Question

If \(Z \sim N(0,1)\), derive the density of \(Y=Z^{2}\). Although \(Y\) is determined by \(Z\), show they are uncorrelated.

Short answer

\(Y = Z^2\) follows a Chi-squared(1) distribution, and \(Z\) and \(Y\) are uncorrelated.

Step-by-step solution

Identifying the Problem

We are given that \(Z\) follows a standard normal distribution, \(N(0,1)\). We need to derive the probability density function (pdf) of \(Y = Z^2\) and show that \(Z\) and \(Y\) are uncorrelated.

Deriving the Distribution of \(Y\)

Since \(Y = Z^2\), where \(Z\) follows \(N(0,1)\), \(Y\) follows a Chi-squared distribution with 1 degree of freedom, also known as Chi-squared(1). The pdf of a Chi-squared distribution with 1 degree of freedom is given by \(f_Y(y) = \frac{1}{\sqrt{2\pi y}} e^{-y/2}\) for \(y > 0\).

Calculating the Expected Values

To show \(Z\) and \(Y\) are uncorrelated, we need \(\mathbb{E}[Z^2] - \mathbb{E}[Z]\mathbb{E}[Y] = 0\). Start by finding \(\mathbb{E}[Z] = 0\). The expectation \(\mathbb{E}[Y] = \mathbb{E}[Z^2] = 1\), since the variance of \(Z\) is 1.

Finding Covariance of \(Z\) and \(Y\)

The covariance is given by \(\text{Cov}(Z, Y) = \mathbb{E}[ZY] - \mathbb{E}[Z]\mathbb{E}[Y]\). Because \(Z\) is symmetric about 0, \(\mathbb{E}[ZY] = \mathbb{E}[Z^3] = 0\) since \(Z^3\) is also symmetric about 0. Therefore, \(\text{Cov}(Z, Y) = 0 - 0 \cdot 1 = 0\).

Conclusion on Uncorrelatedness

Since \(\text{Cov}(Z, Y) = 0\), \(Z\) and \(Y\) are uncorrelated. Despite \(Y\) being a function of \(Z\), their correlation is zero.

Key concepts

Correlation and Uncorrelation

Understanding the concepts of correlation and uncorrelation is essential when analyzing relationships between random variables. Correlation between variables indicates that changes in one variable are associated with changes in another variable. For two variables to be uncorrelated, their covariance must be zero.
In mathematical terms, if you have two variables, say \(Z\) and \(Y\), they are uncorrelated if their covariance \(\text{Cov}(Z, Y)\) is zero. Covariance measures how much two variables change together. The formula for covariance is:

• \(\text{Cov}(Z, Y) = \mathbb{E}[ZY] - \mathbb{E}[Z]\mathbb{E}[Y]\)

If \(\text{Cov}(Z, Y) = 0\), it indicates no linear relationship, even if one variable is a function of the other, like \(Y = Z^2\). In our example, although \(Y\) is derived from \(Z\), they are shown to be uncorrelated since their covariance is zero. Thus, uncorrelated does not mean independent, especially in cases where one variable depends on another.

Probability Density Function

A probability density function (pdf) helps describe the distribution of continuous random variables. In essence, it provides us with the likelihood of a random variable falling within a particular range of values.
Consider a random variable \(Y\) that is \(Z^2\) where \(Z\) follows a standard normal distribution \(N(0,1)\). To find the pdf of \(Y\), we determine that \(Y\) follows a Chi-squared distribution with one degree of freedom. The standard pdf for a Chi-squared distribution with 1 degree of freedom is:

• \(f_Y(y) = \frac{1}{\sqrt{2\pi y}} e^{-y/2}\) for \(y > 0\)

This formula describes how the values of \(Y\) are likely to occur based on the properties of the Chi-squared distribution. The exponential component \(e^{-y/2}\) characterizes the rapid decrease in probability for higher values of \(y\), reflecting the nature of squared normal variables.

Standard Normal Distribution

The standard normal distribution is a fundamental concept in statistics, known for its bell-shaped, symmetric appearance. It is a special type of normal distribution with a mean of 0 and a variance of 1.
The notation \(Z \sim N(0,1)\) signifies that \(Z\) is a standard normal random variable. This means the probability density function for \(Z\) is:

• \(f_Z(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}\)

This function tells us the probability of \(Z\) taking on certain values on the real number line. The distribution is perfectly symmetrical around zero, making both the mean and median equal to zero.
The standard normal distribution is the building block for many statistical techniques and theories. For example, when we derive \(Y = Z^2\), we utilize the standard normal distribution properties to understand how \(Y\)'s distribution (Chi-squared) behaves.