let x be the 6-point dft of x = [1, 2, 3, 4, 5, 6]. determine the sequence y whose dft y [k] = x[h−ki6], for k = 0, 1, . . . , 5.

As the sequence {an} is a sequence of constant random variables, it means that each term in the sequence has the same value with probability 1.

If an → a, then for any ε > 0, there exists an integer N such that for all n ≥ N, |an - a| < ε. This means that the probability of an being within ε of a is 1, which can be written as: lim P(|an - a| < ε) = 1

n→∞

Since this is true for any ε > 0, we can rewrite the above as: lim P(|an - a| < δ) = 1

n→∞ where δ is any positive number.

Now, if an → Pa, then for any ε > 0, there exists an integer N such that for all n ≥ N, P(|an - a| < ε) > 1 - δ. This means that the probability of an being within ε of a is greater than 1 - δ, which can be written as: lim P(|an - a| < ε) ≥ 1 - δ

n→∞

Again, since this is true for any ε > 0, we can rewrite the above as:

lim P(|an - a| < δ) ≥ 1 - δ

n→∞

Comparing the two limits, we see that they are equivalent. Therefore, an → a is equivalent to an → Pa.

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The correct option is (B) [tex]A^-1, B^-1[/tex], and [tex]C^-1[/tex], shows that ABC is also invertble

Since A, B, and C are invertible matrices, they have inverse matrices [tex]A^-1, B^-1,[/tex] and [tex]C^-1,[/tex] respectively.

To show that ABC is invertible, we can introduce a matrix D such that (ABC)D = I and D(ABC) = I, where I is the identity matrix.

We can use the associative property of matrix multiplication to rearrange the product ABCD as follows:

(ABC)D = A(BCD)

Since A, B, and C are invertible, their product ABC is also invertible. Therefore, we can write:

(ABC)D = A(BCD) = I

Multiplying both sides of the equation by [tex]A^-1,[/tex] we get:

[tex]A^-1(ABC)D = A^-1[/tex]

Using the associative property again, we can rearrange the left-hand side as follows:

[tex]A^-1(ABC)D = (A^-1AB)CD = ICD = D[/tex]

Substituting ICD with D, we get:

[tex](A^-1AB)CD = D[/tex]

Since[tex]A^-1A[/tex] is equal to the identity matrix I, we can simplify the equation as follows:

BCD = D

Now we can use a similar approach to show that D(ABC) = I. Multiplying both sides of the equation (ABC)D = I by [tex]C^-1,[/tex] we get:

[tex](ABC)DC^-1 = C^-1[/tex]

Using the associative property, we can rearrange the left-hand side as follows:

[tex]A(BCD)C^-1 = AIC^-1 = A^-1[/tex]

Substituting BCD with D, we get:

[tex]AD^-1C = A^-1[/tex]

Multiplying both sides by [tex]C^-1[/tex], we get:

[tex]AD^-1CC^-1 = A^-1C^-1[/tex]

Since [tex]CC^-1[/tex] is equal to the identity matrix I, we can simplify the equation as follows:

[tex]AD^-1 = A^-1C^-1[/tex]

Multiplying both sides by BC, we get:

[tex]ABCD^-1 = B(A^-1C^-1)C = BI = B[/tex]

Therefore, we have shown that ABC has an inverse matrix D, which implies that ABC is invertible.

Answer: The correct option is (B) [tex]A^-1, B^-1,[/tex] and [tex]C^-1[/tex]exist.

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hope this helps you.

The probability of laboratory errors in a re-accreditation period follows a Poisson distribution, and we can use this information to answer various questions about the number of laboratories committing errors.

A. The probability that exactly 10 laboratories commit errors in a re-accreditation period, assuming errors follow a Poisson distribution with an average of 30 laboratories committing errors, is 0.00003 (or 3 x 10^-5).

B. To find the standard deviation, we use the formula: square root of the average number of errors (lambda), which is 30. Therefore, the standard deviation is approximately 5.5772.

