Exponential distribution

Question 1

State the PDF of the Exponential distribution

Answer 1

λexp{−λx} for x>0 and 0 otherwise

Question 2

State the expectation of X ∼ Exp(λ) for λ > 0

Answer 2

1/λ

Question 3

Suppose X ∼ Exp(λ). What is the distribution of λX ?

Answer 3

Exp(1)

Question 4

Suppose H_i ∼ Exp(λ). Let J_n = sum_{i=0}^n H_i. What is the distribution of J_n ?

Answer 4

Gamma(n, λ)

Question 5

Define the lack of memory property

Answer 5

P(X > x+y | X > x) = P(X > y)

If a r.v has the lack of memory property, it follows an exponential distribution.

Question 6

Consider H_i ∼ Exp(λ_i). Set H := min{H1, . . . , Hn}.
What distribution does H follow?
What is P(H=H_k) ?

Answer 6

H ∼ Exp(sum_{i=1}^{n} λ_i)
For any k = 1, . . . , n, P(H = Hk) = λ_k / (sum_{i=1}^{n} λ_i)

PS 4-22