Let Z1 and Z2 be independent standard normal random...

Let $Z_{1}$ and $Z_{2}$ be independent standard normal random variables and $U_{1}=Z_{1}$ and $U_{2}=Z_{1}+Z_{2}.$ a. Derive the joint density of $U_{1}$ and $U_{2}$ b. Use Theorem 5.12 to give $E\left(U_{1}\right), E\left(U_{2}\right), V\left(U_{1}\right), V\left(U_{2}\right),$ and $\operatorname{Cov}\left(U_{1}, U_{2}\right)$ c. Are $U_{1}$ and $U_{2}$ independent? Why? d. Refer to Section $5.10 .$ Show that $U_{1}$ and $U_{2}$ have a bivariate normal distribution. Identify all the parameters of the appropriate bivariate normal distribution.

Let $Z_{1}$ and $Z_{2}$ be independent standard normal random variables and $U_{1}=Z_{1}$ and $U_{2}=Z_{1}+Z_{2}.$
a. Derive the joint density of $U_{1}$ and $U_{2}$
b. Use Theorem 5.12 to give $E\left(U_{1}\right), E\left(U_{2}\right), V\left(U_{1}\right), V\left(U_{2}\right),$ and $\operatorname{Cov}\left(U_{1}, U_{2}\right)$
c. Are $U_{1}$ and $U_{2}$ independent? Why?
d. Refer to Section $5.10 .$ Show that $U_{1}$ and $U_{2}$ have a bivariate normal distribution. Identify all the parameters of the appropriate bivariate normal distribution.

Key Concepts

Covariance and Independence

Covariance measures the degree to which two random variables change together. It provides a framework for assessing independence, especially when dealing with normal distributions. Zero covariance indicates uncorrelatedness, but in the context of linear combinations of normals, it also implies independence. This concept is essential for validating whether the transformed variables retain independence or exhibit dependency.

Joint Density Function

The joint density function represents the probability distribution of two or more random variables simultaneously. In this context, it involves determining the density of U? and U? by applying a transformation from the original independent variables, taking into account the Jacobian determinant. This is a key tool in multivariate probability theory for analyzing how a transformation affects the probability structure.

Linear Transformation of Random Variables

Linear transformations of random variables, particularly of independent normal variables, are fundamental because any linear combination of normally distributed variables remains normally distributed. This concept underpins the derivation of the joint distribution of U? and U?, and it is crucial for computing moments such as mean, variance, and covariance, as well as for establishing the form of the multivariate distribution.

Bivariate Normal Distribution

A bivariate normal distribution describes a pair of variables that are jointly normally distributed, complete with a mean vector and a covariance matrix. Recognizing that U? and U? have a bivariate normal distribution is important because it allows the use of numerous theoretical results about normal vectors, such as the marginal and conditional distributions, which further simplify the computations of their properties and help in identifying all the parameters characterizing their joint behavior.

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