Problems

01

Poisson satisfies $F(+\infty )=1$

Show that a Poisson variable $X\sim \mathrm{Pois}(\alpha )$ satisfies the total probability rule for a CDF, namely that ${\lim }_{x\to \infty }{F}_X(x)=1$.

Solution

Solutions---5130-01

02

Expectation of Poisson

Derive the formula $E[X]=\alpha$ for a Poisson variable $X\sim \mathrm{Pois}(\alpha )$.

Solution

Solutions---5130-02

03

Application of Poisson: meteor shower

The UVA astronomy club is watching a meteor shower. Meteors appear at an average rate of $4$ per hour.

(a) Write a short explanation to justify the use of a Poisson distribution to model the appearance of meteors. Why should appearances be Poisson distributed?

(b) What is the probability that the club sees more than 2 meteors in a single hour?

(c) Suppose we find out that over a four hour evening, 13 meteors were spotted. What is the probability that none of them happened in the first hour, conditioned on that information?

04

Silver dimes

Suppose 1 out of 500 dimes in circulation is made of silver. Consider a tub of dimes worth $40.

(a) Find a formula for the exact probability that this collection contains at least 2 silver dimes. (Do not use the complement event, you must perform a summation.) Can your calculator evaluate this formula?

(b) Estimate the probability in question using a Poisson approximation.

(This topic for HW only, not for tests.)

05

Application of Poisson approximation of binomial

Let $X\sim \mathrm{Bin}(10,\frac{1}{10})$ and consider the Poisson approximation to $X$.

(a) Estimate the possible error of the approximation (for an arbitrary probability).

(b) Compute the exact error of the approximation for the specific probability $P[X\le 1]$.

(This topic for HW only, not for tests.)

Solution

Solutions---5130-05