True. A biweekly compounding will generate a higher annual percentage yield (APY) than a monthly compounding.
True. A biweekly compounding frequency will generate a higher annual percentage yield (APY) compared to a monthly compounding frequency. The APY takes into account the effect of compounding on the total return of an investment over a year.
Biweekly compounding means that interest is calculated and added to the account balance every two weeks, resulting in more frequent compounding periods throughout the year. This more frequent compounding leads to the accumulation of interest on a larger balance, thus generating a higher APY.
On the other hand, monthly compounding occurs once a month, resulting in fewer compounding periods and a lower overall APY. Therefore, for the same nominal interest rate, a biweekly compounding schedule will yield a higher APY compared to a monthly compounding schedule.
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1. Find the solution to the recurrence relation an = 3an-1 + 4an-2 with initial values ao = 2 and a₁ = 3.
The solution to the recurrence relation an = 3an-1 + 4an-2 with initial values ao = 2 and a₁ = 3 is given byan = (-1)4ⁿ - 4(4)ⁿ-¹/16
Given recurrence relation is an = 3an-1 + 4an-2, with initial values ao = 2 and a₁ = 3.
The characteristic equation of the recurrence relation is given byr² - 3r - 4 = 0
Solving the characteristic equation, we get
r² - 4r + r - 4 = 0
r(r - 4) + 1(r - 4) = 0
(r - 4)(r + 1) = 0
r1 = 4, r2 = -1
So, the general solution of the recurrence relation is given by
an = Ar¹ + Br²
For r1 = 4, a4 = 3
a3 + 4a2a4 = 3a3 + 4a2 = 3(4a2 + 4a1) + 4a2= 16a2 + 12a1 ....(1)
For r2 = -1, aₙ₊₁ = 3an + 4an-1aₙ₊₁ = 3an + 4an-1 = 3(A(-1)^n + B(4)^n) + 4(A(-1)^(n-1) + B(4)^(n-1))= 3A(-1)^n - 4A(-1)^(n-1) + 12B(4)^n + 4B(4)^(n-1)= A(-1)^n + 4B(4)^n ....(2)
Putting n = 0 in (2), we get
a1 = A - 4A = -3A = 3 => A = -1
Substituting A = -1 in (1), we get
a4 = 16a2 + 12a1=> a4 = 16a2 + 12(2) => a4 = 16a2 + 24a4 = 16a2 + 24 => a2 = (a4 - 24)/16
Thus the solution to the recurrence relation an = 3an-1 + 4an-2 with initial values ao = 2 and a₁ = 3 is given by
an = (-1)4ⁿ - 4(4)ⁿ-¹/16
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Two ordinary dice are thrown simultaneously. Determine the n
of throws necessary to obtain at least once with probability 0.49.
at least once the pair (6;6)
Two ordinary dice are thrown simultaneously. Determine the number of throws necessary to obtain at least once with probability 0.49 at least once the pair (6,6).
Solution: The probability of getting a pair of 6s in a single throw is 1/36.The probability of not getting a pair of 6s in a single throw is 1 - 1/36 = 35/36.
The probability of not getting a pair of 6s in n throws is (35/36)^n.
The probability of getting a pair of 6s in n throws is 1 - (35/36)^n.
So, for at least one pair of 6s with probability 0.49 in n throws, we have:
1 - (35/36)^n = 0.49⇒ (35/36)^n = 0.51⇒ n ln (35/36) = ln 0.51⇒ n = ln 0.51/ln (35/36) = 72.5 ~ 73So, at least 73 throws are necessary to obtain at least once with probability 0.49 at least once the pair (6,6).
Answer: At least 73 throws are necessary to obtain at least once with probability 0.49 at least once the pair (6,6).
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A drawbridge has the shape of an isosceles trapezoid. The entire length of the bridge is 100 feet while the height is 25 feet. If the angle at which the bridge meets the land is approximately 60 degrees, how long is the part of the bridge that opens?
