# 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 $$\mathrm{Var}(X)$$.