Question

Compute the median of an Exp(λ) distribution.

Compute the median of a Par(1) distribution.

Answer 1

Compute the median of an Exp(λ) distribution.

f(x) = λ exp(-λx)

= ∫(from 0 to ∞) x λ exp(-λx) dx =
= (1/λ) ∫(from 0 to ∞) u exp(-u) du =
= 1/λ

This is the mean of the exponential distribution.

The median is the value for which

F(x) = ∫(from 0 to x) f(t) dt = 1/2

F(x) = ∫(from 0 to x) λ exp(-λx) dt =
= ∫(from 0 to λx) exp(-u) du =
= 1 - exp(-λx) = 1/2
=>
exp(-λx) = 1/2
=>
x = ln(2)/λ

This is the median.

Compute the median of a Par(1) distribution.