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The average number of defects per wafer (defect density) is 3. The redundancy into the design allows for up to 4 defects per wafer. What is the probability that the redundancy will not be sufficient if the defects follow a Poisson distribution?
Correct answer is '0.1847'. Can you explain this answer?
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The average number of defects per wafer (defect density) is 3. The red...
Poisson distribution:
A Poisson random variable describes the total number of events that happen in a certain time period.
A discrete random variable X is said to have a Poisson distribution with parameter λ(λ > 0) if the pdf of X is,
Now,
The average number of defects (λ) = 3
For k defects:
Where X be the number of defects per wafer.
The redundancy will not be sufficient when X > 4
P(X > 4) = 1 − P(X = 0) − P(X = 1) − P(X = 2) − P(X = 3) − P(X = 4)
=0.1847
Hence, the correct answer is 0.1847.
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The average number of defects per wafer (defect density) is 3. The red...
Defect Density:
The given defect density is 3 defects per wafer. This means that, on average, there are 3 defects per wafer.
Redundancy:
The design allows for up to 4 defects per wafer. This is the maximum number of defects that can be tolerated before the redundancy is considered insufficient.
Poisson Distribution:
The defects are assumed to follow a Poisson distribution. The Poisson distribution is commonly used to model the number of events occurring in a fixed interval of time or space, given the average rate of occurrence.
Calculating the Probability:
To calculate the probability that the redundancy will not be sufficient, we need to find the probability of having more than 4 defects per wafer.
Poisson Probability Formula:
The probability mass function (PMF) of the Poisson distribution is given by the formula:
P(x; λ) = (e^(-λ) * λ^x) / x!
where:
- P(x; λ) is the probability of having x events occur with an average rate of λ.
- e is the base of the natural logarithm (approximately 2.71828).
Calculating the Probability of More than 4 Defects:
To calculate the probability of having more than 4 defects per wafer, we can sum the probabilities of having 5, 6, 7, and so on defects.
P(X > 4) = 1 - P(X <=>
We can use the Poisson PMF formula to calculate the individual probabilities and then sum them up.
P(X > 4) = 1 - [P(0) + P(1) + P(2) + P(3) + P(4)]
Plugging in the average defect density of 3 into the Poisson PMF formula, we get:
P(X > 4) = 1 - [e^(-3) * 3^0 / 0! + e^(-3) * 3^1 / 1! + e^(-3) * 3^2 / 2! + e^(-3) * 3^3 / 3! + e^(-3) * 3^4 / 4!]
Calculating each term and summing them up, we get:
P(X > 4) ≈ 0.1847
Therefore, the probability that the redundancy will not be sufficient is approximately 0.1847 or 18.47%.
The given defect density is 3 defects per wafer. This means that, on average, there are 3 defects per wafer.
Redundancy:
The design allows for up to 4 defects per wafer. This is the maximum number of defects that can be tolerated before the redundancy is considered insufficient.
Poisson Distribution:
The defects are assumed to follow a Poisson distribution. The Poisson distribution is commonly used to model the number of events occurring in a fixed interval of time or space, given the average rate of occurrence.
Calculating the Probability:
To calculate the probability that the redundancy will not be sufficient, we need to find the probability of having more than 4 defects per wafer.
Poisson Probability Formula:
The probability mass function (PMF) of the Poisson distribution is given by the formula:
P(x; λ) = (e^(-λ) * λ^x) / x!
where:
- P(x; λ) is the probability of having x events occur with an average rate of λ.
- e is the base of the natural logarithm (approximately 2.71828).
Calculating the Probability of More than 4 Defects:
To calculate the probability of having more than 4 defects per wafer, we can sum the probabilities of having 5, 6, 7, and so on defects.
P(X > 4) = 1 - P(X <=>
We can use the Poisson PMF formula to calculate the individual probabilities and then sum them up.
P(X > 4) = 1 - [P(0) + P(1) + P(2) + P(3) + P(4)]
Plugging in the average defect density of 3 into the Poisson PMF formula, we get:
P(X > 4) = 1 - [e^(-3) * 3^0 / 0! + e^(-3) * 3^1 / 1! + e^(-3) * 3^2 / 2! + e^(-3) * 3^3 / 3! + e^(-3) * 3^4 / 4!]
Calculating each term and summing them up, we get:
P(X > 4) ≈ 0.1847
Therefore, the probability that the redundancy will not be sufficient is approximately 0.1847 or 18.47%.
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The average number of defects per wafer (defect density) is 3. The redundancy into the design allows for up to 4 defects per wafer. What is the probability that the redundancy will not be sufficient if the defects follow a Poisson distribution?Correct answer is '0.1847'. Can you explain this answer?
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