If X has a binomial distribution with n = 150 and the success probability p = 0.4 find the following probabilities approximately

a. P48 S X <66)
b. P(X> 69)
c. P(X 2 65)
d. P(X < 60)

The greatest common divisor of the pair of integers 19 and 1919 is 19.

the greatest common divisor (GCD) of the pair of integers you provided. The pair of integers in question is 19 and 1919.

To find the GCD, you can use the Euclidean algorithm:

1. Divide the larger integer (1919) by the smaller integer (19) and find the remainder.
  1919 ÷ 19 = 101 with a remainder of 0.

2. Since there is no remainder, the smaller integer (19) is the greatest common divisor.

So, the greatest common divisor of the pair of integers 19 and 1919 is 19.

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Answer:

x < - 4

Step-by-step explanation:

the open circle at - 4 indicates that x cannot equal - 4

the arrow and shaded part of the line to the left of - 4 indicates values that are solutions of the inequality, then the inequality representing the graph is

x < - 4

(a)Based on the given equation the value of  h = 2.8 × 10⁻⁹ m, (b) µ = 4π × 10⁻⁷ H/m, (c) µr = 1.0031, (d) m = 0.4 yaz A/m, (e) j = 0.4 yaz t, (f) jb = 0.028 y A/m², (g) jt = 0.028 t A/m²

(a) The formula to find h is h = (2 * m)/(χm * µ₀), where m is the magnetic dipole moment, χm is the magnetic susceptibility, and µ₀ is the permeability of free space. Plugging in the given values, we get h = 2.8 × 10⁻⁹ m.

(b) The formula to find µ is µ = µ₀ * (µr + χm), where µr is the relative permeability. Plugging in the given values, we get µ = 4π × 10⁻⁷ H/m.

(c) Using the same formula as in (b), we can find µr by rearranging the terms as µr = (µ/µ₀) - χm. Plugging in the values we obtained in (b), we get µr = 1.0031.

(d) The formula to find m is m = VB, where V is the volume of the material and B is the magnetic field strength. The given expression for B can be rewritten as B = 0.4 yaz A/m. Assuming the material is a cube of side length a, we get V = a³ and B = 0.4 y(a/a)z A/m = 0.4 yaz A/m. Substituting this value, we get m = 0.4 yaz A/m.

(e) The formula to find j is j = I/A, where I is the current passing through the material and A is its cross-sectional area. Since the material is a cube, its cross-sectional area is a². Using Ohm's law, we can express I as I = V/R, where V is the potential difference across the material and R is its resistance.

Assuming the material has a resistivity of ρ, we get R = (ρa)/a² = ρ/a. The potential difference across the material can be expressed as V = Bl, where l is the length of the material. Using the given expression for B, we get V = 0.4 yaz lt. Substituting these values, we get j = 0.4 yaz t.

(f) The formula to find jb is jb = σb, where σ is the conductivity of the material. The given expression for B can be rewritten as B = 0.4 y(a/a)z A/m = 0.4 yaz A/m.

Using Ohm's law, we can express σ as σ = 1/ρ, where ρ is the resistivity. Assuming the material has a cross-sectional area of a², we get jb = (1/ρ) * 0.4 yaz A/m². Substituting the given value of χm, we get jb = 0.028 y A/m².

(g) The formula to find jt is jt = σj, where σ is the conductivity of the material. Using Ohm's law, we can express σ as σ = 1/ρ, where ρ is the resistivity. Assuming the material has a cross-sectional area of a², we get jt = (1/ρ) * 0.4

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The given values modulo 45 are 192 (mod 45) = 12, 194 (mod 45) = 14, 198 (mod 45) = 18, and 1916 (mod 45) = 1.

To find the value of modulo of 192 (mod 45),

we can divide 192 by 45 and take the remainder

192 = 4 (45) + 12

So, 192 (mod 45) = 12.

To find 194 (mod 45),

we can divide 194 by 45 and take the remainder

194 = 4 (45) + 14

So, 194 (mod 45) = 14.

To find 198 (mod 45),

we can divide 198 by 45 and take the remainder

198 = 4 (45) + 18

So, 198 (mod 45) = 18.

To find 1916 (mod 45),

we can first reduce 1916 by reducing each digit

1916 = 1 (mod 45)

Therefore, 1916 (mod 45) = 1.

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The calculated value of the area of the figure  is 11326.5 sq units

From the question, we have the following parameters that can be used in our computation:

Composite figure

The shapes in the composite figure are

This means that

Area = Rectangle + Triangle

Using the area formulsa on the dimensions of the individual figures, we have

Area = 62 * 180 + 1/2 * (62 - 25) * 9

Evaluate

Area = 11326.5

Hence, the area of the figure below  is 11326.5 sq units

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Given:

AE = AC + CE = 4 + 12 = 16

BE = BD + DE = 4⅔ + 14 = 14/3 + 14 = 56/3

To Prove:

AB || CD

Now,

By converse of ∆ proportionality theorem

EC/CA = ED/DB

12/4 = 14/4⅔

3 = 14 ÷ 14/3

3 = 14 × 3/14

3 = 3

L H S = R H S

HENCE PROVED!

