Question

1. The exponential distribution (?)E(λ) with density ?(?)=??−??f(x)=λe−λx for all ?>0x>0. Suppose that ?=2λ=2.
   1. Find the median of ?X.
   2. Find the expected value and variance of ?X.
   3. Find P(?>3)

Answer 1

(a)

Probability Density Function of Exponential Distribution is given by:

                      for x > 0

Median is got as follows:

between limits 0 to x.

Applying limits, we get:

Thus, we get:

Taking logarithm on both sides, we get:

i.e.,

So,

Median is given by:

(b)

(i)

Expected Value E(X) is given by:

between limits 0 to .

Applying limits, we get:

(ii)

E(X²) is got as follows:

between limits 0 to .

Applying limits, we get:

So,

Variance is got as follows:

Variance is given by:

(c)

between limits 3 to .

Applying limits, we get:

So,

Answer is: