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Category Archives: Exponential Random Variables
Geometric Branching Process
Consider a branching process with a geometric offspring distribution \( P(X=k) = (1-p)p^k\), for \(k=0,1,2,\dots\) . Show that the ultimate extinction is certain if \(p \leq \frac12\) and that the probability of extinction is \((1-p)/p \) if \(p > \frac12\).
[Meester ex. 6.6.5]
Memory and the Exponential
Let \(X\) have an exponential distribution with parameter \(\lambda\). Show that
\[ P( X> t+ s \,|\, X>s) = P(X>t) \]
for all \(s,t >0\). Explain why one might call this property of the exponential “the lack of memory”.
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]
Joint arrival times
Let \(T_1\) and \(T_5\) be the times of the first and fifth arrival in a Poisson process with rate \(\lambda\). Find joint density of \(T_1\) and \(T_5\).
[Pitman p355 #12]
Expectation of min of exponentials
There are \(15\) stock brokers. The returns (in thousands of dollars) on each brokers is modeled as a separate independent exponential distribution \(X_1 \sim \mbox{Exp}(\lambda_1),…,X_{15} \sim \mbox{Exp}(\lambda_{15})\). Define \(Z = \min\{X_1,…,X_{15}\}\).
What is \(\mathbf{E}(Z)\) ?
Strontium
Assume we have a large number of particles \(N\) of Strontium. The decay model for Strontium is exponential in that \(\mathbf{P}(T > t) = e^{- \lambda t}\), this states the probability of a an atom surviving until time \(T\).
- The half-life of a substance is the amount of time it takes for an appreciable amount of the substance to be reduced in half. If the half life of strontium is 28 years what is the decay parameter of the exponential ?
- What is the probability Strontium lasts at least 50 years, \(\mathbf{P}(T > 50) \) ?
- Suppose we have \(5\) radioactive substances, the decay of each of which can be modeled by five exponential random variables \(X_1,…,X_5\) with parameters \(\lambda_1,…,\lambda_5\). Assume the five distributions are independent. What is the pdf for \(\min\{X_1,…,X_5\}\).
Polonium data
Look at the following link to the following table summarizing the radioactive decay counts of polonium recorded by Rutherford and Geiger (1910) representing the number of scintillations in 2608 1/8 minute intervals. For example, there were 57 frequencies of zero counts. The counts can be thought of as being approximately Poisson distributed.
- Use the fact that for the Poisson distribution \( \mathbf{E}[X] = \lambda \) to estimate the rate parameter. This is using the methods of moments to estimate a parameter.
- Maximize the likelihood to estimate \( \lambda\).
