Question
In: Math
The waiting times X and Y (in minutes) of two clients A and B who are...
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?
Solutions
Expert Solution
(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
milcah answered 6 months agoRelated Solutions
The values listed below are waiting times? (in minutes) of customers at two different banks. At...
The values listed below are waiting times? (in minutes) of
customers at two different banks. At Bank? A, customers enter a
single waiting line that feeds three teller windows. At Bank? B,
customers may enter any one of three different lines that have
formed at three teller windows. Answer the following questions.
Bank A 6.36.3 6.66.6 6.76.7 6.86.8 7.17.1 7.37.3 7.47.4 7.87.8
7.87.8 7.87.8 Bank Upper BBank B 4.24.2 5.45.4 5.85.8 6.26.2 6.76.7
7.77.7 7.77.7 8.68.6 9.39.3 10.010.0 Construct aa...
The values listed below are waiting times (in minutes) of customers at two different banks. At...
The values listed below are waiting times (in minutes) of
customers at two different banks. At Bank A, customers enter a
single waiting line that feeds three teller windows. At Bank B,
customers may enter any one of three different lines that have
formed at three teller windows. Answer the following questions.
Bank A 6.4 6.6 6.7 6.8 7.1 7.3 7.4 7.7 7.7 7.7 Bank B 4.1 5.3 5.9
6.2 6.8 7.6 7.6 8.4 9.4 10
Construct a 99% confidence...
The values listed below are waiting times (in minutes) of customers at two different banks. At...
The values listed below are waiting times (in minutes) of
customers at two different banks. At Bank A, customers enter a
single waiting line that feeds three teller windows. At Bank B,
customers may enter any one of three different lines that have
formed at three teller windows. Answer the following questions.
Bank A
6.46.4
6.66.6
6.76.7
6.86.8
7.17.1
7.27.2
7.57.5
7.87.8
7.87.8
7.87.8
Bank Upper BBank B
4.24.2
5.35.3
5.85.8
6.16.1
6.76.7
7.87.8
7.87.8
8.48.4
9.49.4
10.010.0
LOADING...
Click...
The values listed below are waiting times (in minutes) of customers at two different banks. At...
The values listed below are waiting times (in minutes) of
customers at two different banks. At Bank A, customers enter a
single waiting line that feeds three teller windows. At Bank B,
customers may enter any one of three different lines that have
formed at three teller windows. Answer the following questions.
Bank A 6.5 6.6 6.7 6.8 7.1 7.3 7.6 7.9 7.9 7.9
Bank Upper B 4.3 5.3 5.9 6.2 6.8 7.8 7.8 8.4 9.2 10.0
Construct a 95%...
The values listed below are waiting times (in minutes) of customers at two different banks. At...
The values listed below are waiting times (in minutes) of
customers at two different banks. At Bank A, customers enter a
single waiting line that feeds three teller windows. At Bank B,
customers may enter any one of three different lines that have
formed at three teller windows. Answer the following questions.
Bank A
6.5
6.6
6.7
6.8
7.1
7.2
7.5
7.9
7.9
7.9
Bank B
4.2
5.4
5.8
6.2
6.8
7.7
7.7
8.5
9.3
10.00
a) Using Chi-Square critical...
The values listed below are waiting times (in minutes) of customers at two different banks. At...
The values listed below are waiting times (in minutes) of
customers at two different banks. At Bank A, customers enter a
single waiting line that feeds three teller windows. At Bank B,
customers may enter any one of three different lines that have
formed at three teller windows. Answer the following questions.
Bank A
6.46.4
6.66.6
6.76.7
6.86.8
7.17.1
7.37.3
7.47.4
7.97.9
7.97.9
7.97.9
Bank B
4.14.1
5.35.3
5.85.8
6.26.2
6.66.6
7.87.8
7.87.8
8.48.4
9.39.3
10.010.0
Construct a 99% confidence...
Waiting times (in minutes) of customers at a bank where all customers enter a single waiting...
Waiting times (in minutes) of customers at a bank where all
customers enter a single waiting line and a bank where
customers wait in individual lines at three different teller
windows are listed below. Find the coefficient of variation for
each
of the two sets of data, then compare the variation.
Bank A (single line): 6.5 6.6 6.7 6.9 7.1 7.3 7.4 7.6 7.7 7.8
Bank B (individual lines): 4.4 5.4 5.7 6.2 6.7 7.7 7.8 8.4 9.4
9.9
The...
Waiting times (in minutes) of customers at a bank where all customers enter a single waiting...
Waiting times (in minutes) of customers at a bank where all
customers enter a single waiting line and a bank where customers
wait in individual lines at three different teller windows are
listed below. Find the coefficient of variation for each of the two
sets of data, then compare the variation. Bank A (single line):
6.6 nbsp 6.6 nbsp 6.7 nbsp 6.8 nbsp 7.0 nbsp 7.3 nbsp 7.5 nbsp 7.7
nbsp 7.7 nbsp 7.8
Bank B (individual lines): 4.4 nbsp...
Waiting times (in minutes) of customers in a bank where all customers enter a single waiting...
Waiting times (in minutes) of customers in a bank where all
customers enter a single waiting line and a bank where customers
wait in individual lines at three different teller windows are
listed below. Find the mean and median for each of the two
samples, then compare the two sets of results. Single Line 6.5 6.6
6.7 6.8 7.0 7.1 7.5 7.6 7.6 7.6 Individual Lines 4.2 5.4 5.8 6.2
6.5 7.6 7.6 8.6 9.2 9.9 The mean waiting time...
Waiting times (in minutes) of customers at a bank where all customers enter a single waiting...
Waiting times (in minutes) of customers at a bank where all
customers enter a single waiting line and a bank where customers
wait in individual lines at three different teller windows are
listed below. Find the coefficient of variation for each of the two
sets of data, then compare the variation. Bank A (single line):
6.5 6.6 6.7 6.8 7.1 7.4 7.4 7.6 7.7 7.8 Bank B (individual lines):
4.2 5.4 5.8 6.2 6.7 7.7 7.8 8.6 9.3 9.9
The coefficient...
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