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Questions and Answers

What condition allows the Poisson distribution to approximate the Binomial distribution?

When n approaches zero and p is moderate

When n is moderate and p approaches one

When n is large and p is small (correct)

When n is small and p is large

What does the parameter λ in a Poisson distribution represent?

The total number of trials conducted

The maximum possible number of occurrences

The average rate of occurrence of an event (correct)

The probability of success on each trial

Which of the following statements about the variance of a Poisson random variable is true?

The variance is equal to λ (correct)

The variance is independent of the average rate

The variance is always equal to the number of trials

The variance is equal to the probability of success

In a Poisson distribution, what does the factorial K! in the probability mass function represent?

The number of ways to arrange K events (B) Signup and view all the answers

Which of the following scenarios is best modeled by a Poisson distribution?

The number of emails received each day (A) Signup and view all the answers

What is the expectation E(X) of a Poisson random variable X in relation to λ?

E(X) = λ (C) Signup and view all the answers

To find the probability of no more than two interruptions in a Poisson distribution, which of the following calculations is necessary?

P(X = 0) + P(X = 1) + P(X = 2) (D) Signup and view all the answers

Which formula correctly represents the probability mass function of a Poisson distribution?

$P(X = K) = rac{λ^K e^{-λ}}{K!}$ (C) Signup and view all the answers

Flashcards

Poisson Distribution

A probability distribution that describes the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known average rate and independently of the time since the last event.

Poisson Parameter (λ)

The average number of events in a given interval of time or space.

Poisson Probability Mass Function

A formula for calculating the probability of exactly 'K' events occurring in a given interval, given the parameter λ.

Binomial Distribution Approximation

The Poisson distribution can approximate the binomial distribution when the number of trials (n) is large and the probability of success (p) is small.

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Expected Value (E(X)) of Poisson

The average number of events expected to occur in a given interval.

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Variance of Poisson

The measure of the spread of the distribution around the mean.

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Applications of Poisson Distribution

Used to model events such as customer arrivals, system failures, or email receptions. The events occur randomly, independently, and with a known average rate.

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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) = (λ^k * 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.