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Box-Muller I

Let \(U_1\) and \(U_2\) be independent random variables distributed uniformly on \( (0,1) \).

define \((Z_1,Z_2)\) by

\[Z_1=\sqrt{ -2 \log(U_1) }\cos( 2 \pi U_2) \]

\[Z_2=\sqrt{ -2 \log(U_1) }\sin( 2 \pi U_2) \]

  1. Find the joint density of \((Z_1, Z_2)\).
  2. Are \(Z_1\) and \(Z_2\) independent ? why ?
  3. What is the marginal density of \(Z_1\) and \(Z_2\) ? Do you recognize it ?
  4. 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.

  1. 3Z+5
  2. \(|Z|\)
  3. \(Z^2\)
  4. \(\frac1Z\)
  5. \(\frac1{Z^2}\)

[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.

  1. State the formulae for the probabilities that the sequence came from the normal with mean \(1\) or mean \(10\).
  2. If you know the mean of the normal is \(1\) then what is the variance of \(S = \sum_i X_i\) and  \( \hat{\mu} = \frac{1}{n} \sum_i X_i\).
  3. 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.

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