Tag Archives: JCM_math230_HW9_S15

Closest Point

Consider a Poisson random scatter of points in a plane with mean intensity \(\lambda\) per unit area. Let \(R\) be the distance from zero to the closest point of the scatter.

  1. Find a formula for the c.d.f. and the density of \(R\) and sketch their graphs.
  2. Show that \(\sqrt{2 \lambda \pi} R\) has the Rayleigh distribution.
  3. Find the mean and mode of \(R\).

 

[pitman p 389, # 21]

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

  1. the density of \(Y=\min(R_1,R_2)\);
  2. the density of \(Y^2\)

[Pitman p. 336 #21]

Min, Max, and Exponential

Let \(X_1\) and \(X_2\) be random variables and let \(M=\mathrm{max}(X_1,X_2)\) and \(N=\mathrm{min}(X_1,X_2)\).

  1. Argue that the event \(\{ M \leq x\}\) is the same as the event   \(\{X_1 \leq x, X_2 \leq x\}\) and similarly that t the event \(\{ N > x\}\) is the same as the event   \(\{X_1 > x, X_2 > x\}\).
  2. Now assume that the \(X_1\) and \(X_2\) are independent and distributed with c.d.f. \(F_1(x)\) and \(F_2(x)\) respectively . Find the c.d.f. of \(M\) and the c.d.f. of \(N\) using the proceeding observation.
  3. Now assume that \(X_1\) and \(X_2\) are independently and exponentially  distributed with parameters \(\lambda_1\) and \(\lambda_2\) respectively. Show that \(N\) is distributed exponentially and identify the parameter  in the exponential distribution of \(N\).
  4. The route to a certain remote island contains 4 bridges. If the time to collapse of each bridge is exponential distributed with mean 20 years and is independent of the other bridges, what is the distribution of the time until the road is impassable because one of the  bridges has collapsed.

 

[Jonathan Mattingly]

Order statistics I

Suppose \(X_1, … , X_n \stackrel{iid}{\sim} U(0,1) \). How large must \(n\) be to have that \({\mathbf{P}}(X_{(n)} \geq .95) \geq 1/2\) ?

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]

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 \(\alpha>0\).

  1. Show that \(T^\alpha\) has an \(\text{exponential}(\lambda)\) distribution.
  2. 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 :

  1. \(U^2\) if \(U\) is uniform(0,1).
  2. \(U^2\) if \(U\) is uniform(-1,1).
  3. \(U^2\) if \(U\) is uniform(-2,1).