Exercise 2.5: "Binomial" or "Poisson"?

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Characteristics of  $z_1$  and  $z_2$

Consider two discrete random variables  $z_1$  and  $z_2$  that can take  (at least)  all integer values between  $0$  and  $5$  (including these limits).

The probabilities of these random variables are given in the adjacent table.  However,  one of the two random variables is not limited to the given range of values.

Furthermore it is known that

• one of the variables is binomially distributed,  and

• the other describes a Poisson distribution.

However,  it is not known which of the two variables  $(z_1$  or $z_2)$  is binomially distributed and which is Poisson distributed.

Hints:

• This exercise belongs to the chapter  Poisson distribution.
• But also refers to the previous chapter  Binomial Distribution.
• To check your results you can use the interactive HTML5/JavaScript applet  Binomial and Poisson distribution.

Questions

1

Find out from the probabilities,  means,  and rms values whether  $z_1$  or  $z_2$  is Poisson distributed.

	$z_1$  is Poisson distributed and  $z_2$  is binomially distributed.
	$z_1$  is binomially distributed and  $z_2$  is Poisson distributed.

2

What rate  $\lambda$  does the Poisson distribution exhibit?

$\lambda \ = \ $

3

The values of the Poisson distribution are not limited to the range  $0$, ... , $5$ .
What are the probabilities that the Poisson distributed size is exactly equal to  $6$  or is greater than  $6$ ?

${\rm Pr}(z_{\rm Poisson} = 6) \ = \ $

${\rm Pr}(z_{\rm Poisson} > 6) \ = \ $

4

Now consider the binomial distribution.  Give its characteristic probability&.

$p \ = \ $

5

What is the size of the parameter  $I$  of the binomial distribution?  Check your result using the probability  $\rm Pr(0)$.

$I \ = \ $

Musterlösung

Solution

(1)  Richtig ist der Lösungsvorschlag 1:

• Bei der Poissonverteilung sind Mittelwert  $m_1$  und Varianz  $\sigma^2$  gleich.
• Die Zufallsgröße  $z_1$  erfüllt diese Bedingung im Gegensatz zur Zufallsgröße  $z_2$.

(2)  Bei der Poissonverteilung ist zudem der Mittelwert gleich der Rate.  Deshalb muss  $\underline{\lambda = 2}$  gelten.

(3)  Die entsprechende Wahrscheinlichkeit lautet mit  $z_{\rm Poisson} = z_1$:

$${\rm Pr}(z_1 = 6)=\frac{2^6}{6!}\cdot e^{-2}\hspace{0.15cm} \underline{\approx 0.012}$$

$${\rm Pr}(z_1 > 6)=1 -{\rm Pr}(0) -{\rm Pr}(1) - \ \text{...} \ - {\rm Pr}(6)\hspace{0.15cm} \underline{\approx 0.004}.$$

(4)  Für die Varianz der Binomialverteilung gilt:

$$\sigma^{2}= I\cdot p\cdot (1- p)= m_{\rm 1}\cdot ( 1- p).$$

• Die charakteristische Wahrscheinlichkeit der Binomialverteilung ergibt sich aus der Varianz  $\sigma^2 = 1.095$  und dem Mittelwert  $m_1 = 2$  gemäß der Gleichung:

$$ 1- p = \frac{\sigma^{2}}{m_1}= \frac{1.2}{2} = 0.6\hspace{0.3cm}\Rightarrow \hspace{0.3cm} p \hspace{0.15cm} \underline{= 0.4}.$$

(5)  Aus dem Mittelwert  $m_1 = 2$  folgt weiterhin  $\underline{I= 5}.$

• Die Wahrscheinlichkeit für den Wert "0" müsste mit diesen Parametern wie folgt lauten:

$${\rm Pr}(z_2 = 0)=\left({5 \atop {0}}\right)\cdot p^{\rm 0}\cdot (1 - p)^{\rm 5-0}=0.6^5=0.078.$$

• Das bedeutet:   Unsere Ergebnisse sind richtig.