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A traffic office imposes on an average 5 number of penalties daily on traffic violators. Assume that the number of penalties on different days is independent and follows a Poisson distribution. The probability that there will be less than 4 penalties in a day is ___________
Correct answer is between '0.26,0.27'. Can you explain this answer?
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A traffic office imposes on an average 5 number of penalties daily on ...
**Given information:**
- The average number of penalties imposed daily is 5.
- The number of penalties on different days is independent.
- The number of penalties follows a Poisson distribution.
**Poisson Distribution:**
The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, assuming that these events occur with a known constant mean rate and are independent of the time since the last event.
The probability mass function (PMF) of the Poisson distribution is given by:
P(X = k) = (e^(-λ) * λ^k) / k!
Where:
- X is the random variable representing the number of events (penalties).
- λ is the average rate of events (penalties) per interval.
**Calculating the probability:**
In this case, we need to calculate the probability that there will be less than 4 penalties in a day.
To calculate this probability, we sum the probabilities of having 0, 1, 2, or 3 penalties in a day using the Poisson PMF.
P(X < 4)="P(X" =="" 0)="" +="" p(x="1)" +="" p(x="2)" +="" p(x="" />
Using the Poisson PMF formula:
P(X = k) = (e^(-λ) * λ^k) / k!
Substituting λ = 5 and k = 0, 1, 2, 3:
P(X < 4)="(e^(-5)" *="" 5^0)="" 0!="" +="" (e^(-5)="" *="" 5^1)="" 1!="" +="" (e^(-5)="" *="" 5^2)="" 2!="" +="" (e^(-5)="" *="" 5^3)="" />
Calculating each term separately and summing them up:
P(X < 4)="" ≈="" 0.0337="" +="" 0.1687="" +="" 0.4218="" +="" />
P(X < 4)="" ≈="" />
Since the probability cannot be greater than 1, the correct answer lies between 0.26 and 0.27.
- The average number of penalties imposed daily is 5.
- The number of penalties on different days is independent.
- The number of penalties follows a Poisson distribution.
**Poisson Distribution:**
The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, assuming that these events occur with a known constant mean rate and are independent of the time since the last event.
The probability mass function (PMF) of the Poisson distribution is given by:
P(X = k) = (e^(-λ) * λ^k) / k!
Where:
- X is the random variable representing the number of events (penalties).
- λ is the average rate of events (penalties) per interval.
**Calculating the probability:**
In this case, we need to calculate the probability that there will be less than 4 penalties in a day.
To calculate this probability, we sum the probabilities of having 0, 1, 2, or 3 penalties in a day using the Poisson PMF.
P(X < 4)="P(X" =="" 0)="" +="" p(x="1)" +="" p(x="2)" +="" p(x="" />
Using the Poisson PMF formula:
P(X = k) = (e^(-λ) * λ^k) / k!
Substituting λ = 5 and k = 0, 1, 2, 3:
P(X < 4)="(e^(-5)" *="" 5^0)="" 0!="" +="" (e^(-5)="" *="" 5^1)="" 1!="" +="" (e^(-5)="" *="" 5^2)="" 2!="" +="" (e^(-5)="" *="" 5^3)="" />
Calculating each term separately and summing them up:
P(X < 4)="" ≈="" 0.0337="" +="" 0.1687="" +="" 0.4218="" +="" />
P(X < 4)="" ≈="" />
Since the probability cannot be greater than 1, the correct answer lies between 0.26 and 0.27.
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A traffic office imposes on an average 5 number of penalties daily on traffic violators. Assume that the number of penalties on different days is independent and follows a Poisson distribution. The probability that there will be less than 4 penalties in a day is ___________Correct answer is between '0.26,0.27'. Can you explain this answer?
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A traffic office imposes on an average 5 number of penalties daily on traffic violators. Assume that the number of penalties on different days is independent and follows a Poisson distribution. The probability that there will be less than 4 penalties in a day is ___________Correct answer is between '0.26,0.27'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A traffic office imposes on an average 5 number of penalties daily on traffic violators. Assume that the number of penalties on different days is independent and follows a Poisson distribution. The probability that there will be less than 4 penalties in a day is ___________Correct answer is between '0.26,0.27'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A traffic office imposes on an average 5 number of penalties daily on traffic violators. Assume that the number of penalties on different days is independent and follows a Poisson distribution. The probability that there will be less than 4 penalties in a day is ___________Correct answer is between '0.26,0.27'. Can you explain this answer?.
A traffic office imposes on an average 5 number of penalties daily on traffic violators. Assume that the number of penalties on different days is independent and follows a Poisson distribution. The probability that there will be less than 4 penalties in a day is ___________Correct answer is between '0.26,0.27'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about A traffic office imposes on an average 5 number of penalties daily on traffic violators. Assume that the number of penalties on different days is independent and follows a Poisson distribution. The probability that there will be less than 4 penalties in a day is ___________Correct answer is between '0.26,0.27'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A traffic office imposes on an average 5 number of penalties daily on traffic violators. Assume that the number of penalties on different days is independent and follows a Poisson distribution. The probability that there will be less than 4 penalties in a day is ___________Correct answer is between '0.26,0.27'. Can you explain this answer?.
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