Question

Michael will sell his bike because he will move and he has decided to sell it to the first person who offers at least 200 $. Suppose that each price offer given for Michael's bike is independent and exponentially distributed RVs(random variable) with mean $ 100 each.
a)   Michael sold his bike on the Kth offer. Find PMF(probability mass function) and mean of K
b)   Let the amount of offer which is sold by Michael be X $. Find PDF(Probability density function) of X and mean.

Answer 1

Answer:-

Given that:-

k is the offer

Mean of K the expected number of offer

The offer on the bike follows iid expo

where success mean that the offer is atleast $200

So, the number of offers that are two low is Geam(p)

Including the succeessful offer, the expected number of offers is,

=7.407

(b) The amount of money michael gets for the bike is x.

N be the number of offers ,So the sale price of bike is

Since the successful offer is an exponential for which we have the value is atleast 200$.

To Compute this we make know the memoryless property of exponential for any a>0. if Then the distribution of x-a given x>a is itself , so