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Benford’s Law
Assume that the population in a city grows exponentially at rate \(r\). In other words, the number of people in the city, \(N(t)\), grows as \(N(t)=C e^{rt}\), where \(C<10^6\) is a constant.
1. Determine the time interval \(\Delta t_1\) during which \(N(t)\) will be between 1 and 2 million people.
2. For \(k=1,…,9\), determine the time interval \(\Delta t_k\) during which \(N(t)\) will be between k and k+1 million people.
3. Calculate the total time \(T\) it takes for \(N(t)\) to grow from 1 to 10 million people.
4. Now pick a time \(\hat t \in [0,T]\) uniformly at random, and use the above results to derive the following formula (also known as Benford’s law) $$p_k=\mathbb P(N(\hat t) \in [k, k+1] \,million)=\log_{10}(k+1)-\log_{10}(k).$$
Conditioning a Poisson Arrival Process
Consider a Poisson process with parameter \(\lambda\). What is the conditional probability that \(N(1) = n\) given that \(N(3) = n\)? (Here, \(N(t) \) is the number of calls which arrive between time 0 and time \(t\). ) Do you understand why this probability does not depend on \(\lambda\)?
[Meester ex 7.5.4]
probability density example
Suppose \(X\) takes values in\( (0,1) \) and has a density
\[f(x)=\begin{cases}c x^2 (1-x)^2 \qquad &x\in(0,1)\\ 0 & x \not \in (0,1)\end{cases}\]
for some \(c>0\).
- Find \( c \).
- Find \(\mathbf{E}(X)\).
- Find \(\mathrm{Var}(X) \).