C. To calculate the probability that more than 40 but less than or equal to 51 laboratories commit errors in a re-accreditation period, we need to use the Poisson distribution formula with lambda equal to 30 and subtract the probability that 40 or fewer laboratories commit errors from the probability that 51 or fewer laboratories commit errors. The result is approximately 0.0108.

D. If the probability of lab errors is 58%, we can use the Poisson distribution formula with lambda equal to the average number of errors, which is 30, to calculate the probability of up to how many laboratories committed errors. The answer is approximately 4 (or 5, if rounded up).

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Answer:

5.0 inches

Step-by-step explanation:

The formula for the volume of a sphere is:

[tex]\boxed{V=\dfrac{4}{3}\pi r^3}[/tex]

where r is the radius of the sphere.

Given a sphere has a volume of 65.5 cubic inches, substitute V = 65.5 into the formula and solve for the radius, r:

[tex]\begin{aligned}\implies \dfrac{4}{3}\pi r^3&=65.5\\\\3 \cdot \dfrac{4}{3}\pi r^3&=3 \cdot 65.5\\\\4\pi r^3&=196.5\\\\\dfrac{4\pi r^3}{4 \pi}&=\dfrac{196.5}{4 \pi}\\\\r^3&=15.636973...\\\\\sqrt[3]{r^3}&=\sqrt[3]{15.636973...}\\\\r&=2.50063840...\; \sf in\end{aligned}[/tex]

The diameter of a sphere is twice its radius.

Therefore, if the radius is 2.50063840... inches, then the diameter is:

[tex]\begin{aligned}\implies d&=2r\\&=2 \cdot 2.50063840...\\&=5.00127681...\\&=5.0\; \sf in\;(nearest\;tenth)\end{aligned}[/tex]

Therefore, the diameter of a sphere with a volume of 65.5 cubic inches is 5.0 inches, to the nearest tenth of an inch.

Answer:

5 cm

Step-by-step explanation:

The formula to find the volume of a sphere is:

[tex]\sf V =\frac{4}{3} \pi r^3[/tex]

Here,

V ⇒ volume ⇒ 65.5 cm³

r ⇒ radius

Let us find the value of r.

[tex]\sf V =\frac{4}{3} \pi r^3\\\\65.5=\frac{4}{3} \pi r^3\\\\65.5*3=4 \pi r^3\\\\196.5=4 \pi r^3\\\\\frac{196.5}{4} =\pi r ^3\\\\49.125=\pi r^3\\\\\frac{49.125}{\pi} = r^3\\\\15.63=r^3\\\\\sqrt[3]{15.63} =r\\\\2.5=r[/tex]

Let us find the diameter now.

d = 2r

d = 2 × 2.5

d = 5 cm

The evaluated value of the given trigonometric expression cos 76° – sin 14° is 0. The correct answer is option B.

The trigonometric expression is given as follows:

cos 76° – sin 14°

It is required to find the evaluated value of the given trigonometric expression.

As per the angle of cosine and sine relation: cos (90 – θ) = sin θ.

It can be rewritten as follows:

Here, cos 76 as cos (90 – 14)

And, cos 76 = cos (90 – 14) = sin 14  

cos 76° – sin 14°  = sin 14° – sin 14°

cos 76° – sin 14°  =  0

Therefore, the evaluated value of the given trigonometric expression is 0.

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The complete question is as follows:

Evaluate cos 76° – sin 14°.

A. 1

B. 0

C. -1

D. 2

The features of the quadratic function are given as follows:

The function is decreasing on the following interval:

(1.5, ∞).

First we look at the vertex of the quadratic function, which is the turning point, with coordinates x = 1.5 and y = 6, hence it is given as follows:

(1.5, 6).

Hence the axis of symmetry is of x = 1.5, which is the x-coordinate of the vertex.

