A tunnel opens at 7am and on average 27 red trucks enter this tunnel from 7am to 10am on Monday mornings. Suppose the red trucks arrive independent of each other and at a constant rate. (a) (1 point) Let X be the number of red trucks that pass through the tunnel between 7am and 10am over the next Monday. What is the distribution of X? (b) (2 points) Again let X be the number of red trucks that pass through the tunnel between 7am and 10am next Monday. How many red trucks would you expect to pass through the tunnel between 7am and 10am next Monday? (c) (5 points) What is the probability that 8 red trucks pass through the tunnel between 8am and Sam? State the appropriate distribution and any parameter values for any random variable(s) you use to model the situation. Write the probability statement and show your work in order to solve the problem. (d) (4 points) Suppose it takes a half hour for a red truck to pass through the tunnel. If there are no red trucks in the tunnel when it enters the tunnel at 7:35am on a Monday, what is the probability it will be the only red truck in the tunnel the whole time it spends in the tunnel? State the appropriate distribution and any parameter values for any random variable(s) you use to model the situation. Write the probability statment and show your work to receive full credit. (e) (5 points) Let W represent the amount of time in hours it takes for the g red truck to arrive at the tunnel on Monday morning. What time do you expect the red truck to arrive at the Tunnel on Mondny morning to the nearest 10 minutes)? Recall the tunnel opens at 7 am. Your final answer should be a time.

(a) X follows a Poisson distribution with parameter lambda = 273 = 81.

(b) We would expect 81 red trucks to pass through the tunnel between 7am and 10am next Monday.

(c) The number of red trucks passing through the tunnel between 8am and 10am follows a Poisson distribution with parameter lambda = 272 = 54.

The probability that 8 red trucks pass through the tunnel between 8am and 10am is P(X = 8) = 0.0634.

(d) The appropriate distribution is a geometric distribution with parameter [tex]p = e^{-1} = 0.3679.[/tex]

The probability that the truck will be the only one in the tunnel is P(X = 1) = 0.3679.

(e) The expected time of arrival for the first red truck can be modeled by an exponential distribution with parameter lambda = 27/3 = 9.

We expect the red truck to arrive at the tunnel around 7:06 am.

(a) Since the red trucks arrive independently at a constant rate, the number of red trucks passing through the tunnel between 7am and 10am follows a Poisson distribution with parameter λ = 27, denoted as X ~ Poisson(λ=27).

(b) The expected value of a Poisson distribution is equal to its parameter. Therefore, we would expect 27 red trucks to pass through the tunnel between 7am and 10am next Monday.

(c) Let Y be the number of red trucks that pass through the tunnel between 7am and 8am.

Since the red trucks arrive independently at a constant rate, Y follows a Poisson distribution with parameter λ = 27/3 = 9, denoted as Y ~ Poisson(λ=9).

We want to find the probability that 8 red trucks pass through the tunnel between 8am and 10am.

Let Z be the number of red trucks that pass through the tunnel between 8am and 10am.

Since Y and Z are independent Poisson random variables, the distribution of Z is also Poisson with parameter λ = 27-9 = 18, denoted as Z ~ Poisson(λ=18).

Therefore, we want to find P(Z=8), which can be calculated as:

P(Z=8) = (e^(-18) * 18^8) / 8!

= 0.0948 (rounded to four decimal places)

Therefore, the probability that 8 red trucks pass through the tunnel between 8am and 10am is 0.0948.

(d) Let T be the time in hours that the red truck spends in the tunnel. Since the time it takes for a red truck to pass through the tunnel is exponentially distributed with parameter λ = 2 (since it takes 0.5 hours to pass through the tunnel, the rate parameter is 1/0.5 = 2), we have T ~ Exp(λ=2).

We want to find the probability that the red truck is the only one in the tunnel for the entire time it spends in the tunnel, given that there are no other red trucks in the tunnel when it enters at 7:35am.

Let t be the time in hours from 7:35am that the red truck enters the tunnel.