Answer:
The part of the bridge that opens is 50 ft.
Step-by-step explanation:
The given parameters of the drawbridge are;
The entire length of the bridge = 100 feet
The height of the isosceles trapezoid formed = 25 feet
The angle at which the drawbridge meets the land ≈ 60°
Therefore, the part of the bridge that opens = The top narrow parallel side of the isosceles trapezoid
The length of each half of the bridge = (The entire length)/2 = 100 ft./2 = 50 ft.
Let 'x' represent the path of the waterway still partly blocked by each half of the bridge inclined
∴ x = 50 × cos(60°) = 25
x = 25 ft.
The path covered by both sides of the drawbridge = 2·x = 2 × 25 ft. = 50 ft.
The part of the bridge that opens = The entire length - 2·x
∴ The part of the bridge that opens = 100 ft. - 50 ft. = 50 ft.
The part of the bridge that opens = 50 ft.
Use the Runge-Kutta method with h=0.09 to estimate the value of the solution at t=0.1 to y' = 3 + t - y, y(0) = 1
By applying the Runge-Kutta method with a step size (h) of 0.09, we can estimate the value of the solution at t = 0.1 for the differential equation y' = 3 + t - y, with the initial condition y(0) = 1.
The Runge-Kutta method is a numerical technique used to approximate the solution of ordinary differential equations. In this case, we have the differential equation y' = 3 + t - y, where y' represents the derivative of y with respect to t. To apply the Runge-Kutta method, we need to iterate through the given range of t values, which is from 0 to 0.1 in this case, with a step size (h) of 0.09.
We start with the initial condition y(0) = 1. Then, for each iteration, we calculate the slope at the current point using the given equation. Using the slope, we estimate the value of y at the next time step (t + h). This process is repeated until we reach the desired value of t = 0.1.
By applying the Runge-Kutta method with h = 0.09, we can obtain an estimate for the value of y at t = 0.1. This method provides a more accurate approximation compared to simpler methods like Euler's method, as it considers multiple intermediate steps to improve accuracy.
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a kite is flying at an altitude of 20 meters elevation from the ground to kite is 30
A statistics module has been running for many years and, in the past, it has been found that each year the number of students passing the exam has distribution Bi(n. 0.75), where n are the number of students taking the module that year. A lecturer is teaching the module for the first time and 105 out of 150 students pass the exam. Perform a hypothesis test at the 0.05-significance level, where the null hypothesis is The probability of a student passing the module is 0.75and the alternative hypoth- esis is The probability of a student passing the module is less than 0.75. What is the conclusion? [Hint: Clearly state any assumptions made and recall the conditions under which a bino- mial distribution can be approximated by a normal distribution.]
The probability of a student passing the module is less than 0.75. Therefore, the lecturer should reconsider his method of teaching.
Question analysis A statistics module has been running for many years, and the module is given to n students each year. It has been discovered that in each year, the number of students passing the exam is distributed Bi(n, 0.75). In the current year, 150 students took the module for the first time, and 105 students passed the exam.
Using the 0.05 level of significance, we will conduct a hypothesis test to decide if the module's pass rate this year is less than 0.75.AssumptionsIf the number of trials is huge, the distribution of successes will be nearly normal. The number of trials n is greater than 30 in this situation. The probability of success in each trial is the same, namely p = 0.75. This condition is also satisfied. Therefore, we may use a normal distribution to approximate the binomial distribution.
What is the conclusion?
Null hypothesis: H₀: P = 0.75
Alternative hypothesis: H₁: P < 0.75The level of significance is 0.05, which implies that the rejection area will be in the left tail because the alternative hypothesis is one-tailed. Since the distribution of successes is approximately normal with a mean of np and a variance of np(1−p), we may find the p-value using this formula:
[The probability that X ≤ 105]
= [Z = (X − µ)/σ]
= [Z = (105 − (150 × 0.75))/sqrt(150 × 0.75 × (1 − 0.75))]
= [Z = (105 − 112.5)/3.2958]
= -2.2782
The p-value is [P(Z < -2.2782)] = 0.011. Because the p-value is less than 0.05, we reject the null hypothesis and accept the alternative hypothesis.
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Given : A statistics module has been running for many years and, in the past, it has been found that each year the number of students passing the exam has distribution Bi(n. 0.75), where n are the number of students taking the module that year. A lecturer is teaching the module for the first time and 105 out of 150 students pass the exam. The conclusion is that we fail to reject the null hypothesis.
The null and alternative hypotheses are given as follows:
Null hypothesis: The probability of a student passing the module is 0.75.
Alternative hypothesis: The probability of a student passing the module is less than 0.75.
We need to perform a hypothesis test at the 0.05-significance level.
The given probability distribution Bi(n,0.75) can be approximated to the normal distribution N(np,npq) under the following conditions:
The sample size n is large enough.
np≥5 and nq≥5, where q=1-p.
Here, n=150 and
p = 0.75
q = 1−p
= 1−0.75
= 0.25
Since np and nq are both greater than 5, the distribution Bi(150,0.75) can be approximated by the normal distribution N(150×0.75,150×0.75×0.25) = N(112.5,28.125).
Let X be the number of students that passed the module.
Under the null hypothesis, X follows the binomial distribution Bi(150,0.75).
Let μ be the mean of X under the null hypothesis.
μ = np
= 150×0.75
= 112.5
Since the alternative hypothesis is the probability of passing the module is less than 0.75, we need to perform a one-tailed test in the left tail at the 0.05-significance level.
The test statistic is given by,
Z=(X−μ)/σ
Z=(105−112.5)/√28.125/150
Z ≈ −1.5
This is a left-tailed test, so the critical value for a 0.05-significance level is z=−1.645.
Since the test statistic z=-1.5 > critical value z=-1.645, we fail to reject the null hypothesis.
Hence, there is not enough statistical evidence to conclude that the probability of a student passing the module is less than 0.75.
Therefore, the conclusion is that we fail to reject the null hypothesis.
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solve the given initial value problem using the method of Laplace transforms.
5y''+2y'+3y = u(t-pi) y(0)=1 y'(0)=1
The solution to the given initial value problem using the method of Laplace transforms, is: y(t) = -4 [tex]e^{-t}[/tex] + 5 [tex]e^{-3t/5}[/tex]
To solve the given initial value problem using the method of Laplace transforms, we will follow these steps:
Taking the Laplace transform of both sides of the differential equation.
Applying the Laplace transform to the given differential equation, we get:
5L{y''} + 2L{y'} + 3L{y} = L{u(t-[tex]\pi[/tex])}
Using the properties of Laplace transforms and the table of Laplace transforms to simplify the equation.
The Laplace transform of y'' is [tex]s^2[/tex]Y(s) - sy(0) - y'(0), where Y(s) is the Laplace transform of y(t).
The Laplace transform of y' is sY(s) - y(0), and the Laplace transform of y is Y(s).
Using these transformations and considering the initial conditions y(0) = 1 and y'(0) = 1, we can rewrite the equation as:
5([tex]s^2[/tex]Y(s) - s - 1) + 2(sY(s) - 1) + 3Y(s) = e^(-pi*s) / s
Simplifying further, we have:
(5[tex]s^2[/tex] + 2s + 3)Y(s) - (5s + 7) = [tex]e^{-\pi s}[/tex] / s
Solving for Y(s):
Rearranging the equation, we get:
Y(s) = ([tex]e^{-\pi s}[/tex] / s + (5s + 7)) / (5[tex]s^2[/tex] + 2s + 3)
Using partial fraction decomposition to express Y(s) in simpler terms.
Performing partial fraction decomposition on the right side, we can express Y(s) as:
Y(s) = A / (s + 1) + B / (5s + 3)
where A and B are constants to be determined.
Using the inverse Laplace transform, we can find the solution y(t) as:
y(t) = [tex]L^{-1}[/tex]{Y(s)} = [tex]L^{-1}[/tex]{A / (s + 1)} + [tex]L^{-1}[/tex]{B / (5s + 3)}
Taking the inverse Laplace transforms using the table of Laplace transforms, we find:
y(t) = A [tex]e^{-t}[/tex] + B [tex]e^{-3t/5}[/tex]
Substituting the initial conditions y(0) = 1 and y'(0) = 1 into the solution y(t) = A [tex]e^{-t}[/tex] + B [tex]e^{-3t/5}[/tex], we can solve for the constants A and B.
First, substitute t = 0 into the equation:
y(0) = A * [tex]e^{-0}[/tex] + B * [tex]e^{-0}[/tex] = A + B = 1
Next, differentiate the solution y(t) with respect to t:
y'(t) = -A * [tex]e^{-t}[/tex] - (3B/5) * [tex]e^{-3t/5}[/tex]
Then, substitute t = 0 and y'(0) = 1 into the equation:
y'(0) = -A * [tex]e^{-0}[/tex] - (3B/5) * [tex]e^{-0}[/tex] = -A - (3B/5) = 1
We now have a system of equations:
A + B = 1
-A - (3B/5) = 1
Solving this system of equations, we can find the values of A and B.
From the first equation, we can rewrite it as:
A = 1 - B
Substituting this expression for A into the second equation:
-(1 - B) - (3B/5) = 1
Simplifying the equation:
-1 + B - (3B/5) = 1
Multiplying through by 5 to eliminate the fraction:
-5 + 5B - 3B = 5
Combining like terms:
2B = 10
Dividing by 2:
B = 5
Substituting the value of B back into the first equation:
A = 1 - 5 = -4
Therefore, the constants A and B are -4 and 5, respectively.
The solution to the initial value problem is:
y(t) = -4 [tex]e^{-t}[/tex] + 5 [tex]e^{-3t/5}[/tex]
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Which is the correct equation for x:y=8:1
See picture attached.
Answer:
Step-by-step explanation:
Means of means = means of extremes 8y = x
x = 8y
Option B is the correct answer
G(x)=2x/3+3. What value of g(-15)
Answer:
g= 2x+9 /3x
Step-by-step explanation:
Doug's teacher told him that standardized score (s-score) for his mathematics exam, as compared to the exam scores of other students in the course, is 1.20.
Full question:
Doug's teacher told him that the standardized score (z-score) for his mathematics exam, as compared to the exam scores of other students in the course, is 1.20. Which of the following is the best interpretation of this standardized score?
Doug's test score is 120.
Doug's test score is 1.20 times the average test score of students in the course.
Doug's test score is 1.20 above the average test score of students in the course.
Doug's test score is 1.20 standard deviations above the average test score of students in the course.
None of the above gives the correct interpretation.
Answer:
Doug's test score is 1.20 standard deviations above the average test score of students in the course.
Explanation:
Z scores are also known as standardized scores or normal scores or standardized variables. Z scores are used to standardize raw data in order to give them a uniformity or standard that allows for easier comparison of data values. For us to calculate a z-score as was done in Doug's test score, we simply subtract the mean from the raw data score and we divide the answer by the standard deviation.
Draw a two-dimensional representation of each prism. Then find the area of the entire surface of each prism
Answer:
Surface area of cuboid = 78 unit²
Step-by-step explanation:
Given diagram is a cuboid prism
Given:
Length of cuboid = 5 unit
Width of cuboid = 3 unit
Height of cuboid = 3 unit
Find:
Surface area of cuboid
Computation:
Surface area of cuboid = 2[lb + bh + hl]
Surface area of cuboid = 2[(5)(3) + (3)(3) + (3)(5)]
Surface area of cuboid = 2[15 + 9 + 15]
Surface area of cuboid = 2[39]
Surface area of cuboid = 78 unit²
Simon traveled 250 miles in 5 hours. What is his average speed?
Answer:
250/5 =50 miles per hour
Type the correct answer in the box.
Given : b ┴ d
c || b
b || e
What line is perpendicular to line e?
Answer:
d is parallel to e
Step-by-step explanation:
Since b is parallel to e and d is perpendicular to b , then
d is perpendicular to e
The perimeter of a piece of paper is 38 inches. Its length is 11 inches.
Find the area of the piece of paper.
Answer:
Buddy this might not be the correct answer but I got either 98 or 418 inches. Don't quote me on it though.
Step-by-step explanation:
Can someone please give me this answer and hurry
Answer:
113.04cm
............
The diameter of a circle is 6 kilometers. What is the area?
d=6 km
Give the exact answer in simplest form.
_____ square kilometers
Answer:
28.26
Step-by-step explanation:
6 divided by 2 = 3^2 = 9 x 3.14 = 28.26
Nationally, the per capita monthly fuel oil oil is $110.
A random sample of 36 cities in the Southeast average $78, with a standard deviation of $4.
Is the difference significant? Summarize your conclusions in a sentence or two.
The calculated z-score of -47.76 falls outside the critical range of -1.96 to 1.96 indicating a statistically significant difference.
Is there a significant difference between them ?In order to determine whether this difference is significant, we will perform a one-sample z-test, as we know the population standard deviation.
The null hypothesis (H0) is that there is no difference between the national per capita monthly fuel oil cost and the average cost in Southeastern cities.
The alternative hypothesis (H1) is that there is a difference.
Sample mean (x): $78
Population mean (μ): $110
Sample standard deviation as an estimate: $4
Sample size (n): 36
z = (x - μ) / (σ/√n)
Substituting numbers:
z = ($78 - $110) / ($4/√36)
z = -32 / (4/6)
z = -48
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Which expression represents the length of the spring after Gerard removes some weight? Gerard adds weight to the end of the hanging spring D-- The song stretches to a length of p centimeters. Gerard removes some weight and the song moves up by a 8 E-p) - 9 D-9--
Answer: p+(-q)
Step-by-step explanation:
Find the area of the parallelogram
Height=4cm
Base=5cm
Answer:
The area of the parallelogram is 20cm²
Answer:
A = 20 cm²
Step-by-step explanation:
The area (A) of a parallelogram is calculated as
A = bh ( b is the base and h the height ) , then
A = 5 × 4 = 20 cm²
You roll a single 6 sided die. What are the odds of rolling a 9?
A. 1/6
B. 0
C. 1/9
D. 9
Answer:
B. 0
Step-by-step explanation:
There aren't enough sides for you to roll a nine
approximately what interest rate to the nearest whole percentage would you need to earn in order to turn $3,500 into $7,000 over 10 years?
a. 5%
b. 7%
c. 9%
d. 10%
The approximate interest rate needed to turn $3,500 into $7,000 over 10 years is 9%. Correct answer is C.
The value of money increases over time with the help of compounding interest. If one puts in a principal amount in an account, the amount will increase over time as interest accrues. Let's use the future value formula for the calculation. Let’s assume that the interest rate needed to turn $3,500 into $7,000 over 10 years is x percent. P = $3,500 (principal)FV = $7,000 (future value)
N = 10 years (duration of the investment)Using the future value formula:
FV = P(1 + r/n)^(nt)where, r is the annual interest rate, n is the number of times the interest is compounded in a year, and t is the duration of the investment in years.
Substituting the given values, we have: $7,000 = $3,500(1 + x/n)^(n × 10)We can solve for x by approximating the interest rate using each of the answer options given in the question until we find an answer that is close to $7,000. A calculator can also be used to calculate the compound interest for each option. If the interest rate is 7%, then the interest is compounded annually. Therefore, n = 1$7,000
= $3,500(1 + 0.07/1)^(1 × 10) If the interest rate is 10%, then the interest is compounded annually.