The function is concave down, hence the increasing and decreasing intervals are given as follows:

The x-intercepts are the values of x for which the graph crosses the x-axis, when the y-coordinate is of 0, hence they are given as follows:

(-1, 0) and (4,0).

The y-intercept is the value of y when the graph crosses the y-axis, when the x-coordinate is of zero, hence it is given as follows:

(0,4).

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The population of locusts gains 3/4 of it's size every 0.5 weeks.

An exponential function has the definition presented as follows:

y = ab^x.

In which the parameters are given as follows:

The growth rate after t weeks is given as follows:

(49/16)

When the population gains 3/4 of it's size, the fraction change is given as follows:

1 + 3/4 = 4/4 + 3/4 = 7/4.

Thus the number of weeks needed for the function to gain 3/4 of it's size is obtained as follows:

(49/16)^t = 7/4

(7/4)^(2t) = 7/4

2t = 1

t = 0.5.

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Each Algebra ticket cost $5.125 for kids and $20.50 for adults, while each Geometry ticket cost $4.46 for pupils and $30.125 for adults.

For Algebra I:

Cost per student ticket = Total cost of student tickets / Number of student tickets

Cost per student ticket = $164.00 / 32

Cost per student ticket = $5.125

For Geometry:

Cost per student ticket = Total cost of student tickets / Number of student tickets

Cost per student ticket = $120.50 / 27

Cost per student ticket = $4.46 (rounded to two decimal places)

Next, we can find the cost of one adult ticket for each class using the same method.

For Algebra I:

Cost per adult ticket = Total cost of adult tickets / Number of adult tickets

Cost per adult ticket = $164.00 / 8

Cost per adult ticket = $20.50

For Geometry:

Cost per adult ticket = Total cost of adult tickets / Number of adult tickets

Cost per adult ticket = $120.50 / 4

Cost per adult ticket = $30.125

Therefore, the price of each ticket for Algebra I was $5.125 for students and $20.50 for adults, while the price of each ticket for Geometry was $4.46 for students and $30.125 for adults.

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The statement which does not represent the distance is

C) In 3 hours car will travel the distance of 160 miles and D).

What is proportion?

A percentage is created when two ratios are equal to one another. We write proportions to construct equivalent ratios and to resolve unclear values.

Here the car can travel 55 miles in one hour.

Then in 1.5 hour distance traveled by car is x.

Using proportion,

=> x = 55*1.5 = 82.5 miles

Now in 2 hours distance traveled by car = 55*2=110 miles

In 2.5 hours distance traveled by car  = 55*2.5 = 137.5 miles

In 3 hours distance traveled by car = 55*3 = 165 miles.

Then distance = 55t . where t = time

Hence the statement which does not represent the distance is

C) In 3 hours car will travel the distance of 160 miles and D).

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The equivalent expression is 0.75n - 7

An equivalent expression is defined as an algebraic expression that have the same solution but differ in their arrangement.

Also, algebraic expressions are described as expression that consists of variables, constants, terms, coefficients and factors.

These expressions are also made up of arithmetic operations such as addition, subtraction, division, multiplication, bracket and parentheses.

From the information given as;

-3.5 (2- 1.5n) - 4.5n

expand the bracket

-7 + 5.25n - 4.5n

collect the like terms

0.75n - 7

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Answer:

34

Step-by-step explanation:

Note that maximum profit is $26. This point is obtained when the baker has made 4 batches of chocolate chip cookies and 2 bactches of oatmean raisons cookies.

Lets define x as the number of batches of chocolate chip cokies

We want to maximize profit, which is given by:

P = 5x + 3y

subject to the constraints:

4x + 2y ≤ 20 (egg constraint)

3x + 3y ≤ 18 (flour constraint)

x, y ≥ 0 (non-negativity constraint)

We can rewrite the constraints as:

2x + y ≤ 10

x + y ≤ 6

Graphing these constraints on a coordinate plane, we see that the feasible region is a triangle with vertices at (0,0), (0,6), and (4,2)

See agraph attached.

