X and Y are two continuous random variables whose joint pdf f(x, y) = kr² over the region 0≤x≤1 and 0 ≤ y ≤ 1, and zero elsewhere. Calculate the covariance Cov(X,Y).
Answers
The covariance Cov(X, Y) can be calculated for the given joint probability density function (pdf) f(x, y) = kr² over the region 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.
To calculate the covariance Cov(X, Y), we need to determine the joint probability density function (pdf) of X and Y and apply the formula for covariance.
First, we need to find the constant k by integrating the joint pdf over its entire range to ensure it integrates to 1 (since it represents a probability density function).
The integral of f(x, y) over the region 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 is given by:
∫∫ f(x, y) dy dx = ∫∫ kr² dy dx.
Integrating with respect to y first, we get:
∫[0,1] ∫[0,1] kr² dy dx = k∫[0,1] r² [y=0 to y=1] dx
= k∫[0,1] r² dx
= k[r²x] [x=0 to x=1]
= k(r² - 0)
= kr².
Since the integral of the joint pdf over its entire range equals 1, we have kr² = 1, which implies k = 1/r².
Now, we can calculate the covariance Cov(X, Y) using the formula:
Cov(X, Y) = E[XY] - E[X]E[Y],
where E denotes the expected value.
Since X and Y are continuous random variables with a uniform distribution over the range [0,1], we have E[X] = E[Y] = 1/2.
To calculate E[XY], we integrate the product XY over the range [0,1] for both x and y:
E[XY] = ∫∫ xy f(x, y) dy dx
= ∫∫ xy kr² dy dx
= k∫∫ xyr² dy dx
= k∫[0,1] ∫[0,1] xyr² dy dx.
Integrating with respect to y first, we get:
E[XY] = k∫[0,1] ∫[0,1] xyr² dy dx
= k∫[0,1] [(1/2)xr² [y=0 to y=1]] dx
= k∫[0,1] (1/2)xr² dx
= (k/2)∫[0,1] xr² dx
= (k/2)[(1/3)x³r² [x=0 to x=1]]
= (k/2)(1/3)r²
= (1/2)(1/3)r²
= 1/6r².
Finally, we can calculate the covariance:
Cov(X,Y) = E[XY] - E[X]E[Y]
= 1/6r² - (1/2)(1/2)
= 1/6r² - 1/4.
Therefore, the covariance Cov(X, Y) for the given joint pdf f(x, y) = kr² over the region 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 is 1/6r² - 1/4.
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Related Questions
What is the solution to the system of equations y =- 3x 2 5x 2y 15?
Answers
The solution for the system of equations is (-19,55).
As given the equations in the question
y = –3x – 2
Simplify the above
y + 3x = -2
5x + 2y = 15
Multiply y + 3x = -2 by 2 and subtracted from 5x + 2y = 15.
2y - 2y + 5x -6x = 15 + 4
-x = 19
x = -19
Putting the value of x in the equation y + 3x = -2.
y + 3 × - 19 = -2
y - 57 = -2
y = -2 + 57
y = 55
Therefore the solution for a system of equations is (-19,55).
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A map has a scale of 4 in. = 15 mi. If you measured the distance between two cities to be 19 in. on the map, how many miles would it actually be?
Answers
Answer:
71.25 (or 71 1/4) mi
Step-by-step explanation:
15/4 = 3.75
1 in = 3.75 mi
3.75 × 19 = 71.25 mi
at the various activity levels shown, harper company incurred the following costs. units sold 20 40 60 80 100 a. rental cost per unit of merchandise sold $36.00 $18.00 $12.00 $9.00 b. Total phone expense80.00 100.00 120.00 140.00 c. cost per unit of supplies 1.001.00 1.00 1.00 d. Total insurace cost 480.00 480.00 480.00 480.00 e. Total salary cost 1,200.00 1,600.00 2,000.00 2,400.00 f. Total cost of goods sold 1,800.00 3,600.00 5,400.00 7,200.00 g. Depreciation cost per unit 240.00 120.00 800.00 60.00 h. Total rent cost 3,200.00 3,200.00 3,200.00 3,200.00 i. Total cost of shopping bags 2.00 4.00 6.00 8.00 j. Cost per unit of merchandise sold 90.00 90.00 90.00 90.00
Answers
a. Mixed cost - the cost per unit decreases with increase in units sold.
b. Variable cost - the expense increases with increase in units sold
c. Fixed cost - expense is constant with increase in units sold.
d. Fixed cost - cost is constant with increase in units sold.
e. Variable cost - cost increases with increase in units sold
f. Variable cost - cost increases with increase in units sold
g. Mixed cost- depreciation cost decreases with increase in units sold.
h. Fixed cost - cost is constant with increase in units sold.
i. Variable cost - cost increases with increase in units sold
j. Fixed cost- cost is constant with increase in units sold.