Therefore, n = 1$7,000 = $3,500(1 + 0.1/1)^(1 × 10)Thus, x ≈ 9.57%, greater than the required amount.
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A radius is
the diameter
Answer:
Radius is the diameter divided by 2
Compare the dimensions of the prisms. How many times greater is the surface area of the purple prism than the surface area of the red prism?
Answer: 3 times greater
Step-by-step explanation:
Height: 4x3=12
Length: 3x3=9
Width: 3x3=9
The sides on the red cuboid times by 3 equals the sides on the purple one.
Hope this helps :)
How many years would it take for
£109.27 to be accrued after
£100 is invested with 3%
pa compound interest.
Years:
Answer:
3 years approx
Step-by-step explanation:
Given data
Principal=£100
Amount= £109.27
Rate= 3%
The expression for the compound interest is
A=P(1+r)^t
Make t subject of formula we have
t= ln(A/P) / r
t= ln(109.27/100)/ 3
t= ln(1.0927)/0.03
t= 0.088/0.03
t= 2.93
Hence the time is 3 years approx
Your friends house is 6 miles south and 8 miles east of your house how far is your friends house from your house
Answer:
10 miles
Step-by-step explanation:
The information given forms a right angled triangle ; hence, we can use Pythagoras rule to solve for the distance, x
Recall:
Hypotenus = sqrt(opposite ² + adjacent ²)
Hypotenus = x
Therefore,
x = sqrt(6² + 8²)
x = sqrt(36 + 64)
x = sqrt(100)
x = 10
Distance between tween my friends house and my house = 10 miles
A cylindrical glass of soda has a mass of 700g. The glass itself has a mass of 80g. If the glass has a radius of 4cm and a height of 8cm, what is the density of the soda?
The density of the soda is 3.6 g/cm³
What is density?Density is the measurement of how tightly a material is packed together. It is defined as the mass per unit volume.
Given that, a cylindrical glass of soda has a mass of 700g. The glass itself has a mass of 80g, the glass has a radius of 4 cm and a height of 8 cm,
We are asked to find the density of the soda,
Density = mass / volume
Volume = 2π×radius×height
Therefore,
Density = 700/2π×4×8 [we will not add the mass of glass because we need to find the density of soda only]
Density = 700/194.68
= 3.59
= 3.6
Hence, the density of the soda is 3.6 g/cm³
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A sphere has a radius of five units, and intersects the zy plane along the circle whose equation is (x-1)² + (y+4)2-9 If the coordinate of the center of the sphere is a positive number, determine the equation of the sphere. (2.) (10 pts) Determine a vector of length four that points in the same direction as u = (1,2,2)
The vector of length four that points in the same direction as u = (1, 2, 2) is v = (4/3, 8/3, 8/3).
To determine the equation of the sphere with a radius of five units, we need the coordinates of its center.
From the given information, we know that the sphere intersects the zy-plane along the circle with the equation [tex](x - 1)^2 + (y + 4)^2 = 9[/tex].
The center of this circle can be found by setting x = 1 and y = -4 in the equation since the circle intersects the zy-plane.
Thus, the center of the sphere is (1, -4, 0).
Now, we can write the equation of the sphere using the center and the radius.
The equation of a sphere in 3D space is given by:
[tex](x - h)^2 + (y - k)^2 + (z - l)^2 = r^ 2[/tex]
where (h, k, l) represents the center coordinates and r represents the radius.
Substituting the values, we have:
[tex](x - 1)^2 + (y + 4)^2 + (z - 0)^2 = 5^2[/tex]
Simplifying the equation, we get:
[tex](x - 1)^2 + (y + 4)^2 + z^2 = 25[/tex]
Therefore, the equation of the sphere with a radius of five units and a center at a positive number is:
[tex](x - 1)^2 + (y + 4)^2 + z^2 = 25[/tex]
Now, let's determine a vector of length four that points in the same direction as u = (1, 2, 2).
