Let \(U_1\) and \(U_2\) be independent random variables distributed uniformly on \( (0,1) \).
define \((Z_1,Z_2)\) by
\[Z_1=\sqrt{ -2 \log(U_1) }\cos( 2 \pi U_2) \]
\[Z_2=\sqrt{ -2 \log(U_1) }\sin( 2 \pi U_2) \]
- Find the joint density of \((Z_1, Z_2)\).
- Are \(Z_1\) and \(Z_2\) independent ? why ?
- What is the marginal density of \(Z_1\) and \(Z_2\) ? Do you recognize it ?
- Reflect on the implications of the previous answer for generating an often needed class of random variable on a computer.
Hint: To eliminate \(U_1\) write the formula for \(Z_1^2 + Z_2^2\).
