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Assume that the poisson distribution applies and that the mean number of hurricanes in a certain area is 7.2 per year.
a. find the probability​ that, in a​ year, there will be 6 hurricanes.
b. in a 45​-year ​period, how many years are expected to have 6 ​hurricanes?
c. how does the result from part​ (b) compare to a recent period of 45 years in which 6 years had 6 ​hurricanes? does the poisson distribution work well​ here?
a. the probability is 0.144. ​(round to three decimal places as​ needed.)
b. the expected number of years with 6 hurricanes is 6.5. ​(round to one decimal place as​ needed.)
c. the result from part​ (b) is ▼ the number of hurricanes in the recent period of 45 ​years, so the poisson distribution ▼ appear to work well in the given situation

Respuesta :

nobillionaireNobley
Part A:

The poisson distribution is given by 

[tex]P(x;\mu)=e^{-\mu}\cdot \frac{\mu^x}{x!} [/tex]

Thus, given that the mean number of hurricanes in a certain area is 7.2 per year.

The probability​ that, in a​ year, there will be 6 hurricanes is given by

[tex]P(6;7.2)=e^{-7.2}\cdot \frac{7.2^6}{6!} \\ \\ =0.000747\times \frac{139314.0695}{720} \\ \\ =0.000747\times193.492 \\ \\ =0.1445[/tex]


Part B:

In a 45​-year ​period, the number of years that are expected to have 6 ​hurricanes is given by 0.1445 x 45 = 6.5 years.


Part C:

The poisson distribution worked here because, the observed value of the number of years that hurricanes occurred in a 45 year period which is 6 is very close to the calculated value which is 6.5

a) 0.1445

b) 6.5 years

c) Refer the below solution for better understanding.

Step-by-step explanation:

Given :

The mean number of hurricanes in a certain area is 7.2 per year.

Calculation :

a) Poisson's Distribution is given by

[tex]\rm P(x;\mu)=e^-^\mu \dfrac{\mu^x}{x!}[/tex]

The probability​ in a​ year, there will be 6 hurricanes is,

[tex]\rm P(6;7.2)=e^-^7^.^2\times \dfrac{7.2^6}{6!}[/tex]

[tex]\rm P(6;7.2)=0.000727\times\dfrac{139314.0695}{720}[/tex]

[tex]\rm P(6;7.2)=0.1445[/tex]

b) In a 45​-year ​period, the number of years that are expected to have 6 ​hurricanes is

[tex]\rm 0.1445\times 45 = 6.5 \; years[/tex]

c) Yes, the poisson distribution work well here because the observed value of the number of years that hurricanes occurred in a 45 year period which is 6 is very close to the calculated value which is 6.5 years.

For more information, refer the link given below

https://brainly.com/question/23017717?referrer=searchResults

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