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Questions and Answers
What condition allows the Poisson distribution to approximate the Binomial distribution?
What condition allows the Poisson distribution to approximate the Binomial distribution?
What does the parameter λ in a Poisson distribution represent?
What does the parameter λ in a Poisson distribution represent?
Which of the following statements about the variance of a Poisson random variable is true?
Which of the following statements about the variance of a Poisson random variable is true?
In a Poisson distribution, what does the factorial K! in the probability mass function represent?
In a Poisson distribution, what does the factorial K! in the probability mass function represent?
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Which of the following scenarios is best modeled by a Poisson distribution?
Which of the following scenarios is best modeled by a Poisson distribution?
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What is the expectation E(X) of a Poisson random variable X in relation to λ?
What is the expectation E(X) of a Poisson random variable X in relation to λ?
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To find the probability of no more than two interruptions in a Poisson distribution, which of the following calculations is necessary?
To find the probability of no more than two interruptions in a Poisson distribution, which of the following calculations is necessary?
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Which formula correctly represents the probability mass function of a Poisson distribution?
Which formula correctly represents the probability mass function of a Poisson distribution?
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Study Notes
Poisson Distribution
- The Poisson distribution is a special case of the binomial distribution where the number of trials (n) approaches infinity and the probability of success (p) on each trial is very small.
- It's used to model the number of events occurring in a fixed interval of time or space, where events occur independently and at a known average rate.
- Probability mass function: P(X = k) = (λk * e-λ) / k!, where:
- k is the number of occurrences of an event.
- λ (lambda) is the average rate of occurrences.
- e is the mathematical constant approximately equal to 2.718.
- k! is the factorial of k (k! = k × (k-1) × (k-2) ×... × 2 × 1).
When to Use Poisson Distribution
- Suitable for count data, like:
- Number of customer arrivals per hour.
- System failures per day.
- Number of emails received in a day.
- Number of car crashes on a highway.
Using Poisson Distribution
- Example: Interruptions per day on a wi-fi network. Average rate of interruptions (λ) = 0.9 per day.
- Find the probability of no more than two interruptions (P(X ≤ 2)). This involves calculating P(X = 0) + P(X = 1) + P(X = 2) using the Poisson formula.
Poisson Approximation to Binomial Distribution
- Use Poisson when n is large and p is small. Here X = np.
- The Poisson probability is approximately equal to the binomial probability.
Examples
- Example: Defective hard drives (probability of a defective hard drive is 0.02).
- The number of defective hard drives from 100 produced is shown as an example of binomial distribution.
- Calculating the probability of X > 3 based on 100 trials using the binomial formula versus using the complement of X < 3.
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