Respuesta :
Answer:
Step-by-step explanation:
Given that the number of drivers who travel between a particular origin and destination during a designated time period has a Poisson distribution with parameter μ = 20 (suggested in the article "Dynamic Ride Sharing: Theory and Practice"†).
a) [tex]P(X\leq 15) = 0.1565=0.157[/tex]
b) [tex]P(X>26) =1-F(26)\\= 1-0.9221\\=0.0779=0.078[/tex]
c) [tex]P(15\leq x\leq 26)\\=F(26)-F(14)\\=0.9221-0.1049\\=0.8172=0.817[/tex]
d) 2 std dev = 2(20) =40
Hence 2 std deviation means
20-40, 20+40
i.e. (0,60)
[tex]P(0<x<60)\\=F(60)-F(0)\\=1-0.00000206\\=0.99999794=1.000[/tex]
The probability that the number of drivers will be within 2 standard deviations of the mean value is 100%.
What is Poisson Distribution?
The Poisson Distribution is a discrete probability distribution that describes the likelihood of a specific number of events occurring in a specified span of time or space at a known constant mean rate, regardless of the time since the last occurrence.
As it is given that the number of drivers who traveled between a particular origin and destination during a designated time period has a Poisson distribution with parameter μ = 20.
A.) The probability that the number of drivers will be at most 15.
[tex]\begin{aligned}P(X\leq 15)&= P(X < 15)\\&=0.1565\\&=15.7\% \end{aligned}[/tex]
B.) The probability that the number of drivers will exceed 26.
[tex]\begin{aligned}P(X > 26)&= 1 - P(X < 26)\\&=1-0.9221\\&=0.0779\\&=7.8\% \end{aligned}[/tex]
C.) The probability that the number of drivers will be strictly between 15 and 26.
[tex]\begin{aligned}P(15\leq x\leq 26)&= P(X < 26)-P(x < 15)\\&=0.9221-0.1049\\&=0.8172\\&=81.72\% \end{aligned}[/tex]
D.) The probability that the number of drivers will be within 2 standard deviations of the mean value.
[tex]\begin{aligned}\sigma_2 &= 2\mu\\&=2(20)\\&=40\end{aligned}[/tex]
[tex]20 \pm 40\\(0, 60)[/tex]
Now, the probability can be written as,
[tex]\begin{aligned}P(0 < X < 60)&=P(X < 60)-P(X < 0)\\&=1-0.00000206\\&=0.99999794\approx 1.000 \end{aligned}[/tex]
Hence, the probability that the number of drivers will be within 2 standard deviations of the mean value is 100%.
Learn more about Poisson Distribution:
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