Customers arrive at a shop according to a Poisson process at a mean rate of 2 customers every ten minutes. The shop opens at 9am. (a) Let X denote the waiting time (in hours, counted from the shop ope
Answers
(a) The pdf for X is \(f(x) = 12e^(-12x).\)
(b) The probability that the first customer arrives within the first hour is approximately 0.632.
(c) The expected value or mean waiting time for the first customer to arrive is 1/12 hours, approximately 5 minutes.
(a) Let X denote the waiting time (in hours, counted from the shop opening at 9am) for the first customer to arrive. We are given that customers arrive at a mean rate of 2 customers every ten minutes, which can be converted to a rate of 12 customers per hour.
Since the arrival rate follows a Poisson process, the probability density function (pdf) for X can be expressed as:
\(f(x) = λe^(-λx)\)
Where λ is the arrival rate and x is the waiting time.
In this case, λ = 12 customers per hour. Therefore, the pdf for X is:
\(f(x) = 12e^(-12x)\)
(b) To find the probability that the first customer arrives within the first hour (0 ≤ X ≤ 1), we need to calculate the integral of the pdf within this range:
\(P(0 ≤ X ≤ 1) = ∫[0,1] 12e^(-12x) dx\)
Integrating this expression gives us:
\(P(0 ≤ X ≤ 1) \\= [-e^(-12x)] from 0 to 1P(0 ≤ X ≤ 1) \\= -e^(-12) + 1\)
Therefore, the probability that the first customer arrives within the first hour is -e^(-12) + 1, which is approximately 0.632.
(c) To find the expected value or mean of X, we need to calculate the integral of xf(x) over the entire range of X:
\(E(X) = ∫[-∞,+∞] x * 12e^(-12x) dx\)
Integrating this expression gives us:
\(E(X) = [-xe^(-12x) + (1/12)e^(-12x)] from 0 to ∞\\E(X) = [0 - 0 + (1/12)] - [0 - 0 + (1/12)e^(-12∞)]\\E(X) = 1/12\)
Therefore, the expected value or mean waiting time for the first customer to arrive is 1/12 hours, which is approximately 5 minutes.
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Related Questions
uses
Katherine has 27 cups of yogurt to make smoothies. Each smoothie
1 cup of yogurt. What is the maximum number of smoothies that
Katherine can make with the yogurt?
Answers
Answer:
27 smoothies
Step-by-step explanation:
if you have 27 cups and one requires one cup, it would be 27 smoothies Katherine can make :)
A triangular prism is 12 meters long and has a triangular face with a base of 8 meters and a height of 9 meters. What is the volume of the triangular prism?
Answers
The volume of the triangular prism is 216 cubic meters.
To find the volume of a triangular prism, we need to multiply the area of the triangular base by the length of the prism.
Calculate the area of the triangular base.
The base of the triangular face is given as 8 meters, and the height is 9 meters. We can use the formula for the area of a triangle: area = (1/2) * base * height.
Plugging in the values, we get:
Area = (1/2) * 8 meters * 9 meters = 36 square meters.
Multiply the area of the base by the length of the prism.
The length of the prism is given as 12 meters.
Volume = area of base * length of prism = 36 square meters * 12 meters = 432 cubic meters.
Adjust for the shape of the prism.
Since the prism is a triangular prism, we need to divide the volume by 2.
Adjusted Volume = (1/2) * 432 cubic meters = 216 cubic meters.
Therefore, the volume of the triangular prism is 216 cubic meters.
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The selling price for a classic painting is $16,000. What was the painting's original price if it is currentlyselling for $500 more than two times its original price?AnswerKeypadKeyboard Shortcuts$
Answers
To answer this question, we will set and solve a linear equation.
Let x be the original price of the painting, with the given information we can set the following equation:
\(2x+500=16000.\)Subtracting 500 from the above equation we get:
\(2x=16000-500=15500.\)Dividing by 2 we get:
\(\begin{gathered} \frac{2x}{2}=\frac{15500}{2}, \\ x=7750. \end{gathered}\)Therefore, the original price of the painting was
\(7750\)dollars.
Answer: $7750.
The Vance family is saving money to buy a new car that costs $12,000. They plan to save $715 per month (m), and they have already saved $645. Which of the following inequalities show the number of months (m) the Vance family could save in order to buy the new car? Select all that apply. A. 715m≥11,355 B. 715m≤11,355 C. 12,000≤715m+645 D. 12,645≤715m
Answers
Answer:
C. 12,000≤715m+645
Step-by-step explanation:
You want to have either equal to or more than 12,000
Answecr:
C
Step-by-step explanation:
Solve.
3.5d + 8.25 = 3 + 5.25d
The solution is d = ____
Answers
Answer:
d=3
Step-by-step explanation:
.......................
Answer: 2
Step-by-step explanation:
3.5d + 8.25 = 3 + 5.25d
-1.75d = -5.25
d = 3
Which of these graphs represents a function?
W.A diagonal curve rises through (negative 5, negative 2), (negative 3, 0), (negative 2, 1), (0, 3), (1, 4), (2, 5) and (3, 6) on the x y coordinate plane.
X.A vertically linear curve descends through (5, 5), (5, 4), (5, 3), (5, 2), (5, 1), (5, 0), (5, negative 1), (5, negative 2), (5, negative 3), (5, negative 4) and (5, negative 5) on the x y coordinate plane.
YA curve declines from (negative 5, 6) through (4, 4), further declines through (0, 3), and turns and declines through (5, 2) on the x y coordinate plane.
Z.A curve declines diagonally from (3, 6) through (negative 2, 1) and further declines through (2, negative 3, (3, negative 4), and (5, negative 6) on the x y coordinate plane.
A. W
B. X
C. Y
D. Z
Answers
The only graph that represents functions is grap Y, hence choosing option C is the right response.
The graph that represents a function is graph C, Y. A function is a relation in which each input (x-value) is paired with exactly one output (y-value). In graph Y, each x-value corresponds to only one y-value, satisfying the definition of a function.
Graphs W, X, and Z do not represent functions because they fail the vertical line test, which states that no vertical line should intersect the graph at more than one point.
Graph W has multiple y-values for the x-value of 0, graph X has multiple y-values for the x-value of 5, and graph Z has multiple y-values for several x-values.
Therefore, the only graph that represents a function is graphs C, Y and the correct answer is option C.
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suppose you are the owner of a sari sari store and you have 8 pieces of different canned goods [ligo, 555,mega,Young`s town,saba ,blue bay,and century]]and you areonly allowed to diplay 7 canned goods on the shelflist down all the possible combinations?
PLSS!!!! HELP ME WITH THIS I NEED SOLUTION...
Answers
All you have to do to solve this is find the amount of different options (8) and the spaces allowed (7) and multiply them.
7 x 8 = 56
a sample of 90 adult randolph county residents showed that 59 own a home. what is the risk of owning a home?
Answers
The risk of owning a home in Randolph County based on the given sample can be calculated as:
Risk of owning a home = Number of residents who own a home / Total number of residents in the sample
Risk of owning a home = 59 / 90
Therefore, the risk of owning a home in Randolph County based on the given sample is approximately 0.656 or 65.6%.
is the weight of a hamburger a discrete random variable, continuous random variable, or not a random variable?
Answers
The weight of a hamburger can be considered a continuous random variable.
A random variable is a mathematical representation of a random phenomenon.
In probability and statistics, the weight of a hamburger can be used to illustrate the concepts of discrete and continuous random variables.
A discrete random variable is one that can only take on specific, distinct values.
For example, the number of hamburgers sold at a fast food restaurant in one day could be considered a discrete random variable, as the number of hamburgers sold can only be a whole number.
On the other hand, a continuous random variable is one that can take on an infinite number of values within a given range.
The weight of a hamburger could be considered a continuous random variable, as the weight of a hamburger can take on any value within a certain range, such as between 0.1 and 0.5 kilograms.
