which is 2).
• F-distribution: An F-distribution arises from the ratio of two independent chi-square variables, each divided by their respective degrees of freedom. It is used in ANOVA and other tests comparing variances, but it is not the distribution of a single squared standard normal variate.

Based on the definition and properties, the square of a standard normal variate is indeed a chi-square distribution with 1 degree of freedom.