Exponential Distribution Properties

Question 1

What distribution does the sum of n i.i.d. Exponential(λ) variables follow?

Answer 1

Gamma(n, λ)

Question 2

How can the sum of exponentials be transformed into a chi-squared distribution?

Answer 2

Multiply the sum by 2λ to get χ²₍₂ₙ₎

Question 3

What is the relationship between the chi-squared and Gamma distributions?

Answer 3

χ²₍k₎ ~ Gamma(k/2, 1/2)

Question 4

What distribution does the sample mean of n Exponential(λ) variables follow?

Answer 4

Gamma(n, nλ)

Question 5

How can the sample mean of exponentials be transformed into a chi-squared distribution?

Answer 5

Multiply by 2nλ to get χ²₍₂ₙ₎

Question 6

What is the distribution of the minimum of n i.i.d. Exponential(λ) variables?

Answer 6

Exponential(nλ)

Question 7

What is the CDF of a single Exponential(λ) variable?

Answer 7

F(x) = 1 - e^(-λx)

Question 8

What is the PDF of the minimum of n Exponential(λ) variables?

Answer 8

f(y) = nλ * e^(-nλy)

Question 9

Why is transforming variables to chi-squared important in statistics?

Answer 9

Chi-squared is central to many statistical tests and procedures.

Question 10

What is the shape parameter of a Gamma distribution formed by summing n Exponential(λ) variables?

Answer 10

n

Question 11

What property allows multiplication to affect the second parameter of a Gamma distribution?

Answer 11

Scaling a Gamma variable by c divides the rate parameter by c.

Question 12

What is the purpose of transforming exponential variables in hypothesis testing?

Answer 12

To utilize chi-squared distribution properties for test construction.