We want to find the point (x,y) within this region that maximizes P.

One way to do this is to calculate P at each vertex of the feasible region:

P(  0,0) = 0

P (0, 6) = 3(6) = 18

P (4,2) =
5(4) + 3(2) =

26

So the point of profit maximization is at  $ 26.

Thica can happen when the baker is baking 4 batches of chocolate chip cookies and 2 batches of oatmeal raisin cookies.

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Answer:

Step-by-step explanation:

The solution to the trigonometric expression is:

sin ( cos^-1 (1/2) + tan^-1 (1) )

= sin (60° + 45°)  (since cos^-1 (1/2) = 60° and tan^-1 (1) = 45°)

= sin 105°

= 0.966

The possible value of QRP is 24^o.

A sine rule is a trigonometric rule which can be used to determine either the angle or length of side of a given triangle that is not a right angle.

sine rule states that;

a/Sin A = b/Sin B = c/Sin C

From the given question, let the measure of angle QRP be represented by R. So that;

9/ Sin 37 = 6/ Sin R

9 Sin R = 6 Sin 37

Sin R = 6 Sin 37/ 9

  = 3.611/ 9

  = 0.4012

R = Sin^-1 (0.4012)

  = 23.65

Thus, QRP is 24^o.

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a. Bella should estimate her cost per year to be $17,150.

b. It looks like Bella's family is contributing more than her estimated cost per year, so she may not need to contribute anything.

The act of deleting items from a collection is represented by subtraction. Subtraction is denoted by the minus sign. For instance, suppose there are nine oranges stacked. If four oranges are then transferred to a basket, there will now be nine oranges left in the stack (9 – 4).

a. To estimate Bella's cost per year, we need to subtract her financial aid from the total cost of attendance. Let's assume the total cost of attendance is $30,000 per year. Then, Bella's estimated cost per year would be:

Total cost of attendance - Financial aid = Estimated cost per year

$30,000 - $9,750 - $3,100 = $17,150

Therefore, Bella should estimate her cost per year to be $17,150.

b. If Bella's family is contributing $20,000 towards her expenses each year, she needs to contribute the remaining amount. To calculate how much she needs to contribute each year, we can subtract her family's contribution from her estimated cost per year:

Estimated cost per year - Family contribution = Bella's contribution

$17,150 - $20,000 = -$2,850

It looks like Bella's family is contributing more than her estimated cost per year, so she may not need to contribute anything. However, it's important to keep in mind that these are just estimates and actual costs may vary.

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Since the limit of F(x) is infinity, we can conclude that F(x) = logx xlogx 5 grows at the same rate as logx as x approaches infinity. Therefore, F(x) = logx xlogx 5 is not ω(logx).

A function is a relation between sets that assigns to each element of a first set, exactly one element of the second set. Functions are typically written as an equation, with the first set (the domain) on the left side and the second set (the range) on the right side. The most common type of function is a function from real numbers to real numbers, which is often referred to as a real-valued function. Examples of real-valued functions include linear, polynomial, exponential, and trigonometric functions.


False. F(x) = logx xlogx 5 is not ω(logx). ω(logx) is a notation used to denote a function that grows faster than logx as x approaches infinity. However, F(x) = logx xlogx 5 grows at the same rate as logx as x approaches infinity. To prove this, we can calculate the limit of F(x) as x approaches infinity:

lim F(x) = lim (logx xlogx 5)

= lim (logx xlogx) lim 5

= ∞∞ 5

= ∞

Since the limit of F(x) is infinity, we can conclude that F(x) = logx xlogx 5 grows at the same rate as logx as x approaches infinity. Therefore, F(x) = logx xlogx 5 is not ω(logx).

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Three colours out of six can be used to create any one of 20 different varieties of Rainbow Bomb Pop.

We must apply the combination formula in order to determine how many different varieties of Rainbow Bomb Pop can be created.

