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Traffic on a highway is moving at a rate 360 vehicles per hour at a location. If the number of vehicles arriving on this highway follows Poisson distribution, the probability (round off to 2 decimal places) that the headway between successive vehicles lies between 6 and 10 seconds is _________
Correct answer is '0.18'. Can you explain this answer?
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Traffic on a highway is moving at a rate 360 vehicles per hour at a lo...
λ = 360 veh/hr
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Traffic on a highway is moving at a rate 360 vehicles per hour at a lo...
Given Information:
- Traffic on a highway is moving at a rate of 360 vehicles per hour at a specific location.
- The number of vehicles arriving on this highway follows a Poisson distribution.
To Find:
- The probability that the headway between successive vehicles lies between 6 and 10 seconds.
Approach:
To solve this problem, we need to convert the rate of vehicles per hour to the rate of vehicles per second. Once we have the rate, we can calculate the probability using the Poisson distribution formula.
Step 1: Convert Rate from Hour to Second:
To convert the rate from vehicles per hour to vehicles per second, we need to divide the rate by 3600 (since there are 3600 seconds in an hour).
Rate per second = 360 vehicles per hour / 3600 seconds per hour = 0.1 vehicles per second
Step 2: Calculate the Mean:
In a Poisson distribution, the mean is equal to the rate. Therefore, the mean (λ) for our problem is 0.1 vehicles per second.
Step 3: Calculate the Probability:
To calculate the probability that the headway between successive vehicles lies between 6 and 10 seconds, we need to find the cumulative probability for the range.
Using the Poisson distribution formula, the probability of observing exactly x events in a given time period is given by:
P(x) = (e^(-λ) * λ^x) / x!
To find the cumulative probability, we need to sum the probabilities for all x values from 6 to 10.
P(6 ≤ x ≤ 10) = P(x=6) + P(x=7) + P(x=8) + P(x=9) + P(x=10)
Step 4: Calculate the Individual Probabilities:
Using the formula mentioned above, we can calculate the individual probabilities for each x value:
P(x=6) = (e^(-0.1) * 0.1^6) / 6!
P(x=7) = (e^(-0.1) * 0.1^7) / 7!
P(x=8) = (e^(-0.1) * 0.1^8) / 8!
P(x=9) = (e^(-0.1) * 0.1^9) / 9!
P(x=10) = (e^(-0.1) * 0.1^10) / 10!
Step 5: Calculate the Cumulative Probability:
Now, we can sum up the individual probabilities to get the cumulative probability:
P(6 ≤ x ≤ 10) = P(x=6) + P(x=7) + P(x=8) + P(x=9) + P(x=10)
Step 6: Calculate the Final Probability:
By calculating the above expression, we get the probability that the headway between successive vehicles lies between 6 and 10 seconds.
Calculations:
Performing the calculations as mentioned above, the final probability is found to be approximately 0.18.
Answer:
The probability that the headway between successive vehicles lies between 6 and 10
- Traffic on a highway is moving at a rate of 360 vehicles per hour at a specific location.
- The number of vehicles arriving on this highway follows a Poisson distribution.
To Find:
- The probability that the headway between successive vehicles lies between 6 and 10 seconds.
Approach:
To solve this problem, we need to convert the rate of vehicles per hour to the rate of vehicles per second. Once we have the rate, we can calculate the probability using the Poisson distribution formula.
Step 1: Convert Rate from Hour to Second:
To convert the rate from vehicles per hour to vehicles per second, we need to divide the rate by 3600 (since there are 3600 seconds in an hour).
Rate per second = 360 vehicles per hour / 3600 seconds per hour = 0.1 vehicles per second
Step 2: Calculate the Mean:
In a Poisson distribution, the mean is equal to the rate. Therefore, the mean (λ) for our problem is 0.1 vehicles per second.
Step 3: Calculate the Probability:
To calculate the probability that the headway between successive vehicles lies between 6 and 10 seconds, we need to find the cumulative probability for the range.
Using the Poisson distribution formula, the probability of observing exactly x events in a given time period is given by:
P(x) = (e^(-λ) * λ^x) / x!
To find the cumulative probability, we need to sum the probabilities for all x values from 6 to 10.
P(6 ≤ x ≤ 10) = P(x=6) + P(x=7) + P(x=8) + P(x=9) + P(x=10)
Step 4: Calculate the Individual Probabilities:
Using the formula mentioned above, we can calculate the individual probabilities for each x value:
P(x=6) = (e^(-0.1) * 0.1^6) / 6!
P(x=7) = (e^(-0.1) * 0.1^7) / 7!
P(x=8) = (e^(-0.1) * 0.1^8) / 8!
P(x=9) = (e^(-0.1) * 0.1^9) / 9!
P(x=10) = (e^(-0.1) * 0.1^10) / 10!
Step 5: Calculate the Cumulative Probability:
Now, we can sum up the individual probabilities to get the cumulative probability:
P(6 ≤ x ≤ 10) = P(x=6) + P(x=7) + P(x=8) + P(x=9) + P(x=10)
Step 6: Calculate the Final Probability:
By calculating the above expression, we get the probability that the headway between successive vehicles lies between 6 and 10 seconds.
Calculations:
Performing the calculations as mentioned above, the final probability is found to be approximately 0.18.
Answer:
The probability that the headway between successive vehicles lies between 6 and 10
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Traffic on a highway is moving at a rate 360 vehicles per hour at a location. If the number of vehicles arriving on this highway follows Poisson distribution, the probability (round off to 2 decimal places) that the headway between successive vehicles lies between 6 and 10 seconds is _________Correct answer is '0.18'. Can you explain this answer?
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Traffic on a highway is moving at a rate 360 vehicles per hour at a location. If the number of vehicles arriving on this highway follows Poisson distribution, the probability (round off to 2 decimal places) that the headway between successive vehicles lies between 6 and 10 seconds is _________Correct answer is '0.18'. Can you explain this answer?, a detailed solution for Traffic on a highway is moving at a rate 360 vehicles per hour at a location. If the number of vehicles arriving on this highway follows Poisson distribution, the probability (round off to 2 decimal places) that the headway between successive vehicles lies between 6 and 10 seconds is _________Correct answer is '0.18'. Can you explain this answer? has been provided alongside types of Traffic on a highway is moving at a rate 360 vehicles per hour at a location. If the number of vehicles arriving on this highway follows Poisson distribution, the probability (round off to 2 decimal places) that the headway between successive vehicles lies between 6 and 10 seconds is _________Correct answer is '0.18'. Can you explain this answer? theory, EduRev gives you an
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