GATE Exam > GATE Questions > A traffic office imposes on an average 5 numb...
Start Learning for Free
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?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
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.
|
Explore Courses for GATE exam
|
|
Similar GATE Doubts
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?
Question Description
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?.
Solutions 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? in English & in Hindi are available as part of our courses for GATE.
Download more important topics, notes, lectures and mock test series for GATE Exam by signing up for free.
Here you can find the meaning of 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? defined & explained in the simplest way possible. Besides giving the explanation of
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 detailed solution 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? has been provided alongside types of 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? theory, EduRev gives you an
ample number of questions to practice 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? tests, examples and also practice GATE tests.
|
|
Explore Courses for GATE exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup