18. Let Z(t) = X(t) – aX(t – s) where X(t) is the Wiener process. (a) Find the pdf of y(t). (b) Find mean and autocovariance functions.
Answers
a. P[Y(t) < y] = P[log Y(t) < log y] = Φ[(log y - log(0)) / √((1 - a²)t)] is the probability density function of Y(t).
b. Mean = - a exp(-λt) E[X(t - s)]
The autocovariance function is Cov(Y(t), Y(t + h)) = exp(-λt) exp(-λh) G(h) - a exp(-λt) exp(-λ(t + h)) G(h - s)
Z(t) = X(t) - aX(t - s), where X(t) is the Wiener process.
(a) The probability density function of Y(t) can be derived as follows:
Y(t) = exp(-λt) Z(t) ⇒ Z(t) = Y(t) exp(λt)
P[Z(t) < z] = P[Y(t) exp(λt) < z] = P[Y(t) < z exp(-λt)]
From the given, we have Z(t) = X(t) - aX(t - s) ⇒ Z(t) has a normal distribution Z(t) ~ N(0, (1 - a²)t)
Y(t) = exp(-λt) Z(t) ⇒ Y(t) has a lognormal distribution Y(t) ~ log N(0, (1 - a²)t)
The probability density function of Y(t) is given by:
P[Y(t) < y] = P[log Y(t) < log y] = Φ[(log y - log(0)) / √((1 - a²)t)], where Φ is the cumulative distribution function of the standard normal distribution.
(b) Mean and autocovariance functions can be obtained as follows:
Mean = E[Y(t)] = E[exp(-λt) Z(t)] = E[exp(-λt) [X(t) - aX(t - s)]]
= exp(-λt) E[X(t)] - a exp(-λt) E[X(t - s)]
From the properties of the Wiener process, E[X(t)] = 0 for all t.
Therefore, Mean = - a exp(-λt) E[X(t - s)]
The autocovariance function is given by:
Cov(Y(t), Y(t + h)) = E[Y(t)Y(t + h)] - E[Y(t)]E[Y(t + h)]
= E[exp(-λt) Z(t) exp(-λ(t + h)) Z(t + h)] - exp(-λt) exp(-λ(t + h)) E[Z(t)] E[Z(t + h)]
= exp(-λt) exp(-λh) E[X(t) X(t + h)] - a exp(-λt) exp(-λ(t + h)) E[X(t) X(t + h - s)]
Let G(h) = E[X(t) X(t + h)] and G(h - s) = E[X(t) X(t + h - s)]
Then, Cov(Y(t), Y(t + h)) = exp(-λt) exp(-λh) G(h) - a exp(-λt) exp(-λ(t + h)) G(h - s)
To learn more about density, refer below:
https://brainly.com/question/29775886
#SPJ11
Related Questions
Simplify 3(x+2)-6+x please!
Answers
A line has equation y = x - 5. The line is shifted 3 units to the left and 1 unit up. Which
points are on the graph of the line after these 2 transformations are applied? Select all the
points that apply.
A. (-3, - 4)
B. (-1, -3)
O C. (0, -7)
D.
(1, - 6)
E. (2, 1)
F. (5,0)
OG. (6,5)
Answers
The points that lie on the graph of the line after the Transformations are:
C. (0, -7) and D. (1, -6).Therefore, the correct answer is C and D.
To shift the line 3 units to the left and 1 unit up, we need to adjust the x and y coordinates accordingly.
The original line equation is y = x - 5.
1. Shifting 3 units to the left: To shift the line 3 units to the left, we need to subtract 3 from the x-coordinate. The updated equation becomes y = (x - 3) - 5 = x - 8.
2. Shifting 1 unit up: To shift the line 1 unit up, we need to add 1 to the y-coordinate. The updated equation becomes y = x - 8 + 1 = x - 7.
Now, let's check which points lie on the graph of the line after these transformations:
A. (-3, -4):
When we substitute x = -3 into the updated equation, we get y = -3 - 7 = -10, so this point is not on the graph.
B. (1, -3):
When we substitute x = -1 into the updated equation, we get y = -1 - 7 = -8, so this point is not on the graph.
C. (0, -7):
When we substitute x = 0 into the updated equation, we get y = 0 - 7 = -7, so this point is on the graph.
D. (1, -6):
When we substitute x = 1 into the updated equation, we get y = 1 - 7 = -6, so this point is on the graph.
E. (2, 1):
When we substitute x = 2 into the updated equation, we get y = 2 - 7 = -5, so this point is not on the graph.
F. (5, 0):
When we substitute x = 5 into the updated equation, we get y = 5 - 7 = -2, so this point is not on the graph.
G. (6, 5):
When we substitute x = 6 into the updated equation, we get y = 6 - 7 = -1, so this point is not on the graph.
Based on these calculations, the points that lie on the graph of the line after the transformations are:
C. (0, -7) and D. (1, -6).
Therefore, the correct answer is C and D.
For more questions on Transformations .
https://brainly.com/question/29788009
#SPJ8
HELP PLEASE IM ALREADY FAILING MATH AND I DONT UNDERSTAND HOW TO DO THIS!! STEPS HAVE TO BE SHOWN OR I GET A 0!!!
Answers
Answer:
I put down my steps in the comment section. If you still don't understand just comment to me.
Step-by-step explanation:
Answer: I answered 6-11
Step-by-step explanation:
can someone help me find the annulus of the shaded area? much appreciated
Answers
Answer:
34.83
Step-by-step explanation:
A 90% confidence interval is constructed based on a sample of data, and it is 74% +3%. A 99% confidence interval based on this same sample of data would have: A. A larger margin of error and probably a different center. B. A smaller margin of error and probably a different center. C. The same center and a larger margin of error. D. The same center and a smaller margin of error. E. The same center, but the margin of error changes randomly.
Answers
As a result, for the same data set, a 99% confidence interval would have a greater margin of error than a 90% confidence interval.
Answer: If a 90% confidence interval is constructed based on a sample of data, and it is 74% + 3%, a 99% confidence interval based on this same sample of data would have a larger margin of error and probably a different center.
What is a confidence interval? A confidence interval is a statistical technique used to establish the range within which an unknown parameter, such as a population mean or proportion, is likely to be located. The interval between the upper and lower limits is called the confidence interval. It is referred to as a confidence level or a margin of error.
The confidence level is used to describe the likelihood or probability that the true value of the population parameter falls within the given interval. The interval's width is determined by the level of confidence chosen and the sample size's variability. The confidence interval can be calculated using the standard error of the mean (SEM) formula
.A 90% confidence interval indicates that there is a 90% chance that the interval includes the population parameter, while a 99% confidence interval indicates that there is a 99% chance that the interval includes the population parameter.
When the level of confidence rises, the margin of error widens. The center, which is the sample mean or proportion, will remain constant unless there is a change in the data set. Therefore, alternative A is the correct answer.
To know more about margin visit:
https://brainly.com/question/15357689
#SPJ11
_____ refers to deliberate efforts to provide full and equal opportunities in employment, education, and other areas for members of traditionally disadvantaged groups and applies only to organizations that receive federal funding.
Answers
Affirmative-Action refers to "deliberate-efforts" to provide full and equal opportunities in education, employment and other areas for members of disadvantaged groups.
The "Affirmative-Action" has the policies and practices that aim to eliminate discrimination and promote diversity by giving preferential treatment to individuals from underrepresented or historically disadvantaged groups.
These policies involve steps such as setting specific recruitment goals, implementing targeted outreach programs, and considering race, ethnicity, gender, or other factors in the selection process.
It is important to note that affirmative action policies are associated with organizations that receive federal funding, as they are subject to certain legal requirements and regulations.
Learn more about Affirmative Action here
https://brainly.com/question/1163395
#SPJ4
if 600 cubic centimeters of water flow into the container in one minute, about how many minutes will it take to fill the container?
Answers
a. The volume of the spherical container is approximately 37,785.41 cubic centimeters.
b. It will take approximately 62.98 minutes to fill the container with a flow rate of 600 cubic centimeters per minute.
a. To find the volume of the spherical container, we can use the formula:
V = (4/3)πr^3
where V is the volume and r is the radius of the sphere. However, we need to first find the radius of the sphere using the given surface area.
The surface area of a sphere is given by the formula:
A = 4πr^2
Given that the surface area is approximately 5,538.96 square centimeters, we can equate the formula for surface area to the given value and solve for r:
4πr^2 = 5,538.96
r^2 = 5,538.96 / (4π)
r^2 ≈ 441.57
Taking the square root of both sides, we get:
r ≈ √441.57 ≈ 21.02
Now that we have the radius, we can substitute it into the formula for volume to find the volume of the container:
V = (4/3)π(21.02)^3 ≈ 37,785.41 cubic centimeters
Therefore, the volume of the container is approximately 37,785.41 cubic centimeters.
b. If 600 cubic centimeters of water flow into the container in one minute, we can calculate the time it takes to fill the container by dividing the volume of the container by the flow rate:
Time = Volume / Flow rate
Time ≈ 37,785.41 / 600 ≈ 62.98 minutes
Therefore, it will take approximately 62.98 minutes to fill the container.
The correct question should be :
A spherical container has a surface area of about 5,538.96 square centimeters.
a. What is the volume of the container?
b. If 600 cubic centimeters of water flow into the container in one minute, about how many minutes will it take to fill the container?
To learn more about volume visit : https://brainly.com/question/463363
#SPJ11
Describe and correct the error in setting up the trigonometric function.
Answers
The value of side length w is 13.75 .
Given right angled triangle,
Perpendicular = w
Hypotenuse = 17
Angle of triangle = 54°
So,
According to the trigonometric ratios,
tanФ = p/b
cosФ = b/h
sinФ = p/h
By using sinФ,
sinФ = p/h
sin 54° = w/ 17
0.809 = w/17
w = 13.75 .
Thus after correction w will be 13.75
Know more about trigonometry,
https://brainly.com/question/25618616
#SPJ1
the time (in hours) required to repair a machine follows an exponential distribution, and the mean repair time is 2.5 hours. (a) what is the probability that a repair will take less than 2 hours?
Answers
0. 63212 is the probability that a repair will take less than 2 hours.
What is the simple definition of probability?
A probability is a number that expresses the possibility or likelihood that a specific event will take place.
In addition to being expressed as percentages ranging from 0% to 100%, probabilities can also be expressed as proportions between 0 and 1.
P( X > 2 ) = 1 - P ( X/ leq2 )
= 1 - F( 2)
= 1 - \(e^{0.5 * 2}\)
= 0. 63212.
Learn more about probability
brainly.com/question/11234923
#SPJ4
1.5(3x + 8) = 4.5 x + 12
Answers
Answer:
Interval Notation: (-∞, ∞)
Any value of x makes the equation always true.
There is no other answer.
Which expressions are equivalent to -
4 (16-8 +12x)?
Select all that apply.
A. 2x
B. 4 + 2 - 3x
C. - 4+2 - 3x
D. - 3x - 2
DE 3x + 2
OF 3x - 2
Answers
Step-by-step explanation:
I hope this helps..........
Students from classes 4-A, 4-B and 4-C- will take part in a group dance. So we need to select an equal number of girls from each class, to have 21 girls in the dance. In the same way we need to select an equal number of boys from each class , to have 15 boys in the dance. How many students do we need to elect from each class
Answers
Answer:
Step-by-step explanation:
we need from each class 7 girls and 5 boys
to get this, we have three classes and we need 21 girls so we will divide 21 by the three classes which is
21/3=7
and for the boys we will divide the 15 boys by the three classes which is
15/3=5
How do you find the equation, domain, and range using a graph?
Answers
Answer/Step-by-step explanation:
For the one on the left, the domain is (-∞,-1)∪[0,0]∪(1,3]
Remember that the domain is what values x are. If the dot is solid, use a bracket, if it is empty, use parentheses.
The range is [-3,-3]∪[-1,-1]∪(2,4]
The equation is a piece-wise function.
When x>-1 f(x)=-3
When x=0 f(x)=-1
When 1<x≤2 f(x)=2x
this is my last question please help
Answers
Answer:
N
Step-by-step explanation:
I hope this helps!!
Leon has already had 4 science quizzes this semester. Going forward, he expects to have 4 quizzes per month.
Write an equation that shows how the total number of science quizzes, y, depends on the number of months that have passed, x.
Answers
The equation that shows how the total number of science quizzes, y, depends on the number of months that have passed, x is y = 4x.
An equation is a mathematical statement that shows the relationship between two or more values.
In this case, Leon has already had 4 science quizzes this semester and expects to have 4 quizzes per month going forward.
We can use an equation to show how the total number of science quizzes, y, depends on the number of months that have passed, x.
The equation for this would be y = 4x, where x is the number of months that have passed and y is the total number of science quizzes.
This equation shows that for each month that passes, the total number of science quizzes increases by 4
To know more about equation here.
https://brainly.com/question/10413253
#SPJ4
Set up for proportion its not 9 or 6 or 2
Answers
The value of x in the similar triangle is 6 units.
How to find the side of similar triangle?Similar triangles are the triangles that have corresponding sides in proportion to each other and corresponding angles equal to each other.
Using the proportional relationship of the similar triangle,
4 / x = x /9
cross multiply
x² = 36
square root both sides
√x² = √36
x = 6
Therefore,
x = 6 units
learn more on similar triangle here: https://brainly.com/question/22137601
#SPJ1
A hot air balloon is ascending from the lift off point (C) at a 60 degree angle. When it is 200 meters from the lift off point, at what Vertical height has the balloon reached? Round your answer to the nearest Whole Number.
Answers
A hot air balloon is ascending from the lift off point (C) at a 60 degree angle. When it is 200 meters from the lift off point, the vertical height it has reached is about 173 meters.
Now, let's have a closer look at how it was derived.
The first step to solving this problem is to understand the given information and interpret it visually. We can represent the balloon's lift-off point as point C, and we know that it's ascending at a 60 degree angle. This means that we can construct a right triangle ABC, where the hypotenuse (AB) is the path of the balloon, angle B is 60 degrees, and angle A is 30 degrees.
Side AC represents the vertical height we're trying to find, while side BC represents the horizontal distance the balloon has traveled. The problem tells us that the balloon has traveled a distance of 200 meters, so we can represent this with BC=200 meters.
Using trigonometric functions, we can now find the value of AC, which represents the height of the balloon.
We know that tan(60 degrees) = AC/BC, which simplifies to sqrt(3) = AC/200.
Multiplying both sides by 200, we get AC = 200*sqrt(3) meters.
Rounding this to the nearest whole number gives us 173 meters, which is the final answer.
To know more about vertical height visit:
https://brainly.com/question/30394460
#SPJ11
For a fundraising project, your math club is publishing a calendar. The cost of the digital images and the permission to use them is $125. In addition to these “one time” charges, the unit cost of printing each calendar is $3.25.
Part a: Write an equation for the total cost of the calendars.
Part b: Write a model that gives the average cost A per calendar as a function of the number of calendars printed.
Part c: Use the graph of the model shown below to estimate the number of calendars that must be sold before the average cost drops to $5 per calendar.
Answers
The require answers are,
Part a: The equation for the total cost of the calendars is C = 125 + 3.25*100 = 350
Part b: The model that gives the average cost A per calendar as a function of the number of calendars printed is A = 125 + 3.25x
Part c: The estimate number of calendars that must be sold before the average cost drops to $5 per calendar is 70.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions by connecting them with the equal sign = .
Now it is given that,
Cost of the digital images and the permission = $125
“one time” charges = $3.25
Part a: The equation for the total cost of the calendars
THere are 100 calendars printed.
So, “one time” charge = 3.25*100
So, the total cost of the calendars C is given as,
C = 125 + 3.25*100
C = 125 + 325 = 450
Thus, the equation for the total cost of the calendars is given as,
C = 125 + 3.25*100 = 450
Part b: The model that gives the average cost A per calendar as a function of the number of calendars printed
Let there are x calendars printed
Total “one time” charges = 3.25x
So, Average cost is given as,
A = 125 + 3.25x
this is the model that gives the average cost A per calendar
Part c: The estimate number of calendars that must be sold before the average cost drops to $5 per calendar
Now from the graph it must be seen that the average cost drops to $ 5 per calender at 70.
Thus, number of calendars that must be sold before the average cost drops to $5 per calendar = 70
Thus, the require answers are,
Part a: The equation for the total cost of the calendars is C = 125 + 3.25*100 = 350
Part b: The model that gives the average cost A per calendar as a function of the number of calendars printed is A = 125 + 3.25x
Part c: The estimate number of calendars that must be sold before the average cost drops to $5 per calendar is 70.
To learn more about equations :
https://brainly.com/question/13482978
#SPJ1
pls help asap if you can!!!!
Answers
The statement that proves that angle XWY is equal to angle ZYW is
A. If two parallels are cut by a transverse, then alternate interior angles are congruent
What are alternate interior anglesAlternate interior angles are a pair of angles that are formed on opposite sides of a transversal line when two parallel lines are intersected by the transversal.
When a transversal intersects two parallel lines, it creates eight angles. Among these angles, the alternate interior angles are located on the inside of the parallel lines and on opposite sides of the transversal.
In a parallelogram, the two opposite sides are parallel to each other hence the line crossing them will lead to formation of alternate interior angles
Learn more about alternate interior angles at
https://brainly.com/question/20344743
#SPJ1
Marina can bicycle 19. 5 miles in the same time it takes her to run 6 miles. She bikes 9 miles per hour faster than she runs. At what speed does Marina run? Distance (mi) Rate (mph) Time (hr) Bicycling 19. 5 r 9 StartFraction 19. 5 Over r 9 EndFraction Running 6 r StartFraction 6 Over r EndFraction 4 mph 5 mph 9 mph 13 mph.
Answers
Marina runs at a speed of 4 mph.
Let's denote the speed at which Marina runs as "r" mph.
Given that Marina can bike 19.5 miles in the same time it takes her to run 6 miles, we can set up the equation:
19.5 / (r + 9) = 6 / r
To solve this equation, we can cross-multiply:
19.5r = 6(r + 9)
19.5r = 6r + 54
Subtracting 6r from both sides:
19.5r - 6r = 54
13.5r = 54
Dividing both sides by 13.5:
r = 54 / 13.5
r = 4
To learn more about equation visit;
https://brainly.com/question/10413253
#SPJ11
Help
2х+6y=3
3х - 7у=6
Answers
Solve for X. Assume that lines which appear tangent are tangent
Answers
Answer:
x = 16 and x = 15
Step-by-step explanation:
Given 2 chords intersecting then the product of the parts of one chord is equal to the product of the parts of the other chord.
(3)
7x = 14 × 8 = 112 ( divide both sides by 7 )
x = 16
(1)
8x = 12 × 10 = 120 ( divide both sides by 8 )
x = 15
Number 4. HURRY PLEASE
Answers
Answer:
its b
Step-by-step explanation:
becasue i had the same typ eof questions 2 days ago glad to help pls mark it 5 starts its 100% corect i am an A+ student
A recently televised broadcast of a popular television show had a 15 share, meaning that among 5000 monitored households with TV sets in use, 15% of them were tuned to the show. A 0.01 significance level is used to test an advertiser’s claim that among the households with TV sets in use, less than 20% were tuned in to the show. Find the P-value.
1.9998
0.9999
0.0001
0.0002
Answers
The p-value of the given hypothesis is; 0.9999
How to find the p-value of the statistics?The formula for the z-score of proportions is;
z = (p^ - p)/√(p(1 - p)/n)
where;
p^ is sample proportion
p is population proportion
n is sample size
We are given;
p^ = 15% = 0.15
p = 20% = 0.2
n = 5000
Thus;
z = (0.15 - 0.2)/√(0.2(1 - 0.2)/5000)
z = -8.8388
From p-value from z-score calculator, we have;
P(Z < -8.8388) = 1 - 0.0001 = 0.9999
Read more about p-value at; https://brainly.com/question/4621112
#SPJ1
The way an individual perceives stimuli and the general manner in which he or she responds to it is a __________-_______ style
Answers
Answer:
decision-making
Step-by-step explanation:
8pie = 1/3 pie r squared x 6
Answers
Solution of the given equation is r = 2
What is equation?Equations are mathematical statements that shows equality, containing two algebraic expressions on both sides of an 'equal to (=)' sign.
Given equation,
8π = 1/3πr²×6
⇒ 2r² = 8
⇒ r² = 8/2
⇒ r = √4
⇒ r = 2
Hence, r = 2 is solution for the given equation.
Learn more about equation here:
https://brainly.com/question/12381311
#SPJ1
ordered pair of the solution
y=4x+9
Answers
The solution of the equation has ordered pairs (0, 9) and (1, 13)
How to determine the ordered pair in the solution?The representation of the equation is given as
y = 4x + 9
There is only one equation in the system (i.e. y = 4x + 9)
This implies that we assume any value for x and then substitute this value in the equation y = 4x + 9
Take for instance, we set x = 0
So, we have the following representation
y = 4 x 0 + 9
This gives
y = 0 + 9
Evaluate the sum
y = 9
When represented as an ordered pair, we have (0, 9)
Using the above as a guide, the ordered pair (-2, -3) cannot be in the solution.
This is so because:
-3 = 4 x -2 + 9
-3 = 1 -- the equation is false
Next, we test another ordered pair, we have
Take for instance, we set x = 1
So, we have the following representation
y = 4 x 1 + 9
This gives
y = 4 + 9
Evaluate the sum
y = 13
So, we have
(0, 9) and (1, 13)
Hence, the ordered pairs in the solution are (0, 9) and (1, 13)
Read more about equations at
https://brainly.com/question/2476251
#SPJ1
Please can someone help me with calculating area and finding volume to cubes/rectangular prisms
Answers
Write the fraction that represents 3 feet of rope divided into 8 equal parts.
Answers
Step-by-step explanation: Divide 3 ft by 8
How many points need to be removed from this graph so that it will be a function?