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consider the following random sample of income and education: income(y) education(x) 100 18 65 14 47 14 112 16 34 12 85 16 92 18 83 14 67 16 29 12 a. find sample means, variances, and standard deviations of x and y. b. find sample covariance, cov(y,x), and correlation corr(y,x). c. find cov(2y,3x), and correlation corr(2y,3x) d. find covariance of y and y and corr
Answers
The sample means are Mean(Y) = 61.4 and Mean(X) = 15, the sample variances are Variance(Y) ≈ 1124.89 and Variance(X) ≈ 2.67, the sample standard deviations are Standard deviation(Y) ≈ 33.56 and Standard deviation(X) ≈ 1.63, the sample covariance is Covariance(X, Y) ≈ -131.69, and the sample correlation is Correlation(X, Y) ≈ -0.77.
a. To find the sample means, variances, and standard deviation, we can use the following formulas:
Sample mean of income (Y):
Mean(Y) = (100 + 65 + 47 + 112 + 34 + 85 + 92 + 83 + 67 + 29) / 10 = 614 / 10 = 61.4
Sample mean of education (X):
Mean(X) = (18 + 14 + 14 + 16 + 12 + 16 + 18 + 14 + 16 + 12) / 10 = 150 / 10 = 15
Sample variance of income (Y):
Variance(Y) = [(100 - 61.4)^2 + (65 - 61.4)^2 + (47 - 61.4)^2 + (112 - 61.4)^2 + (34 - 61.4)^2 + (85 - 61.4)^2 + (92 - 61.4)^2 + (83 - 61.4)^2 + (67 - 61.4)^2 + (29 - 61.4)^2] / 9 ≈ 1124.89
Sample variance of education (X):
Variance(X) = [(18 - 15)^2 + (14 - 15)^2 + (14 - 15)^2 + (16 - 15)^2 + (12 - 15)^2 + (16 - 15)^2 + (18 - 15)^2 + (14 - 15)^2 + (16 - 15)^2 + (12 - 15)^2] / 9 ≈ 2.67
Sample standard deviation of income (Y):
Standard deviation(Y) = √Variance(Y) ≈ √1124.89 ≈ 33.56
Sample standard deviation of education (X):
Standard deviation(X) = √Variance(X) ≈ √2.67 ≈ 1.63
b. To find the sample covariance and correlation, we can use the following formulas:
Sample covariance:
Covariance(X, Y) = [(18 - 15)(100 - 61.4) + (14 - 15)(65 - 61.4) + (14 - 15)(47 - 61.4) + (16 - 15)(112 - 61.4) + (12 - 15)(34 - 61.4) + (16 - 15)(85 - 61.4) + (18 - 15)(92 - 61.4) + (14 - 15)(83 - 61.4) + (16 - 15)(67 - 61.4) + (12 - 15)(29 - 61.4)] / 9 ≈ -131.69
Sample correlation:
Correlation(X, Y) = Covariance(X, Y) / (Standard deviation(X) * Standard deviation(Y)) ≈ -131.69 / (1.63 * 33.56) ≈ -0.77
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Correct question-
Consider the following random sample of income and education: Income(Y) Education(X) 100 18 65 14 47 14 112 16 34 12 85 16 92 18 83 14 67 16 29 12 a. Find sample means, variances and standard deviation. b. Find sample covariance and correlation.
g provide a table and an appropriate graph for each question, except indicated otherwise. what percentage of people survived? did women had a better survival rate than men? (suggestion stacked bar plot) how does class affect survival rate? (suggestion: a stacked bar plot, facet wrapped by class)
Answers
The total number of third-class people was 709, and 402 of them survived.
In order to create a table and an appropriate graph for each question,
we need data first.
Assuming the data is about the Titanic, let's proceed with the questions.
Percentage of people survived :
To answer this question, we need to have the total number of people who were aboard the Titanic, and how many of them survived.
Let's say the total number of people was 2224, and 722 of them survived.
Using this data, we can create a table as follows:
| | Survived | Not Survived | Total ||-------------------|-------------------|----------------------|-----------|| Number of People | 722 | 1502 | 2224 || Percentage | 32.46% | 67.54% | ||-------------------|-------------------|----------------------|-----------|
To create an appropriate graph, we can use a pie chart as follows:
Did women have a better survival rate than men:
To answer this question, we need to have the total number of men and women who were aboard the Titanic, and how many of them survived.
Let's say the total number of women was 887, and 342 of them survived.
Also, the total number of men was 1337, and 380 of them survived.
Using this data, we can create a table as follows:
| | Survived | Not Survived | Total ||------------------|-------------------|---------------------|-----------|| Women | 342 | 545 | 887 || Men | 380 | 957 | 1337 ||------------------|-------------------|---------------------|-----------|
To create an appropriate graph, we can use a stacked bar plot as follows:
How does class affect the survival rate.
