Exercise 2.5Z: Flower Meadow

Flower meadow – another example of the Poisson distribution

A farmer is happy about the splendor of flowers on his land and wants to know how many dandelions are currently blooming on his meadow.

He knows that the meadow has an area of $5000$ square meters and he also knows from the agricultural school that the number of flowers in a small area is always Poisson distributed.
He stakes out ten squares, each with an edge length of $\text{25 cm}$, randomly distributed over the entire meadow and counts the flowers in each of these squares:

$$\rm 3, \ 4, \ 1, \ 5, \ 0, \ 3, \ 2, \ 4, \ 2, \ 6.$$

Consider these numerical values as random results of the discrete random variable $z$.

It is obvious that the sample size $(10)$ is very small but – this much is revealed – the farmer is lucky. First consider how you would proceed to solve this task, and then answer the following questions.

Hints:

This exercise belongs to the chapter Poisson Distribution.
Reference is also made to the chapter Moments of a Discrete Random Variable.

Questions

$m_z \ =$

$\sigma_z\ = \ $

	Actually, one would have to use considerably more than ten random numbers (squares) for the moment calculation.
	The random variable $z$ is in fact Poisson distributed.
	The rate $\lambda$ of the Poisson distribution is equal to the standard deviation $\sigma_z$.
	The rate $\lambda$ of the Poisson distribution is equal to the mean $m_z$.

$B\ = \ $

$\ \text{thousand}$

${\rm Pr}(z = 0) \ = \ $

$\ \%$

Solution

(1) The linear mean of these ten numbers gives

$$\underline{m_z = 3}.$$

(2) For the second moment of the random variable $z$ applies accordingly:

$$m_{\rm 2\it z}=\frac{1}{10}\cdot (0^2+1^2+ 2\cdot 2^2+ 2\cdot 3^2+2\cdot 4^2+ 5^2+6^2)=12.$$

According to Steiner's theorem, the variance is:

$$\sigma_z^2 =12 -3^2 = 3,$$

and thus the standard deviation:

$$\underline{\sigma_z \approx 1.732}.$$

(3) Correct are solutions 1, 2, and 4:

Mean and standard deviation agree here. This is indicative of the Poisson distribution with rate $\lambda = 3$
(equal to the mean and equal to the variance, not equal to the standard deviation).
Naturally, it is questionable to make this statement on the basis of only ten values.
However, in the case of moments, a smaller sample number is less serious than, for example, in the case of probabilities.

(4) In total, there are $80000$ such squares, each with three flowers in the mean.

This suggests a total of $\underline{B = 240}$ thousand flowers.

(5) According to the Poisson distribution, this probability results in

$${\rm Pr}(z = 0) = \frac{3^0}{0!} \cdot{\rm e}^{-3}\hspace{0.15cm}\underline{\approx 5\%}.$$

However, the small sample size $N = 10$ on which this task was based would have indicated probability ${\rm Pr}(z = 0) = { 10\%}$
since only in a single square no single flower was counted.