# probability and statistics 2 , poisson distribution

the numbers of vehicles, X, sailing using the ferry were as follows;

20,24,24,22,23,21,21,22,21,23,21,20,22,23,22,22,20,22,20,24

in fact, 20 0f those vehicles belongs to commuters who use the ferry everyday . the random variable Y is the number of vehicles other than those 20 those who are using the ferry evryday.

The question; Investigate whether Y can be modelled by a poisson distribution?
How is a poisson distribution parameterised, can you calculate it? Then does the data roughly follow the distribution?
To investigate whether the random variable Y, representing the number of vehicles other than those belonging to daily commuters, can be modeled by a Poisson distribution, we need to check if the Poisson assumptions hold true for this scenario.
The Poisson distribution assumes the following:

1.

The events occur randomly and independently.

2.

The average rate of occurrence is constant.

3.

The probability of more than one event occurring in an infinitesimally small time interval is negligible.

In this case, we'll analyze the data:
Number of observations (n) = 20 Mean (λ) = Average number of non-commuter vehicles per day
To calculate the mean (λ), we'll subtract the number of daily commuter vehicles from the total and then divide by the number of observations:
Total non-commuter vehicles = Sum of all vehicles - Number of daily commuter vehicles
Sum of all vehicles = 20 + 24 + 24 + 22 + 23 + 21 + 21 + 22 + 21 + 23 + 21 + 20 + 22 + 23 + 22 + 22 + 20 + 22 + 20 + 24 = 449
Total non-commuter vehicles = 449 - 20 = 429
Mean (λ) = Total non-commuter vehicles / n = 429 / 20 = 21.45
Now, let's check if the assumptions hold true:

1.

The events (arrival of vehicles) appear to occur randomly and independently.

2.

The average rate of occurrence appears to be relatively constant around 21.45.

3.

The probability of more than one vehicle arriving in an infinitesimally small time interval is likely negligible.

Given that the assumptions seem reasonable, and the average rate of occurrence is relatively constant, we can tentatively conclude that the number of vehicles other than those belonging to daily commuters can be modeled by a Poisson distribution.

However, it's important to note that further analysis and statistical tests may be needed for a more rigorous confirmation of the suitability of the Poisson distribution for this data.
Original post by RobertCsmart
To investigate whether the random variable Y, representing ....

Best not to both post "solutions" (see the sticky at the top of the forum) and adverts for the company/essay mill. However, your posted "solution" is wrong both the sum and how discount the commuters. The data in no way follows a poisson with a mean of 21.45.
(edited 1 month ago)