To answer this question, we need to have the total number of people who were aboard the Titanic, how many of them survived, and their class (first, second, or third).
Let's say the total number of first-class people was 324, and 202 of them survived.
Also, the total number of second-class people was 277, and 118 of them survived.
Finally, the total number of third-class people was 709, and 402 of them survived.
Using this data, we can create a table as follows:| | Survived | Not Survived | Total ||-------------------|-------------------|---------------------|-----------|| First-Class | 202 | 122 | 324 || Second-Class | 118 | 159 | 277 || Third-Class | 402 | 307 | 709 ||-------------------|-------------------|---------------------|-----------|
To create an appropriate graph, we can use a stacked bar plot, facet wrapped by class, as follows:
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Is this relation a function? Justify your answer.
A. Yes because every x- and y-value is positive.
B. No because two points with the same y-value have different x-values.
C. No because two points with the same x-value have different y-values.
D. Yes because the number of x-values is the same as the number of y-values
Answers
Answer:
C. No because two points with the same x-value have different y-values.
Step-by-step explanation:
(view my screenshot attachment too please)
A function can't have same x-values that appear for multiple different y-values.
In the diagram, (10, 6) and (10, 9) indicate that it isn't a function.
This is because the same 10 appears twice (x-value) but with multiple different y-values (6 and 9).
SOMEONE HELP ME PLEASE
Answers
Answer:
m=-2/3
Step-by-step explanation:
two points from graph:(-1,1) and (-4,3)
m=y2-y1/x2-x1
m=3-1/-4-(-1)
m=2/-3
m=-2/3
if you randomly select a card from a well-shuffled standard deck of 52 cards, determine the probability that the card you select is a club write your answer as a decimal, rounded to the nearest thousandth
Answers
The probability of selecting a club card from a well-shuffled standard deck of 52 cards is 0.25 or 25%.
To determine the probability of selecting a club card from a standard deck of 52 cards, we need to consider the number of favorable outcomes (club cards) and the total number of possible outcomes (all cards in the deck).
In a standard deck, there are 13 club cards (2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace) out of the total 52 cards. Therefore, the number of favorable outcomes is 13.
The total number of possible outcomes is simply the total number of cards in the deck, which is 52.
To calculate the probability, we divide the number of favorable outcomes by the total number of possible outcomes:
Probability of selecting a club card = Number of favorable outcomes / Total number of possible outcomes
= 13 / 52
= 0.25
Therefore, the probability of selecting a club card from a well-shuffled standard deck is 0.25, or 25% (rounded to the nearest thousandth).
This means that, on average, there is a 25% chance of randomly selecting a club card from the deck.
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Please help
Charles correctly answered 23 questions on a test receiving a grade of 92% how many questions where on the test
Answers
Answer:
25
Step-by-step explanation:
Cross multiply
23/x=92/100
x=25
There were approximately 25 questions on the test.
To determine the number of questions on the test, we can set up a proportion based on Charles' correct answers and the grade received.
Let x be the total number of questions on the test.
We can set up the proportion:
23 (Charles' correct answers) / x (total number of questions) = 92% (grade as a decimal: 0.92)
This can be written as:
23 / x = 0.92
To solve for x, we can multiply both sides of the equation by x:
23 = 0.92 * x
Dividing both sides by 0.92:
23 / 0.92 = x
x ≈ 25
Therefore, there were approximately 25 questions on the test.
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Determine if the event shows uniform probability or not. You have a spinner with 8 equal spaces on it. Each space is numbered with a different integer between 1 and 8.
Answers
Answer:
it does show uniform probability.
Step-by-step explanation:
because the spaces are all equal, so there is an equal chance of every integer being picked.
On a coordinate plane, parallelogram H I J K is shown. Point H is at (negative 2, 2), point I is at (4, 3), point J is at (4, negative 2), and point K is at (negative 2, negative 3).
HIJK is a parallelogram because the midpoint of both diagonals is __________, which means the diagonals bisect each other.
(1,−1)
(1,1)
(1,0)
(0,1)
Answers
============================================================
Explanation:
Point H is at (-2,2) and J is at (4,-2)
Focusing on the x coordinates, the midpoint is (x1+x2)/2 = (-2+4)/2 = 1.
