Tag Archives: JCM_math230_HW8_S15

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 \(c>0\).

  1. Find \( c \).
  2. Find \(\mathbf{E}(X)\).
  3. Find \(\mathrm{Var}(X) \).