According to the given question;
Mixed costs: That has a high initial cost generally because they contain some fixed components and change in proportion because they contain some variable components.
Fixed costs: Costs that are consistent or constant are known as fixed costs.
Variable costs: These alter as the quantity of units changes.
(a) The price is fixed. This is because even while the price per unit fluctuates, the overall price does not. For instance, the total cost for 20 units is $720 ($36 x 20 units), and the total cost for 100 units is $720 ($7.2 x 100 units).
(b) The overall cost of the phone is mixed. On the basis of unit cost fluctuations, it may be claimed that the first 20 units have a fixed component and that subsequent 20 units have an equal distribution of fixed and variable costs.
(c) The price is vary. Supply costs are fixed regardless of quantity changes, and variable cost per unit is also fixed regardless of quantity changes that affect changes in the aggregate but not in the individual.
(d) The fact that the overall insurance cost remains constant as the quantity changes suggests that it is a fixed cost.
(e) Total salary expenses are mixed costs since they have a higher initial cost due to the inclusion of some fixed costs and ratio changes as a result of the addition of variable costs.
(f) Variable expenses, which rise in line with an increase in quantity, make up the entire cost of items sold.
(g) Depreciation is a fixed cost because, despite the fact that unit cost reduces as quantity increases, total cost remains constant.
(h) The total rent expense is constant despite changes in quantity, indicating that it is a fixed expense.
(i) The price of shopping bags is variable since the price and total fluctuate when the amount changes.
(j) Since individual unit expenses are constant per unit, the cost per unit of goods sold is variable.
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The write question is given below:
from a point a on the ground, the angle of elevation to the top of a tall building is . from a point b, which is ft closer to the building, the angle of elevation is measured to be . find the height of the building.
Answers
To find the height of the building, we need to use trigonometry. Let's call the height of the building "h" and the distance from point a to the building "x". From point a, the angle of elevation to the top of the building is given. Let's call this angle "θ".
Using trigonometry, we can write:
tan(θ) = h/x
We can rearrange this equation to solve for h:
h = x * tan(θ)
Now let's move to point b. We know that it is ft closer to the building than point a, so the distance from point b to the building is (x - ft). We also know the angle of elevation from point b, which we'll call "α".
Using the same equation as before, but with the new values, we get:
h = (x - ft) * tan(α)
Now we can set these two expressions for h equal to each other:
x * tan(θ) = (x - ft) * tan(α)
We can solve this equation for h:
h = x(tan(θ) - tan(α)) / (1 - tan(θ)tan(α))
This gives us the height of the building. We just need to plug in the values we were given for x, ft, θ, and α.
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What is the radius of the circle?
Answers
Answer:
210:28
Step-by-step explanation:
In 1998 there were 305 students who graduated from high school. In 2019, there were 356 students. What is the rate of change in the number of students?
1. About 1 student per year
2. about 51 students per year
3. about 2 students per year
4. about 5 students per year
Answers
Answer:
3. about 2 students per year
Step-by-step explanation:
The rate of change would be the difference in the number of students divided by the number of years passed.
So, rate of change = \(\frac{356-305}{2019-1998}\)
= \(\frac{51}{21} = 2.43\)
So it's around 2 students per year.
I hope this is correct and helps!
Given the graph and the equation y = 2x2 - 8x + 3. which one has the smaller minimum and by how much?
A)
The graph by 6 units
B)
The graph by 70 units
C)
The equation by 2 units
D)
The equation by 5 units
Answers
Answer:
c
Step-by-step explanation:
y=2(2)2-8(2)+3
=8-16+3
=-8+3
=-5
Answer:
c
Step-by-step explanation:
i did the prep
is this right or no?
Answers
1/3 of 12= 4
5/12 of 12= 5
1/4 of 12= 3
find the value of sin 18,cos 18 by without using table or calculater
Answers
The value of the trigonometry expression sin(78)cos(18) - cos(78)sin(18) is √3/2
Finding the value of the trigonometry expressionFrom the question, we have the following parameters that can be used in our computation:
sin(78)cos(18) - cos(78)sin(18)
Using the law of sines, we have
sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
using the above as a guide, we have the following:
sin(78)cos(18) - cos(78)sin(18) = sin(78 - 18)
Evaluate
sin(78)cos(18) - cos(78)sin(18) = sin(60)
When evaluated, we have
sin(78)cos(18) - cos(78)sin(18) = √3/2
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Question
Find the value of sin(78)cos(18) - cos(78)sin(18) by without using table or calculator
We are interested in determining if a new diet is effective in helping people lose weight. We measured the weight of 50 randomly selected dieters before they started the diet and again after 6-weeks on the diet. Once all the measurements were collected, how should the researchers analyze the data?