The y coordinates have a midpoint of (y1+y2)/2 = (2+(-2))/2 = 0.
The midpoint is at (1,0). That's why the answer is choice C.
--------------------------
This next section is optional.
Points I and K are located at (4,3) and (-2,-3) respectively
Averaging the x coordinates gets us (x1+x2)/2 = (4+(-2))/2 = 1
Doing the same for the y coordinates gets us (y1+y2)/2 = (3+(-3))/2 = 0
We end up with the same midpoint (1,0)
The midpoint being the same for both diagonals proves that each diagonal is bisected, ie cut in half.
Answer:
C is the correct answer
Step-by-step explanation:
3. Write the equation of a line that is perpendicular to y =V3 x + 3 and goes through the point (-1, -2).
Answers
Step-by-step explanation:
is that root 3 x..........?
Given: x + 2 <-5.
Choose the solution set.
OXIXER,<-7)
OXIXERX <-3)
OxIXER X<3)
OXIXER X<7)
Answers
Answer:
The solution set is {x : x ∈ R, x < -7} ⇒ A
Step-by-step explanation:
The solution set of the inequality x > a is {x : x ∈ R, x > a}The solution set of the inequality x < a is {x : x ∈ R, x < a}The solution set of the inequality x ≥ a is {x : x ∈ R, x ≥ a}The solution set of the inequality x ≤ a is {x : x ∈ R, x ≤ a}Let us solve the question
∵ x + 2 < -5
→ Subtract 2 from both sides of the inequality
∴ x + 2 - 2 < -5 - 2
∴ x + 0 < -7
∴ x < -7
→ By using the second rule above
∴ The solution set is {x : x ∈ R, x < -7}
What other information is needed to prove that FGE Ijh by the SAS?
Answers
To prove that triangle FGE and triangle IJH we need information like the two sides of each triangle and the included angle to be congruent.
To prove two triangles are similar by the SAS is that you need to show that two sides of one triangle are proportional to two corresponding sides of another triangles, with the included corresponding angles being congruent.
For the SAS postulate you need two sides and the included angle in both triangles.
Side-Angle-Side (SAS) postulate:-
If two sides and the included angles of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. SAS postulate relate two triangles and says that two triangles are congruent if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle.
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I need help with algebra
Answers
Answer:
Hello, my name is 1frogaroni feel free to ask any questions about algebra such as equations you don't understand. I will try my best to help you out with your question :)
Step-by-step explanation:
Oh and welcome to brainly where the community will be glad to help you!
The number of calories consumed from an apple is directly proportional to the number of apples eaten. If 4 apples contain 380 total calories, how many apples would you have to eat to reach a total of 1,045 calories?
Answers
Answer:
Step-by-step explanation:
The direct proportion equation is y = kx which reads, "y is directly proportional to x". k is the constant of proportionality. In our case, we read this is as, "the number of calories is directly proportional to the number of apples eaten." Our equation would look like c = ka. We first are given that a = 4 and = 380. We will use that to solve for k:
380 = k4 so
k = 95. Now that know the value of k, we will use that along with the number of calories (1045) to find the number of apples, a:
1045 = 95a so
a = 11
This means that it will be 11 apples that contain 1045 calories.
To convert 5 to a fraction, what would the denominator be? ОА. 10 Ов. 5 Ос. О Od.1
Answers
If you move from zero to 15 on the number line, you are representing all of the following except
the absolute value of 15
the distance between zero and 15
the opposite of -15
the opposite of 15
Answers
Answer: the opposite of 15
Step-by-step explanation:
a dealer sold a total of 60 of the three appliances listed in the table above for a total of $4,200. if the dealer sold twice as many food processors as mixers, what was the ratio of the number of blenders sold to the number of food processors sold?
Answers
On solving the provided question, we got to know that, Blenders sold as a percentage of food processors sold is 30/20, or 3/2.
What is equation?Since equations are essentially questions, efforts to systematically find answers to those questions have been the driving force behind the development of mathematics. From straightforward algebraic equations that just require addition or multiplication to differential equations, exponential equations that use exponential expressions, and integral equations, there are many different types of equations that range in complexity.
Compared to mixers, the dealer sold twice as many food processors.
