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Delhi averages three murder per week and their occurrences follow a poission distribution
Que: On the average, how many weeks a year can Delhi expect to have no murders ?
- a)1.4
- b)1.9
- c)2.6
- d)3.4
Correct answer is option 'C'. Can you explain this answer?
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Delhi averages three murder per week and their occurrences follow a po...
average number of week, per year with no murder
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Problem Analysis:
- Delhi averages three murders per week.
- The occurrences of murders follow a Poisson distribution.
Solution:
To find the average number of weeks in a year with no murders in Delhi, we need to calculate the probability of having zero murders in a given week.
Step 1: Find the average number of murders per week
Given that Delhi has an average of three murders per week, we can assume that the average rate parameter, λ, is 3.
Step 2: Calculate the probability of having zero murders in a given week
The Poisson distribution formula is given by:
P(x; λ) = (e^(-λ) * λ^x) / x!
In this case, we want to find the probability of having zero murders, so x = 0.
P(0; 3) = (e^(-3) * 3^0) / 0!
Since anything raised to the power of zero is 1, the formula simplifies to:
P(0; 3) = e^(-3)
Using a calculator, we find that e^(-3) is approximately 0.0498.
Step 3: Calculate the average number of weeks with no murders in a year
Since the probability of having zero murders in a given week is 0.0498, the probability of having at least one murder in a given week is 1 - 0.0498 = 0.9502.
To find the average number of weeks with no murders in a year, we can raise the probability of having no murders in a given week to the power of 52 (the number of weeks in a year).
Average number of weeks = (0.0498)^52
Using a calculator, we find that (0.0498)^52 is approximately 0.0063.
Therefore, Delhi can expect to have no murders in approximately 0.0063 weeks, which is equivalent to 2.6 days in a year.
Hence, the correct answer is option 'C' - 2.6.
- Delhi averages three murders per week.
- The occurrences of murders follow a Poisson distribution.
Solution:
To find the average number of weeks in a year with no murders in Delhi, we need to calculate the probability of having zero murders in a given week.
Step 1: Find the average number of murders per week
Given that Delhi has an average of three murders per week, we can assume that the average rate parameter, λ, is 3.
Step 2: Calculate the probability of having zero murders in a given week
The Poisson distribution formula is given by:
P(x; λ) = (e^(-λ) * λ^x) / x!
In this case, we want to find the probability of having zero murders, so x = 0.
P(0; 3) = (e^(-3) * 3^0) / 0!
Since anything raised to the power of zero is 1, the formula simplifies to:
P(0; 3) = e^(-3)
Using a calculator, we find that e^(-3) is approximately 0.0498.
Step 3: Calculate the average number of weeks with no murders in a year
Since the probability of having zero murders in a given week is 0.0498, the probability of having at least one murder in a given week is 1 - 0.0498 = 0.9502.
To find the average number of weeks with no murders in a year, we can raise the probability of having no murders in a given week to the power of 52 (the number of weeks in a year).
Average number of weeks = (0.0498)^52
Using a calculator, we find that (0.0498)^52 is approximately 0.0063.
Therefore, Delhi can expect to have no murders in approximately 0.0063 weeks, which is equivalent to 2.6 days in a year.
Hence, the correct answer is option 'C' - 2.6.
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Delhi averages three murder per week and their occurrences follow a poission distributionQue: On the average, how many weeks a year can Delhi expect to have no murders ?a)1.4b)1.9c)2.6d)3.4Correct answer is option 'C'. Can you explain this answer?
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