18 - Exponential & Poisson Distributions

Question 1

What is the defining characteristic of a Poisson Distribution?

Answer 1

Events occur independently at random at a constant average rate.

Question 2

What is the notation for a Poisson Distribution?

Answer 2

𝑋~𝑃𝑜(𝜆), where 𝜆 is the constant average rate.

Question 3

What are the formulas for the mean and variance of a Poisson Distribution?

Answer 3

𝐸[𝑋] = 𝜆 and 𝑉𝑎𝑟[𝑋] = 𝜆

Question 4

What conditions must be met for a variable to have a Poisson distribution?

Answer 4

Singly, Independently, at a Constant Rate, and Randomly.

Question 5

If 𝑋~𝑃𝑜(𝜆) and 𝑌~𝑃𝑜(𝜇) and are independent, what is the distribution of 𝑋 + 𝑌?

Answer 5

𝑋 + 𝑌~𝑃𝑜(𝜆 + 𝜇)

Question 6

What does the Exponential Distribution represent in relation to the Poisson Distribution?

Answer 6

The time that passes between successive events of a Poisson distribution.

Question 7

What type of distribution is the Exponential Distribution?

Answer 7

Continuous

Question 8

What is the notation for an Exponential Distribution?

Answer 8

𝑋~𝐸𝑥𝑝(𝜆)

Question 9

What is the formula for the cumulative probability of an Exponential Distribution (P (X < x))?

Answer 9

1 − 𝑒−λ𝑥

Question 10

What are the formulas for the mean and variance of the exponential distribution?

Answer 10

Mean 𝐸[𝑋] = 1/𝜆, Variance 𝑉𝐴𝑅[𝑋] = 1/𝜆²

Question 11

How is the parameter λ for the exponential distribution calculated from the Poisson distribution?

Answer 11

λ for the exponential distribution is the interval for the Poisson distribution divided by the λ for the Poisson distribution.