Answers
The researchers should perform a paired t-test on the weight measurements of the 50 randomly selected dieters before and after 6 weeks on the diet.
What is paired t-test?
A paired t-test is suitable for analyzing data when each subject's measurements are taken at two different time points or conditions. In this case, the weight measurements of the dieters were taken before they started the diet (time point 1) and after 6 weeks on the diet (time point 2).
By using a paired t-test, the researchers can compare the mean weight before and after the diet to assess if there is a significant difference. The paired t-test takes into account the within-subject correlation, making it appropriate for paired data analysis.
The paired t-test calculates the t-statistic by comparing the mean difference between the paired measurements to the null hypothesis of no difference (mean difference = 0). The calculated t-value can then be compared to the critical values from the t-distribution to determine the statistical significance of the weight change.
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Help me please !!!!!
Answers
Answer:
Step-by-step explanation:
The way to do this is just to add the two real parts together and add the two imaginary parts together.
2 + 5j + 7 - 3j
2 + 7 + 5j - 3j
9 + 2j
PLS HELP ASAP STUCK ON THIS
Answers
Explanation: 1/3 • 2 you turn the 2 into 2/1 so the problem is 1/3 • 2/1 which is 2/3
Answer:
all of them except b and d, d and b are the only ones that equal a number bigger than or equal to 1
Step-by-step explanation:
a- 1/3 ·2 is 2/3
b- 2÷1/3 is 6
c- 1/4 ·2/3 is 1/6
d- 3/4÷2/3 is 1 and 1/8
e- 2/3·3/4 is 1/2
f- 2/3÷3/4 is 8/9
hope this helps :)
given the equation ax + b = 9x + 21 which set of values for a and b would result in an equation with infinity many solutions
a = 3,b = 7
a = 9, b = 9
a = 9, b = -21
a = 9, b = 21
Answers
Answer:
D. a = 9, b = 21Step-by-step explanation:
Given the equation
ax + b = 9x + 21For the equation to have infinitely many solutions, both sides of the equation should be same.
a = 9 and b = 21 will make both sides equalCorrect option is D
Can someone pleaseeee help and if you’re correct i’ll give brainliest
Answers
Answer: D
Step-by-step explanation: please give me it!
Answer:
maybe the bottom right one?
Step-by-step explanation:
Evaluate the following expression. "5 less than 15"
A. -10
B. 8
C. 20
D. 10
Answers
Answer: 10
Step-by-step explanation: 5 less than 15 is 10 because 15 - 5 is 10. If it was 15 less from 5, only then it would be -10.
Answer:
5 less than 15 is Option D, 10. This is because using mathematical symbols, it says 15-5 which equals to 10.
Find the measure of f.
Answers
Answer:
44 degrees.
Step-by-step explanation:
The 3 small triangles are isosceles so m < a = m < b , therefore
m < a = (180 - 92) / 2 = 44 degrees.
m < f = m < a ( they are both subtended on the circle by the same chord (cd)),
So m < f = 44.
Can you think of a solution of the differential equation y'= -(1/4)y that is not a member of the family in part (b)? A. y = 4 is a solution of y' = -(1/4)y2 that is not a member of the family in part (b). B. y = e4x is a solution of y' = -(1/4)y2 that is not a member of the family in part (b). C. y = 0 is a solution of y' = -(1/4)y2 that is not a member of the family in part (b). D. Every solution of y' = -(1/4)y2 is a member of the family in part (b). E. y = x is a solution of y' = -(1/4)y2 that is not a member of the family in part (b).
Answers
The correct answer is B. y = e4x is a solution of y' = -(1/4)y2 that is not a member of the family in part (b).
An equation is a mathematical statement that contains an equal sign. A solution is a value or set of values that make the equation true. A member refers to a specific solution within a family of solutions.
In this question, we are given a differential equation y' = -(1/4)y and asked to find a solution that is not a member of a given family of solutions. The family of solutions is not provided in the question, but it is implied that it is a set of solutions that have the form y = Ce^(-x/4), where C is a constant.