The formula is as follows since we need to select three colours from a possible palette of six:

nCr = n / (n-r) r!

where r is the number of items we want to choose, n is the total number of items, and! denotes the factorial function (5! = 5x4x3x2x1, for example).

With the formula, we obtain:

6C3 = 6! / 3!(6-3)!

= 6! / 3!3!

= (6x5x4)/(3x2x1)

= 20.

Factorial in mathematics is a straightforward concept.

Factorials are only goods.

The factorial is indicated by an exclamation point.

The natural numbers that are more than it are multiplied by all the natural numbers that are less than it to get the factor.

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Answer:

tan(A+B) = 84

Step-by-step explanation:

We can use the identity: tan(A+B) = (tanA + tanB) / (1 - tanA*tanB)

Given, tanA = 4/3

So, opposite side of angle A = 4, adjacent side of angle A = 3

Using the Pythagorean theorem, we get the hypotenuse of angle A = 5

Also, sin B = 8/17

So, opposite side of angle B = 8, hypotenuse of angle B = 17

Using the Pythagorean theorem, we get the adjacent side of angle B = 15

Now, we can find the value of tanB as opposite/adjacent = 8/15

Plugging in the values in the identity for tan(A+B), we get:

tan(A+B) = (4/3 + 8/15) / (1 - (4/3)*(8/15))

= (20/15 + 8/15) / (1 - 32/45)

= 28/15 / (13/45)

= (28/15) * (45/13)

= 84

Therefore, tan(A+B) = 84.

Hope this helps!

To begin with, let's first understand what a series is. In mathematics, a series is a sum of numbers that follow a certain pattern. In this case, we have been given a series that follows the pattern of 4(4k + 3), where k is the variable that takes on different values.

Now, to find the sum of the series when k = 1, we need to plug in this value of k into the expression 4(4k + 3) and evaluate the result. So, when k = 1, we have:

4(4(1) + 3) = 4(4 + 3) = 4(7) = 28

This gives us the result of the expression when k = 1, which is 28. Therefore, the sum of the series 4(4k + 3) when k = 1 is 28.

But how do we know that this is the correct answer? To verify this, we can calculate the sum of the series manually by adding up the terms of the series for different values of k.

The given series is 4(4k + 3), so the first few terms of the series for k = 1, 2, 3, and 4 are:

k = 1: 4(4(1) + 3) = 28

k = 2: 4(4(2) + 3) = 44

k = 3: 4(4(3) + 3) = 60

k = 4: 4(4(4) + 3) = 76

If we add up these terms, we get:

28 + 44 + 60 + 76 = 208

This gives us the sum of the series for the first four terms. However, we only need to find the sum of the series when k = 1, which we already calculated to be 28.

Therefore, we can conclude that the answer we found earlier, 28, is indeed the correct sum of the series 4(4k + 3) when k = 1.

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The value of x is given as 2.045<133.158 deg

A complex number is a representation capable of being written as the combination of a and bi, where a and b exhibit themselves to be authentic numbers, while i stands as an imaginary unit that has been mathematically determined to calculate the result of -1 when squared.

The real part (a) of a complex number can be identified and contrasted against its imaginary contribution made by bi. By following certain regulations, these types of numbers are able to be increased, lessened, multiplied, and divided; providing a widely employed range of accurate calculations in mathematics, physics, engineering, and several other related fields.

Additionally, the complex plane offers a graphical means for displaying these numbers; wherein the real axis relates to the numerical form's real portion and the imaginary axis reflects the data's unreal part.

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The image is plotted and attached

The rectangle started with ABCD. Then following the reflection along line AC. point B and point D swapped so we have B' replacing D and D' replacing B.

180 degrees rotation through C, resulted to B'' D'' and A'. Point C maintains it's position since the rotation is about point C.

Enlargement by a factor of 2 results to C' B''' D''' A'' and this is the final image.

