Home » Basic probability » Change of Variable
Category Archives: Change of Variable
An example of min and change of variable
Suppose \(R_1\) and \(R_2\) are two independent random variables with the same density function
\[f(x)=x\exp(-{\textstyle \frac12 }x^2)\]
for \(x\geq 0\). Find
- the density of \(Y=\min(R_1,R_2)\);
- the density of \(Y^2\)
[Pitman p. 336 #21]
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]
Change of variable: Weibull distribution
A random variable \(T\) has the \(\text{Weibull}(\lambda,\alpha)\) if it has probability density function
\[f(t)=\lambda \alpha t^{\alpha-1} e^{-\lambda t^\alpha} \qquad (t>0)\]
where \(\lambda >0\) and \(\alpha>0\).
- Show that \(T^\alpha\) has an \(\text{exponential}(\lambda)\) distribution.
- Show that if \(U\) is a \(\text{uniform}(0,1)\) random variable, then
\[ T=\Big( – \frac{\log(U)}{\lambda}\Big)^{\frac1\alpha}\]
has a \(\text{Weibull}(\lambda,\alpha)\) distribution.
Change of Variable: Uniform
Find the density of :
- \(U^2\) if \(U\) is uniform(0,1).
- \(U^2\) if \(U\) is uniform(-1,1).
- \(U^2\) if \(U\) is uniform(-2,1).