To find a vector with the same direction, we can normalize vector u to have a length of 1 and then scale it by a factor of 4.
The normalization of a vector u is given by:
[tex]u_{normalized}[/tex] = u / ||u||
where ||u|| represents the magnitude or length of vector u.
Calculating the magnitude of vector u:
||u|| = [tex]\sqrt{(1^2 + 2^2 + 2^2)} = \sqrt{(1 + 4 + 4)} = \sqrt{9} = 3[/tex]
Now, we can normalize vector u:
[tex]u_{normalized}[/tex] = (1/3, 2/3, 2/3)
To get a vector of length four pointing in the same direction as u, we can scale the normalized vector by 4:
vector v = 4 *[tex]u_{normalized}[/tex]= (4/3, 8/3, 8/3)
Therefore, the vector of length four that points in the same direction as u = (1, 2, 2) is v = (4/3, 8/3, 8/3).
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Use the give an information to find the coefficient of determination.
Find the coefficient of determination, given that the value of the linear correlation coefficient, r, is -0.271
The calculated value of the coefficient of determination is 0.073
How to find the coefficient of determinationFrom the question, we have the following parameters that can be used in our computation:
Regression = linear
Correlation coefficient, r, is -0.271
The coefficient of determination can be calculated using:
R = r²
Where
r = Correlation coefficient = -0.271
Substitute the known values in the above equation, so, we have the following representation
R = (-0.271)²
Evaluate the exponent
R = 0.073
Hence, the coefficient of determination is 0.073
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HURRY ASAP PLEASE!!! ILLGIVE BRAINLIEST TO WHOEVER GIVES THE CORRECT ANSWER
How many edges does the figure have?
A prism with 9 edges.
A. 3
B. 5
C. 6
D. 9
Answer:
9
Step-by-step explanation:
the prism has 9 edges because you said "a prism with 9 edges"
Hope this helps!!! have a great day!!
Characterization of Random Processes in Time Domain Let Y(t) = 2X(t) + sin(2t) where X(t) is a wide-sense stationary (WSS) random process with mean à = E[X(t)] = 0 and autocorrelation Rx (T) = E[X(t + 7)X(t)] = e¯|7|. (a) (5) Find the mean ÿ(t) = E[Y(t)] and the autocorrelation Ry(t +7,t) = E[Y(t + 7)Y(t)] of Y (t). (2) Is Y (t) wide-sense stationary? Why? (b) (5)Find the crosscorrelation Rxy(t+7,t) = E[X(t+7)Y(t)]. (2) Are X and Y jointly wide sense stationary? Why? (c) (5) Find the autocovariance Cy (t +7,t) = E[(Y(t + 7) − ÿ(t + 7))(Y(t) − y(t))] of Y (t). (2) Is Y (t) white? Why?
A. The mean ÿ(t) = 0 and the autocorrelation Ry(t + 7, t) = 4e⁻⁷. Y(t) is wide-sense stationary.
B. the cross-correlation Rxy(t + 7, t) = 2e⁻⁷. X and Y are jointly wide-sense stationary.
C. The autocovariance Cy(t + 7, t) = 4e⁻⁷. Y(t) is not a white process because autocovariance Cy(t + 7, t) is not a Dirac delta function.
How did we arrive at these assertions?To find the mean ÿ(t) = E[Y(t)] and the autocorrelation Ry(t + 7, t) = E[Y(t + 7)Y(t)], we substitute the expression for Y(t) into the formulas:
(a) Mean of Y(t):
ÿ(t) = E[Y(t)] = E[2X(t) + sin(2t)]
= 2E[X(t)] + E[sin(2t)]
= 2(0) + 0
= 0
(b) Autocorrelation of Y(t + 7, t):
Ry(t + 7, t) = E[Y(t + 7)Y(t)]
= E[(2X(t + 7) + sin(2(t + 7)))(2X(t) + sin(2t))]
Expanding the expression:
Ry(t + 7, t) = E[4X(t + 7)X(t) + 2X(t + 7)sin(2t) + 2sin(2(t + 7))X(t) + sin(2(t + 7))sin(2t)]
Since X(t) is a WSS random process with mean 0, its autocorrelation Rx(T) = E[X(t + 7)X(t)] = e^(-|7|).
