Question

2 $f(x)=\frac{2^{x}e^{-2}}{x}$
3 $f(x)=\frac{2^{x}e^{2}}{x!}$
equation #1
b. what is the expected number of occurrences in three time periods?
6
c. select the appropriate poisson probability function to determine the probability of $x$ occurrences in three time periods.
1 $f(x)=\frac{6^{x}e^{6}}{x!}$
2 $f(x)=\frac{6^{x}e^{-6}}{x}$
3 $f(x)=\frac{6^{x}e^{-6}}{x!}$
equation #3
d. compute the probability of two occurrences in one time period (to 4 decimals).
0.2707
e. compute the probability of six occurrences in two time periods (to 4 decimals).
0.1055
f. compute the probability of seven occurrences in three time periods (to 4 decimals).

Explanation:

Step1: Recall Poisson probability formula

The Poisson probability function is $f(x)=\frac{\lambda^{x}e^{-\lambda}}{x!}$, where $\lambda$ is the average number of occurrences in the given interval and $x$ is the actual number of occurrences.

Step2: For part f

We know from part b that for three - time periods $\lambda = 6$. We want to find the probability of $x = 7$ occurrences in three time periods.
Substitute $\lambda=6$ and $x = 7$ into the Poisson formula:
$f(7)=\frac{6^{7}e^{-6}}{7!}$
First, calculate $6^{7}=279936$, $e^{-6}\approx0.002479$, and $7!=5040$.
Then $6^{7}e^{-6}=279936\times0.002479 = 693.961344$.
$\frac{6^{7}e^{-6}}{7!}=\frac{693.961344}{5040}\approx0.1377$

Answer:

$0.1377$