Answer: f(c) = 3.

Step-by-step explanation:

Since f is differentiable on the open interval (1,10), we can apply the Intermediate Value Theorem and Rolle's Theorem to draw some conclusions about the behavior of f on this interval.

I. f has at least 2 zeros.

This statement cannot be determined solely based on the given information. We know that f(2) = -5 and f(9) = -5, which means that f takes on the value of -5 at least twice on the interval (2, 9). However, we cannot conclude that f has at least 2 zeros without additional information. For example, consider the function f(x) = (x - 2)(x - 9), which satisfies the given conditions but has only 2 zeros.

II. The graph of f has at least one horizontal tangent.

This statement is true. Since f(2) = -5 and f(5) = 5, we know that f must cross the x-axis between x = 2 and x = 5. Similarly, since f(5) = 5 and f(9) = -5, we know that f must cross the x-axis between x = 5 and x = 9. Therefore, by the Intermediate Value Theorem, we know that f has at least one zero in the interval (2, 5) and at least one zero in the interval (5, 9). By Rolle's Theorem, we know that between any two zeros of f, there must be a point c where f'(c) = 0, which means that the graph of f has at least one horizontal tangent.

III. For some c, c is greater than 2 but less than 5, f(c) = 3.

This statement is false. We know that f(2) = -5 and f(5) = 5, which means that f takes on all values between -5 and 5 on the interval (2, 5) by the Intermediate Value Theorem. Since the function is continuous on this interval, it must take on all values between its maximum and minimum. Therefore, there is no value of c between 2 and 5 for which f(c) = 3.

Answer:

The volume is 1 1/15 yards^3

Step-by-step explanation:

For the volume of a rectangular prism, you use this formula: L*W*H.

In this case, we're given 2/3, 4/5, and 2.

All you have to do here is 2/3 times 2 first, and you get 4/3.

But, we're not done yet.

We also have 4/5, so we also have to multiply 4/3 by 4/5, which gives you 16/15.

It says that we can do a proper fraction or mixed number, so the answer is 1  and 1/15, or 1 1/15.

The answer is option 5: f has a local maximum at x = -4 and x = 2.

To find the local maximum for the unspecified function f, we need to follow these steps:

1. Set the derivative of the function f, denoted by f′(x), equal to 0. This is because at a local maximum, the slope of the tangent (i.e., the derivative) is 0.
2. Solve for x to find the critical points.

Given the derivative f′(x) = (3x2 + 6x - 24)(1 + g(x)2), let's set it equal to 0 and solve for x:

(3x^2 + 6x - 24)(1 + g(x)^2) = 0

Since 1 + g(x)2 is always positive (squared terms are non-negative and we are adding 1 to them), we can focus on the quadratic part:

3x^2 + 6x - 24 = 0

Now, let's factor the quadratic:

3(x^2 + 2x - 8) = 0
3(x + 4)(x - 2) = 0

Solving for x, we get:

x = -4, 2

So, there are two critical points: x = -4 and x = 2. Since the question asks for local maximum points, the correct answer is:

5. local maximum at x = -4, 2

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The PDF of of two random variable at x = a/2 is 4.851.

To find the marginal PDF of X, we integrate the joint PDF with respect to Y over the range of possible values of Y:

[tex]f_X(x)[/tex]= ∫ f(x,y) dy from y=0 to y=8.15

= ∫ c dy from y=0 to y=8.15

= c * (8.15 - 0)

= 8.15c

Since the total area under the joint PDF must be equal to 1, we know that:

∫∫ f(x,y) dxdy = 1

We can use this to find the constant c:

∫∫ f(x,y) dxdy = ∫∫ c dxdy

= c * ∫∫ dxdy

= c * (a-0) * (8.15-0)

= c * a * 8.15

= 1

Therefore,

c = 1 / (a * 8.15)

Substituting this into our expression for [tex]f_X(x)[/tex], we get:

[tex]f_X(x)[/tex] = 8.15 / a

So, for x = a/2, we have:

[tex]f_X(a/2)[/tex] = 8.15 / (a/2)

= 16.3 / a

= 4.851

Therefore, the PDF of X at x = a/2 is 4.851.

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a) The value of R2 is (1,3).

b) The value of the discriminant D = 4.

c) There is no local maximum.

d) No saddle point.

e) The maximum value of f on R2 is 3.

f) The minimum value of f on R2 is also 3

In mathematics, a saddle point is a point on the surface of a function where there is a critical point in one direction, but a minimum or maximum point in another direction. In other words, it is a point on the surface of a function where the tangent plane in one direction is a minimum, and the tangent plane in another direction is a maximum.

(A) The partial derivatives of f(x,y) are:

fx = 2x - 2

fy = 2y - 6

Setting fx = 0 and fy = 0, we get:

2x - 2 = 0

2y - 6 = 0

Solving these equations, we get the critical point (1,3).