Then, the probability that the red truck is the only one in the tunnel for the entire time it spends in the tunnel is:

[tex]P(T > 2 - t) = e^{-2(2-t)})[/tex]

[tex]= e^{-4+2t}[/tex]

[tex]= e^{(2t-4) }[/tex]

Therefore, we want to find P(T > 2 - t | T > t) using conditional probability:

P(T > 2 - t | T > t) = P(T > 2 - t) / P(T > t)

[tex]= e^{2t-4} / e^{(-2t)}[/tex]

[tex]= e^{(4t-4)}[/tex]

Since we know that the red truck entered the tunnel at 7:35am, we have t = 0.25.

Substituting this value, we get:

[tex]P(T > 1.75 | T > 0.25) = e^{(4(0.25)}-4)[/tex]

[tex]= e^{(-3)[/tex].

= 0.0498 (rounded to four decimal places)

Therefore, the probability that the red truck is the only one in the tunnel for the entire time it spends in the tunnel, given that there are no other red trucks in the tunnel when it enters at 7:35am, is 0.0498.

(e) Let W be the time in hours that it takes for the g-th red truck to arrive at the tunnel.

For similar question on probability.

https://brainly.com/question/30063166

#SPJ11

Correct answers of the following are:

(a) The distribution of X is 81.

(b) 81 trucks would pass through the tunnel between 7am and 10am next Monday.

(c)  Probability that 8 red trucks pass through the tunnel between 8am and 9am is 0.048.

(d)  Probability it will be the only red truck in the tunnel the whole time it spends in the tunnel is 0.0067.

(e) A red truck would arrive at the tunnel on Monday morning at around 7:06:40am.

In this problem, we are given information about the arrival of red trucks at a tunnel from 7am to 10am on Monday mornings. We are asked to find the distribution of the number of trucks that pass through the tunnel, the expected number of trucks, the probability that 8 trucks pass through the tunnel between 8am and 9am, the probability that a single truck entering at 7:35am will be the only truck in the tunnel, and the expected arrival time of a red truck on Monday morning.

(a) The distribution of X, the number of red trucks passing through the tunnel, is a Poisson distribution, since the arrivals are independent and occur at a constant rate. The parameter λ of the Poisson distribution is equal to the average number of red trucks that enter the tunnel per hour times the number of hours the tunnel is open. Therefore, λ = 27*3 = 81.

(b) The expected number of red trucks passing through the tunnel is equal to the parameter of the Poisson distribution, which is λ = 81.

(c) To find the probability that 8 red trucks pass through the tunnel between 8am and 9am, we can use a Poisson distribution with parameter λ = 27*1 = 27, since we are only considering the arrivals between 8am and 9am. The probability can be calculated as:

P(X=8) = (e^-27)*(27^8)/8!

= 0.048

(d) The distribution that models the number of red trucks in the tunnel at any given time is a Poisson distribution with parameter λ = 27/2, since the trucks arrive at a constant rate of 27 per hour and each truck takes half an hour to pass through the tunnel. The probability that a single truck entering the tunnel at 7:35am will be the only truck in the tunnel for its entire time in the tunnel can be calculated as:

P(X=0) = e^(-27/2)

= 0.0067

(e) To find the expected arrival time of a red truck on Monday morning, we can use an exponential distribution with parameter λ = 27/3 = 9, since the red trucks arrive at a constant rate of 27 per hour and we are interested in the time between arrivals. The expected arrival time can be calculated as:

E(W) = 1/λ

= 1/9 hours

= 6.67 minutes

Therefore, we would expect a red truck to arrive at the tunnel on Monday morning at around 7:06:40am (7:00am + 6.67 minutes = 7:06:40am).

To learn more about probability, click here: https://brainly.com/question/13604758

#SPJ11

I need help with everything but A. assignment is worth 150 points

in order to use an expert from a movie in your history class you must pay a royalty fee. the base fee is 50 and an additional .25 cents per minute of the film. the equation for this is?

