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

(1) State CDF of a Poisson distribution:

We know that $P_X(k)=e^{-\alpha }\frac{{\alpha }^k}{k!}$

We know that $F_X(k)=P[X\le k]$.

$$F_X(k)=\sum _{k=0}^{\infty }{e}^{-\alpha }\frac{{\alpha }^k}{k!}$$

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(2) Compute limit as $k\to \infty$:

Note that $e^x={\sum }_{n=0}^{\infty }\frac{x^n}{n!}$

$$\underset{k\to \infty }{\lim }{F}_X(x)=e^{-\alpha }\underset{k\to \infty }{\lim }\sum _{k=0}^{\infty }\frac{{\alpha }^k}{k!}=e^{-\alpha }{e}^{\alpha }=1$$Link to original

02

Expectation of Poisson

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

Solution

Solutions - 5130-02

PMF of a Poisson distribution:

$$P_X(k)=e^{-\alpha }\frac{{\alpha }^k}{k!}$$

Expected value:

$$\begin{array}{c}E[X]=\sum _{k=0}^{\infty }k{P}_X(k)\\ \\ \gg \gg \sum _{k=0}^{\infty }{e}^{-\alpha }\frac{{\alpha }^k}{(k-1)!}\\ \\ \gg \gg \alpha e^{-\alpha }\sum _{k=0}^{\infty }\frac{{\alpha }^{k-1}}{(k-1)!}\\ \\ \gg \gg \alpha e^{-\alpha }{e}^{\alpha }\gg \gg \alpha \end{array}$$Link to original

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

(a)

(1) Define Poisson approximation:

$Y\sim \text{Pois}(np)=\text{Pois}(1)$.

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(2) Estimate error:

$$|P_X(k)-P_Y(k)|\le n{p}^2=10{\left(\frac{1}{10}\right)}^2=\frac{1}{10}$$

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(b)

(1) Compute $P[X\le 1]$ using the binomial distribution:

$$P[X\le 1]=\left(\genfrac{}{}{0px}{}{10}{0}\right){\left(\frac{1}{10}\right)}^0{\left(\frac{9}{10}\right)}^{10}+\left(\genfrac{}{}{0px}{}{10}{1}\right){\left(\frac{1}{10}\right)}^1{\left(\frac{9}{10}\right)}^9\approx 0.7361$$

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(2) Compute $P[X\le 1]$ using the Poisson distribution:

$$P[X\le 1]=e^{-1}\frac{1^0}{0!}+e^{-1}\frac{1^1}{1!}\approx 0.7358$$

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(3) Compute error:

$$\text{error}=|0.7361-0.7358|=0.0003$$Link to original