Question

Suppose that X is exponential with parameter λ. Compute the median of X (i.e. the t for which P(X ≤ t) = 1/2). Is it smaller or larger than the expectation?

Answer 1

(a)

Probability Density Function of Exponential Distribution is given by:

,

                    for x> 0

Median is given by the value of x for which cdf of x,F(x)= 0.5

Thus, we get:

between the limits 0 to x.

Applying limits, we get:

i.e.,

Taking logarithm on both sides, we get:

So,

Thus

Median is given by:

(b)
Median is smaller than expectation because Expectation is given by:

and ln (2) is less than 1.