Let M be the quantity of mixers sold.
Therefore, 2M is the quantity of food processors sold.
A total of 60 of the three appliances were sold by the dealer.
M + 2M = 3M
Thus, 3M is a representation of the whole sales of mixers and food processors.
60 appliances were purchased in total, thus 60 - 3M equals the quantity of blenders purchased.
Everything was sold by the merchant for a total of $4,200.
We get: (30)(M) + (120)(2M) + (50)(60 - 3M) = 4,200
30M + 240M + 3000 - 150M = 4200
130M + 3000 = 4200
120M = 1200
M = 10
The retailer so sold 10 mixers.
This indicates that the vendor sold 20 food processors and 30 food processors.
Blenders sold as a percentage of food processors sold is 30/20, or 3/2.
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2 intersecting lines are shown. A line with points T, R, W intersects a line with points S, R, V at point R. Angle T R V is 95 degrees. mAngle SRW = °
Answers
Answer:
m<SRW = 95degrees
Step-by-step explanation:
If a line with points T, R, W intersects a line with points S, R, V at point R with the point of intersection at R, this means that;
m<TRV = m<SRW (vertically opposite angles)
Since m<TRV = 95 degrees, hence <SRW = 95degrees
What is the value of x in trapezoid ABCD ?
Answers
The unknown value of the variable is 15
Determining the angles of a trapezium.The given diagram is a trapezium. For a trapezium, the sum of the opposite angles is equivalent to 180 degrees, hence;
3x + 9x = 180
Simplify the resulting expression to have:
12x = 180
Divide both sides by 12
12x/12 = 180/12
x = 180/12
x = 15
Hence the value of x from the expression is 15.
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Can someone help me with this question please? Many thanks in advance.
Jack, Alex and Amir jumped a total of 12.69 m in a long
jump competition.
Alex jumped exactly 200 cm further than Jack.
Amir jumped exactly 2,000 mm further than Alex.
What distance did they all jump?
Give your answers in metres.
Answers
Answer:
Step-by-step explanation:
The distances they jumped were
x,y,z
y=x+2
z=x+2+2=x+4
x+y+z=12.69
x+x+2+x+4=12.69
3x+6=12.69
3x=6.69
x=2.23
So Jack jumped 2.23m, Alex jumped 4.23m, and Amir jumped 6.23m.
One leg of a right triangle is 7 inches longer than the other leg, and the hypotenuse is 35 inches. Find the lengths of the legs of the triangle.
Answers
Answer: 21, 28
Step-by-step explanation:
Side #1 = xSide #2 = x + 7Hypotenuse = 35Use the Pythagorean Theorem \(a^{2}+b^{2}=c^{2}\):
a = xb = x + 7c = 35Substitute in the values & solve:
\(x^{2}+(x+7)^{2}=35^{2}\\x^{2}+x^{2}+14x+49=1225\\2x^{2}+14x+49-1225=0\\2x^{2}+14x-1176=0\\2(x^{2}+7x-588)=0\\2(x + 28)(x - 21)=0\\x_{1}=-28, x_{2}=21\)
-28 is not a possible solution since you can't have negative inches...
a = x = 21b = x + 7 = 21 + 7 = 28c = 35k^4/(16k^6 5) use the comparison test or limit comparison test to determine whether the following series converges.
Answers
To use the comparison test or limit comparison test, we need to find a known series that either diverges or converges and is similar to the given series.
Using the comparison test, we can compare the given series to the series 1/n^2, which is a p-series with p=2 and converges.
To see if our series is smaller or larger than 1/n^2, we can simplify the given series by canceling out k^4 from the numerator and denominator:
k^4 / (16k^6 5) = 1 / (16k^2 5)
Now, we can compare 1 / (16k^2 5) to 1/n^2:
1 / (16k^2 5) < 1/n^2 for all k > 1
Therefore, since our series is smaller than a convergent series, it must also converge.
Alternatively, we can use the limit comparison test by finding the limit of the ratio of the given series to 1/n^2 as n approaches infinity:
lim (n→∞) [k^4/(16k^6 5) / (1/n^2)] = lim (n→∞) (n^2 / (16k^6 5))
This limit is zero for all k > 1, which means the given series and the series 1/n^2 have the same behavior and both converge.
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