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Category Archives: Normal/ Gaussian
Box-Muller I
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 \(Z_1^2 + Z_2^2\).
Change of Variable: Gaussian
Let \(Z\) be a standard Normal random variable (ie with distribution \(N(0,1)\)). Find the formula for the density of each of the following random variables.
- 3Z+5
- \(|Z|\)
- \(Z^2\)
- \(\frac1Z\)
- \(\frac1{Z^2}\)
[based on Pitman p. 310, #10]
Two normals
A sequence \(X_1,…,X_n\) is draw iid from either \(\mbox{N}(0,1)\) or \(\mbox{N}(0,10)\) with equal prior probability.
- State the formulae for the probabilities that the sequence came from the normal with mean \(1\) or mean \(10\).
- If you know the mean of the normal is \(1\) then what is the variance of \(S = \sum_i X_i\) and \( \hat{\mu} = \frac{1}{n} \sum_i X_i\).
- What is \(\mbox{Pr}(Z > \max\{x_1,…,x_n\})\) if \(\mu =1\) and \(\mu =10\).
Random stock brokers
There are \(15\) stock brokers. The returns (in thousands of dollars) on the brokers are modeled
\( X_1,…,X_{15} \stackrel{iid}{\sim} \mbox{N}(0,1).\)
What is the probability that given the above random model at least one broker would bring in greater than $1000 dollars.
Standard Normal Tail Bound
As usual define
\[\Phi(z) = \int_{-\infty}^z \phi(x) dx \quad\text{where} \quad \phi(x)=\frac{1}{2\pi} e^{-\frac12 x^2}\]
Some times it is use full to have an estimate of \(1-\Phi(z)\) which rigorously bounds it from above (since we can not write formulas for \(\Phi(z)\) ).
Follow the following steps to prove that
\[ 1- \Phi(z) < \frac{\phi(z)}{z}\,.\]
First argue that
\[ 1- \Phi(z) < \int^{\infty}_z \frac{x}{z}\phi(x) dx\,.\]
Then evaluate the integral on the right hand side to obtain the bound.