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Calls arriving
Assume that calls arrive at a call centre according to a Poisson arrival process with a rate of 15 calls per hour. For \(0 \leq s < t\), let \(N(s,t)\) denote the number of calls which arrive between time \(s\) and \(t\) where time is measured in hours.
- What is \( \mathbf{E}\big(\,N(3,5)\,\big)\) ?
- What is the second moment of \(N(2,4) \) ?
- What is \( \mathbf{E}\big(\,N(1,4)\,N(2,6)\,\big)\) ?
Tail-sum formula for continuous random variable
Let \(X\) be a positive random variable with c.d.f \(F\).
- Show using the representation \(X=F^{-1}(U)\) where \(U\) is \(\textrm{unif}(1,0)\) that \(\mathbf{E}(X)\) can be interpreted as the area above the graph on \(y=F(x)\) but below the line \(y=1\). Using this deduce that
\[\mathbf{E}(X)=\int_0^\infty [1-F(x)] dx = \int_0^\infty \mathbf{P}(X> x) dx \ .\] - Deduce that if \(X\) has possible values \(0,1,2,\dots\) , then
\[\mathbf{E}(X)=\sum_{k=1}^\infty \mathbf{P}(X\geq k)\]
Min, Max, and Exponential
Let \(X_1\) and \(X_2\) be random variables and let \(M=\mathrm{max}(X_1,X_2)\) and \(N=\mathrm{min}(X_1,X_2)\).
- 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 \(\{X_1 > x, X_2 > x\}\).
- 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.
- 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 \(N\).
- 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]
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).
