In: Math
The waiting times X and Y (in minutes) of two clients A and B who are standing in line at two different check outs in the supermarket are modeled as independent, exponential random variables with parameter 1.
(a) Find the cumulative distribution function of the random variable M :=min{X,Y} where min{x,y} is just the smaller value of the two numbers.
(b) Find the probability density function of M. Do you recognize the socalled probability law or probability distribution of the random variable M?
(c) What is the probability that both clients wait more than 2 minutes?
(a)
First we need to find out the distributon of minimum of two. Let
Since X and Y are exponentially distributed so CDF of X will be
and CDF of Y will be
and we have
Let the cumulative distribution function of M is
(b)
The pdf of M will be
It is pdf of exponential distribution with parameter .
(c)
The probability that client A wait more than 2 minutes is:
The probability that client B wait more than 2 minutes is:
SInce both counters are independent so the probability that both clients wait more than 2 minutes is
Answer: 0.0183