To find a solution that is not a member of this family, we need to find a different function that satisfies the differential equation y' = -(1/4)y. Option B, y = e4x, is such a function. We can verify that it is a solution by taking its derivative and plugging it into the differential equation:
y = e4x
y' = 4e4x
y' = -(1/4)y
4e4x = -(1/4)e4x
This equation is true, so y = e4x is indeed a solution of y' = -(1/4)y that is not a member of the given family.
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Problem 8) Compute the unit tangent vector T and the principal unit normal vector N for r (t) = hsin (t) + 2, cos (t) + 10, 6ti
Answers
The unit tangent vector T and the principal unit normal vector N can be computed for the given vector-valued function r(t) = (sin(t) + 2, cos(t) + 10, 6t). The unit tangent vector T represents the direction of the curve at each point, while the principal unit normal vector N is perpendicular to the tangent vector and points towards the center of curvature.
To find the unit tangent vector T, we differentiate r(t) with respect to t and divide by its magnitude:
r'(t) = (cos(t), -sin(t), 6)
||r'(t)|| = sqrt((cos(t))^2 + (-sin(t))^2 + 6^2) = sqrt(1 + 1 + 36) = sqrt(38)
Therefore, the unit tangent vector T is given by:
T = r'(t) / ||r'(t)|| = (cos(t)/sqrt(38), -sin(t)/sqrt(38), 6/sqrt(38))
To find the principal unit normal vector N, we differentiate T with respect to t and divide by its magnitude:
T'(t) = (-sin(t)/sqrt(38), -cos(t)/sqrt(38), 0)
||T'(t)|| = sqrt((sin(t)/sqrt(38))^2 + (-cos(t)/sqrt(38))^2) = sqrt(1/38 + 1/38) = sqrt(2/38) = sqrt(1/19)
Therefore, the principal unit normal vector N is given by:
N = T'(t) / ||T'(t)|| = (-sin(t)/sqrt(19), -cos(t)/sqrt(19), 0)
In summary, the unit tangent vector T for the given vector-valued function is (cos(t)/sqrt(38), -sin(t)/sqrt(38), 6/sqrt(38)), and the principal unit normal vector N is (-sin(t)/sqrt(19), -cos(t)/sqrt(19), 0). These vectors represent the direction and perpendicular direction to the curve defined by r(t).
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Is 40% of 90 closer to 10 or 30?
Answers
Answer:
Closer to 30.
Step-by-step explanation:
To answer this first find 40% of 90.
90*0.4=36
0.4 is the same as 40%
Hope this helps!
If not, I am sorry.
0.05 is 1/10 of a decimal as
Answers
Answer:
1/10 =0.1 in decimal
Step-by-step explanation:
Joe Smith scored 37 of 100 his teams points in a game. What percentage of points did he score?
Answers
Answer:
37% ;-;
Step-by-step explanation:
Answer:
37%
Step-by-step explanation:
It's kinda obvious...
If A=(1 2 3), find 2 A?
Answers
Answer:
Remove parentheses
Step-by-step explanation:
A= 123
Apply De Morgan's law repeatedly to each Boolean expression until the complement operations in the expression only operate on a single variable. For example, there should be no xy¯ or x+y¯ in the expression. Then apply the double complement law to any variable where the complement operation is applied twice. That is, replace x¯¯ with x.
a. 1/ x + yz + u b. 1/x(y + 2)u c. 1/(x + y)(uv + x y) d. 1/xy + yz + xz
Answers
The simplified expression using De Morgan's law are a)x'y'z'u b)x'y'u c): x'y'u and d)x'y'z'+xy'z'+xyz.
The main idea is to simplify each Boolean expression by repeatedly applying De Morgan's law until each complement operation operates on a single variable.
Then, apply the double complement law to simplify the expression further. In the end, the simplified expression should contain only AND and OR operations without any complement operators acting on multiple variables.
a. 1/ x + yz + u can be simplified using De Morgan's law to: (x'y'z')u'. Then, applying the double complement law, we get the simplified expression as: x'y'z'u.
b. 1/x(y + 2)u can be simplified using De Morgan's law to: x'(y'+2')u'. Then, applying the double complement law, we get the simplified expression as: x'y'u.
c. 1/(x + y)(uv + xy) can be simplified using De Morgan's law to: (x'y')(u' + x'y'). Then, applying the double complement law, we get the simplified expression as: x'y'u.
d. 1/xy + yz + xz can be simplified using De Morgan's law to: (x'+y')(y'+z')(x'+z'). Then, applying the double complement law, we get the simplified expression as: x'y'z'+xy'z'+xyz.
In summary, to simplify Boolean expressions, we can apply De Morgan's law repeatedly and then use the double complement law to remove complement operators acting on a single variable twice.