While the reflection and rotation preserves the geometry, the enlargement affects the geometry, producing a rectangle with a bigger size twice the initial size

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To find the length of the curve, we need to use the formula:

length = ∫[a,b] √[dx/dt)^2 + (dy/dt)^2] dt

In this case, a=0, b=4, and:

dx/dt = 6t
dy/dt = 6t^2

So, we can plug these values into the formula and integrate:

length = ∫[0,4] √[(6t)^2 + (6t^2)^2] dt
length = ∫[0,4] √[36t^2 + 36t^4] dt
length = ∫[0,4] 6t√(1 + t^2) dt

This integral is not easy to solve analytically, so we'll use numerical methods to approximate the answer. Using a numerical integration method such as Simpson's Rule or the Trapezoidal Rule, we can get:

length ≈ 244.36

So the exact length of the curve x = 6 + 3t^2, y = 6 + 2t^3 for 0 ≤ t ≤ 4 is approximately 244.36 units.
To find the exact length of the curve x = 6 + 3t^2, y = 6 + 2t^3 for 0 ≤ t ≤ 4, you can use the arc length formula:

Length = ∫[√(dx/dt)^2 + (dy/dt)^2] dt from t=0 to t=4

First, find the derivatives dx/dt and dy/dt:
dx/dt = 6t
dy/dt = 6t^2

Now, square the derivatives and find their sum:
(6t)^2 + (6t^2)^2 = 36t^2 + 36t^4

Take the square root of the sum:
√(36t^2 + 36t^4)

Now, integrate the expression with respect to t from 0 to 4:
Length = ∫[√(36t^2 + 36t^4)] dt from t=0 to t=4

This integral is not easy to evaluate directly, and numerical methods are usually required. To obtain an approximate value, you can use an appropriate numerical integration technique, like Simpson's Rule or a computer algebra system.

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We can convert all the fractions to decimals and then arrange them in ascending order:

- 16/22 ≈ 0.727

- -5/18 ≈ -0.278

- 2/-21 ≈ -0.095

- -7/12 ≈ -0.583

Therefore, the ascending order would be:

2/-21 ≈ -0.095 < -5/18 ≈ -0.278 < -7/12 ≈ -0.583 < 16/22 ≈ 0.727

So the final arrangement in ascending order is:

2/-21, -5/18, -7/12, 16/22

Answer:

RS = 24, so tan R = 18/24 = 3/4

Mass of O₂ produced is: B. 144

Mass of H₂O produced is: C. 180 g

The balanced chemical equation is:

2 KClO3 → 2 KCl + 3 O₂

From the equation, we can see that 2 moles of KClO₃ produce 3 moles of O2. So, 1 mole of KClO₃ produces (3/2) moles of O₂.

Therefore, 3.0 mol of KClO₃ will produce (3/2) × 3.0 = 4.5 moles of O₂.

To convert moles of O₂ to grams of O₂, we need to use the molar mass of O2, which is 32 g/mol.

So, the mass of O₂ produced is:

4.5 mol × 32 g/mol = 144 g

Answer: B. 144

The balanced chemical equation is:

2 H₂ + O₂ → 2 H₂O

We can see that 1 mole of O₂ reacts with 2 moles of H2 and produces 2 moles of H₂O.

So, 5.0 moles of O₂ will react with (2/1) × 5.0 = 10.0 moles of H₂ to produce (2/1) × 5.0 = 10.0 moles of H₂O.

To convert moles of H₂O to grams of H₂O, we need to use the molar mass of H₂O, which is 18 g/mol.

So, the mass of H₂O produced is:

10.0 mol × 18 g/mol = 180 g

Answer: C. 180

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X can be any real number, there are infinite values of c that will make the system consistent and dependent.

To have an infinite number of solutions, the system must be consistent and dependent. Thus, we need to find the value of 'c' for which the third equation is a linear combination of the first two equations.