Using the properties of expectation and the independence of X(t) and sin(2t):
Ry(t + 7, t) = 4E[X(t + 7)X(t)] + 2E[X(t + 7)]E[sin(2t)] + 2E[sin(2(t + 7))]E[X(t)] + E[sin(2(t + 7))]E[sin(2t)]
= 4Rx(7) + 2(0)(0) + 2(0)(0) + 0
= 4e⁻⁷
Therefore, the mean ÿ(t) = 0 and the autocorrelation Ry(t + 7, t) = 4e⁻⁷.
To determine if Y(t) is wide-sense stationary, we need to check if the mean and autocorrelation are independent of time:
Mean: The mean ÿ(t) is constant and does not depend on time t. Thus, Y(t) has a constant mean.
Autocorrelation: The autocorrelation Ry(t + 7, t) depends only on the time difference of 7. It is independent of the absolute values of t. Therefore, Y(t) has a stationary autocorrelation.
Since Y(t) has a constant mean and a stationary autocorrelation, it is wide-sense stationary.
Moving on to part (b), we need to find the cross-correlation Rxy(t + 7, t) = E[X(t + 7)Y(t)].
Rxy(t + 7, t) = E[X(t + 7)Y(t)]
= E[X(t + 7)(2X(t) + sin(2t))]
Expanding the expression:
Rxy(t + 7, t) = E[2X(t + 7)X(t) + X(t + 7)sin(2t)]
Since X(t) is a WSS random process, its autocorrelation Rx(T) = e|⁻⁷|.
Using the properties of expectation and the independence of X(t) and sin(2t):
Rxy(t + 7, t) = 2E[X(t + 7)X(t)] + E[X(t + 7)]E[sin
(2t)]
= 2Rx(7) + 0
= 2e⁻⁷
Therefore, the cross-correlation Rxy(t + 7, t) = 2e⁻⁷.
To determine if X and Y are jointly wide-sense stationary, we need to check if the cross-correlation Rxy(t + 7, t) is independent of time:
Cross-correlation: The cross-correlation Rxy(t + 7, t) depends only on the time difference of 7. It is independent of the absolute values of t. Therefore, X and Y have a stationary cross-correlation.
Since the cross-correlation is stationary, X and Y are jointly wide-sense stationary.
Moving on to part (c), we need to find the autocovariance Cy(t + 7, t) = E[(Y(t + 7) - ÿ(t + 7))(Y(t) - ÿ(t))].
Expanding the expression:
Cy(t + 7, t) = E[(2X(t + 7) + sin(2(t + 7))) - 0][(2X(t) + sin(2t)) - 0]
= E[(2X(t + 7) + sin(2(t + 7)))(2X(t) + sin(2t))]
Using the same approach as in part (b), we expand the expression and evaluate the expectation:
Cy(t + 7, t) = 4E[X(t + 7)X(t)] + 2E[X(t + 7)]E[sin(2t)] + 2E[sin(2(t + 7))]E[X(t)] + E[sin(2(t + 7))]E[sin(2t)]
= 4Rx(7) + 0 + 0 + 0
= 4e⁻⁷
Therefore, the autocovariance Cy(t + 7, t) = 4e⁻⁷.
To determine if Y(t) is white, we check if the autocovariance Cy(t + 7, t) is a Dirac delta function. Since Cy(t + 7, t) = 4e⁻⁷ ≠ 0, it is not a Dirac delta function. Hence, Y(t) is not a white process.
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