Stats Solutions

(Step-by-Step) Suppose that X_1 and X_2 are independent and identically distributed standard normal random variables, that is X_i ~ N(0,1) for i=1,2, with probability

Question: Suppose that \(X_{1}\) and \(X_{2}\) are independent and identically distributed standard normal random variables, that is \(X_{i} \sim N(0,1)\) for \(i=1,2\), with probability density function

\[f_{X_{i}}\left(x_{i}\right)=\frac{1}{\sqrt{2 \pi}} \exp \left\{-\frac{1}{2} x_{i}^{2}\right\} \quad-\infty Consider the bivariate transformation to \(V_{1}\) and \(V_{2}\) such that

\[V_{1}=\frac{1}{\sqrt{2}}\left(X_{1}+X_{2}\right) \text { and } V_{2}=\frac{1}{\sqrt{2}}\left(X_{1}-X_{2}\right)\]

Write down the means and variances of \(V_{1}\) and \(V_{2}\). Now, find the joint density of \(V_{1}\) and \(V_{2}\), and hence the marginal distributions. Are \(V_{1}\) and \(V_{2}\) independent?

Price: $2.99
Solution: The downloadable solution consists of 2 pages
Deliverable: Word Document


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