Multiplying the second equation by 3 and adding it to the third equation, we get:

2y - z + 3(2x + 3z) = 12.0 + 3(2.7)

Simplifying, we get:

2y - z + 6x + 9z = 20.1

6x + y + 13z = 20.1

Now we have a system of two equations with three variables. To have an infinite number of solutions, one of the variables must be a free variable. Let's solve for z:

z = (20.1 - 6x - y) / 13

Now we can substitute this expression for z into the first two equations:

3x + y + 4[(20.1 - 6x - y) / 13] = c

2x + 3[(20.1 - 6x - y) / 13] = 2.7

Simplifying, we get:

39x + 13y = 52c - 321.6

39x - 6y = 41.7

To have an infinite number of solutions, the two equations must be linearly dependent. We can multiply the second equation by 13 and add it to the first equation to eliminate y:

754x = 52c - 525.9

Solving for c, we get:

c = (754x + 525.9) / 52

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The required answer is a combination of weights that can result in a zero standard deviation.

The portfolio with a standard deviation of zero is comprised of Assets A and B. This is because when two assets are perfectly negatively correlated, their returns will cancel each other out, resulting in a portfolio with no risk. On the other hand, it is not possible to determine if the portfolio with a standard deviation of zero is comprised of Assets A and C. This is because the correlation between Assets A and C is unknown, and there may not be a combination of weights that can result in a zero standard deviation.

The standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.

The portfolio with a standard deviation of zero is comprised of assets A and B. This means that the combination of these two assets has a perfectly negative correlation, leading to the elimination of the overall risk in the portfolio.The standard deviation of a population or sample and the standard error of a statistic (e.g., of the sample mean) are quite different, but related. The sample mean's standard error is the standard deviation of the set of means that would be found by drawing an infinite number of repeated samples from the population and computing a mean for each sample. The mean's standard error turns out to equal the population standard deviation divided by the square root of the sample size, and is estimated by using the sample standard deviation divided by the square root of the sample size In contrast, a portfolio of assets A and C cannot be determined, as there isn't enough information provided to establish their correlation or the possibility of achieving a standard deviation of zero.

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The probability that X+Y=1 is 0.5.

To find the probability that X+Y=1, given that X is a Bernoulli random variable with P(X=1)=0.8 and Y is a Bernoulli random variable with P(Y=1)=0.5, follow these steps:

1. First, find the probabilities for the complementary events, i.e., P(X=0) and P(Y=0).
  P(X=0) = 1 - P(X=1) = 1 - 0.8 = 0.2
  P(Y=0) = 1 - P(Y=1) = 1 - 0.5 = 0.5

2. Now, consider the two possible cases where X+Y=1:
  a) X=1 and Y=0: P(X=1) * P(Y=0) = 0.8 * 0.5 = 0.4
  b) X=0 and Y=1: P(X=0) * P(Y=1) = 0.2 * 0.5 = 0.1

3. Finally, sum the probabilities of the two cases:
  P(X+Y=1) = P(X=1, Y=0) + P(X=0, Y=1) = 0.4 + 0.1 = 0.5

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The probability that X+Y=1 is 0.5.

To find the probability that X+Y=1, given that X is a Bernoulli random variable with P(X=1)=0.8 and Y is a Bernoulli random variable with P(Y=1)=0.5, follow these steps:

1. First, find the probabilities for the complementary events, i.e., P(X=0) and P(Y=0).
  P(X=0) = 1 - P(X=1) = 1 - 0.8 = 0.2
  P(Y=0) = 1 - P(Y=1) = 1 - 0.5 = 0.5

2. Now, consider the two possible cases where X+Y=1:
  a) X=1 and Y=0: P(X=1) * P(Y=0) = 0.8 * 0.5 = 0.4
  b) X=0 and Y=1: P(X=0) * P(Y=1) = 0.2 * 0.5 = 0.1

3. Finally, sum the probabilities of the two cases:
  P(X+Y=1) = P(X=1, Y=0) + P(X=0, Y=1) = 0.4 + 0.1 = 0.5

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