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help me find the area pls
Answers
Area = 30
Step-by-step explanation:
a^2 + b^2 = c^2
12^2 + b^2 = 13^2
144 + b^2 = 169
b^2 = 169 - 144
b^2 = 25
b = 5
Area = 1/2 bh
Area = 1/2 (12)(5)
Area = 1/2 (60)
Area = 30
As a nasal decongestant, doctors sometimes prescribe saline solutions with a concentration between 6% and
20%. In "the old days," pharmacists had to create different mixtures, but only needed to stock these
concentrations, since any percentage in between could be obtained using a mixture. An order comes in for a
15% solution. How many milliliters of the 20% solution must be mixed with 10 mL of the 6% solution to
obtain the desired 15% solution?
Answers
Answer:
18mL of the 20% solution must be mixed with 10 mL of the 6% solution to
obtain the desired 15% solution
Step-by-step explanation:
Ignore all of the useless information til the last sentence: How many milliliters of the 20% solution must be mixed with 10 mL of the 6% solution to obtain the desired 15% solution?
Our goal is to find out how many mL of 20% solution is needed to mix with 10mL of 6% solution to achive a 15% solution.
An easy and simple way to do it is to create a system of equation:
We add in x mL of 15% solution(Which has x mL of water and 20%*x mL of salt)
The equation: 15%=(20%x+0.6)/(x+10)
After solving it we get x=18
Therefore:
18mL of the 20% solution must be mixed with 10 mL of the 6% solution to
obtain the desired 15% solution
Hope this helps, have a good day!
Ps. If you have any questions, feel free to ask them in the comment section below-I'll be more than happy to answer.
Based on the family the graph below belongs to, which equation could represent the graph?
y=2^x+3
y=log(2x)+3
y=2x² +2
y=1/2x+2
Answers
I WILL GIVE BRAINLIEST TO RIGHT ANSWER
write an equation of the line in a slope intercept form has a slope of -1/2 and passes through the point (2, 3)
Answers
Answer:
y = -¹/₂ x + 4Step-by-step explanation:
The point-slope form of the equation is y - y₀ = m(x - x₀), where (x₀, y₀) is any point the line passes through and m is the slope:
m = -¹/₂
(2, 3) ⇒ x₀ = 2, y₀ = 3
The point-slope form of the equation:
y - 3 = -¹/₂ (x - 2)
So:
y - 3 = -¹/₂ x + 1 {add 3 to both sides}
y = -¹/₂ x + 4 ← the slope-intercept form of the equation
Solve for X triangle.
Answers
Answer:
x= 12.942
Law of sines :)
HELP #1 *i will give whoever answers brainliest,but your answer has to be correct and you have to explain*
The first term in an arithmetic sequence is 12. The third term in the sequence is 4. The tenth term in the sequence is -24.
A. Write the function that could be used to find the nth term of the arithmetic sequence
B. Daelyn said that the recursive formula for this sequence could be described as, “to find the next term of the sequence, add -4 to the previous term.” Do you agree or disagree that this statement is equivalent to the explicit formula that you created in part A? Explain.
Answers
Answer:
A-> a(n)=a+(n-1)d
B-> Yes, I do agree
Step-by-step explanation:
A= a+(n-1)d
where a=the first term
n=the term's location
d= the difference between two adjacent terms
B= I agree
a3=12+2d=4.........equation 1
2d=-8
d=-4
a10=12+9d=-24.....equation 2
9d=-36
d=-4
Hence, to find the next term of the sequence, add -4 to the previous term.
{I hope this answers your question}
The lateral height of a cone is 4 inches and the area of the base of the cone is 49 in². It requires 2.5 minutes to paint the cone.
The area of the base is doubled.
How long will it take to paint this cone if it can be painted at the same rate? Use ≈ 3.14.
Enter your answer, rounded to the nearest tenth, in the box.
Answers
If the lateral height of a cone is 4 inches and the area of the base of the cone is 49 in². The time it take to paint this cone if it can be painted at the same rate is 3.5 minutes.
How to find the time?The radius of the cone can be found using the formula for the area of a circle, A = πr^2
where:
r= radius
Rearrange this to solve for the radius:
r = √(A/π).
The original cone has a base area of 49 in², so:
r = √(49/π)
r ≈ 3 in.
The doubled cone has a base area of 2 * 49 in² = 98 in², so:
r = √(98/π)
r ≈ 4.5 in.
Hence,
Time = 2.5 * (4.5/3)^(3/2)
Time ≈ 3.5 minutes
Therefore the time it take to paint this cone if it can be painted at the same rate is 3.5 minutes.
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