Write a ratio in three different forms.
Shawn biked 5 miles and ran 2 miles. What is the ratio of the distance Shawn ran to the distance he biked?

Which of the following is not a true statement? *

a. The boundary line of y > x + 1 is a solid line.
b. The boundary line of y < 2x - 3 is not a solid line.
c. 3x + y < 2 is an example of linear inequality
d. (1, 1) is a solution of x + 3y > 2​

Help po plss

1.jane read 10books in one year,while huan rrad 8books.find the ratio jane's to juan's finished books .

2.nadine has 860 female followers out of a total of 1260followes in the social photosharing site find the ratio of female to male followers.

3.a public school has 4415 students and 1200 teachers.what is the ratio of the students to each other teacher.​

FIND THE AREA

WORK PLEASEEE and thank you

LOOK IF YOU DONT KNOW HOW TO PARAPHRASE JUST USE THIS WEBSITE AND TEACHERS NEVER NOTICE BOOK MARK IT IT WILL HELP YOU

A student needs 23 pound of modeling clay for a project. If each package has 18 pound of clay, how many packages of clay does the student need for his projec

PLZ help me i will give u many points

Im giving brainiest to the person who answers it first. Thank you!

Paula is going to hire a makeup artist for a fashion show and is comparing prices. Shelley charges $40 as a booking fee and an additional $30 per hour. Ezra charges $35 per hour, plus a booking fee of $30. Depending on the length of the show, the cost could end up being the same for either artist. What would the cost be?​

Joe determined the mass of 2 boxes using a scale. Since the cost of each box is the same, Joe would like to choose the box with the larger mass. Box A Box B 32.012 kg 32.02 kg Place the correct symbol between the 2 masses to help Joe decide. 32.012 Response area 32.02

If you have 100$ and then lose 25$ ,what is the percent change

#2 Find the value of x. Round to the nearest degree.

How much water do you add to make 75% sugar syrup using 99g sugar?

1. Bella and Caiden each bought a 12-oz can of soda. They are drinking their sodas at the

rates shown. Who is drinking their soda the fastest?

BELLA

CAIDEN

Min

Oz

0

12.

Ounces

20

18

16

14

12

10

8

6

4

2

2

3

4.5

4

2.

1 2 3 4 5 6 7 8 9 10

Minutes

A metallic cylinder of diameter 14 cm and height 49 cm is melted and cast into cubes with side 22 7 cm. How many cubes will be formed​

please help me if you can please

a block that is 2 inches wide,3 inches long, and 2 inches tall. what is the volume of a tower that is built by 3 blocks?​

A figure is shown
9 in.
3 in.
2 in.
3 in.
Which expression shows how to find the area of the figure?
A. 9x2 + 3x3
O C. 9+3+2+3
B. 9x 3 + 3 x 2
OD. 9 X5
What is the area of the figure? Enter the answer in the box.

What amount of tax will be owed on an income of 63,280 if city income tax rate is 1.5% of all income

Please help!! Given that triangle ABC is congruent to triangle DEF, the measure of angle A = 70°, the measure of angle B = 60°, the measure of angle C = 50°, the measure of angle D = (3x + 10)°, the measure of angle E = (y/3 + 20)°, and the measure of angle F = (z^2 + 14)°, find the values of x, y, and z.

fine the equation of the line that goes through the point (2,-9) that is parallel to the line with equation y=-3/4x

can u guys please help! tysmmmmmm

2a+3b=5
b=a-5 help please

Which of the following values are solutions to the inequality 6≥4x−5?

Graph and then find the perimeter of this shape. H(3, 0), I(3, 7), J(6, 7), K(6, 0)

The side lengths of a square are each 4p. By adding 3 to the length and subtracting 3 from the width, a rectangle is made. What is the area of the rectangle?

please help asap!!!!

Write the equation of the line that passes through the points (-4,8) and (-2, -1).
Put your answěr in fully reduced point-slope form, unless it is a vertical or horizontal
line.PLEASEEE HELPPP MEE I’ll give you BRAINLEST