(E) To find the maximum value of f on R2, we need to compare the value of f at the critical point (1,3) with the values of f on the boundary of the region enclosed by R2. The boundary of R2 consists of three line segments:

The line segment from (0,0) to (3,3)

The line segment from (3,3) to (3,6)

The line segment from (3,6) to (0,0)

We can parametrize each line segment and substitute it into f to get its value along the boundary. Alternatively, we can use the fact that the maximum and minimum values of a continuous function on a closed, bounded region occur at critical points or at the boundary.

Since there is only one critical point and it is a local minimum, the maximum value of f on R2 occurs on the boundary. We can calculate the value of f at each vertex of the triangle:

f(0,0) = 3

f(3,3) = 3

f(3,6) = 3

The maximum value of f on R2 is 3.

(F) Similarly, the minimum value of f on R2 occurs on the boundary. Using the same calculations as above, we find that the minimum value of f on R2 is also 3.

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Answer:

8) The distance the jet traveled is the area under the graph.

9) (1/2)(600)(20 + 25) = 13,500 miles

10) (1/2)(600)(5) = 1,500 miles

Improper integral converges and its value is 2.

We can use the integration by parts formula:

∫u dv = uv - ∫v du

where u = x^2 and dv = e^(-x) dx. Then we have

∫[tex]x^2 e^{-x} dx = -x^2 e^{-x} - 2x e^{-x} - 2 e^{-x} + C[/tex]

To evaluate the integral from 0 to infinity, we take the limit as b approaches infinity of the definite integral from 0 to b:

∫_0^∞ [tex]x^2 e^{-x}[/tex] dx = lim┬(b→∞)⁡〖∫_[tex]0^b x^2 e^{-x} dx[/tex]〗

= lim┬(b→∞)[tex][-b^2 e^{-b} - 2b e^{-b} - 2 e^{-b} + 2][/tex]

Since [tex]e^{-b}[/tex] approaches 0 as b approaches infinity, we have

lim┬(b→∞)⁡[tex][-b^2 e^{-b} - 2b e^{-b} - 2 e^{-b} + 2] = 2[/tex]

Therefore, the improper integral converges and its value is 2.

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The required answer is the given size of 23 m is smaller than the actual size.

The given size is 3.0 × 102 cubits by 5.0 × 101 cubits by 5.0 × 101 cubits. To convert cubits to feet, we can use the conversion factor 1 cubit = 1.5 ft. So, the size in feet would be:

3.0 × 102 cubits × 1.5 ft/cubit = 4.5 × 102 ft
5.0 × 101 cubits × 1.5 ft/cubit = 7.5 × 101 ft
5.0 × 101 cubits × 1.5 ft/cubit = 7.5 × 101 ft

Therefore, the size in feet is 4.5 × 102 ft by 7.5 × 101 ft by 7.5 × 101 ft.

To convert feet to meters, we can use the conversion factor 1 ft = 0.3048 m. So, the size in meters would be:

4.5 × 102 ft × 0.3048 m/ft = 137.16 m
7.5 × 101 ft × 0.3048 m/ft = 22.86 m
7.5 × 101 ft × 0.3048 m/ft = 22.86 m

Therefore, the size in meters is 137.16 m by 22.86 m by 22.86 m.
Cubits of various lengths were employed in many parts of the world in antiquity, during the Middle Ages and as recently as early modern times. The term is still used in hedgelaying, the length of the forearm being frequently used to determine the interval between stakes placed within the hedge.

Now, to answer the last part of the question, we have to compare the given sizes in feet and meters with the converted sizes. The given size in feet is 75 ft, which is smaller than the converted size of 4.5 × 102 ft. Therefore, it is true that the given size of 75 ft is smaller than the actual size.
Similarly, the given size in meters is 23 m, which is smaller than the converted size of 137.16 m. Therefore, it is also true that the given size of 23 m is smaller than the actual size.

To solve this question, we will first convert the given dimensions from cubits to feet, and then to meters.

1. Convert dimensions to feet:
- 3.0 × 10^2 cubits = 300 cubits
- 5.0 × 10^1 cubits = 50 cubits

Since 1 cubit = 1.5 ft:
- 300 cubits × 1.5 ft/cubit = 450 ft
- 50 cubits × 1.5 ft/cubit = 75 ft

2. Convert dimensions to meters:
Since 1 ft ≈ 0.3048 meters:
- 450 ft × 0.3048 m/ft ≈ 137.16 m
- 75 ft × 0.3048 m/ft ≈ 22.86 m

The dimensions in feet and meters are approximately 450 ft by 75 ft by 75 ft and 137.16 m by 22.86 m by 22.86 m.
Cubits of various lengths were employed in many parts of the world in antiquity, during the Middle Ages and as recently as early modern times. The term is still used in hedgelaying, the length of the forearm being frequently used to determine the interval between stakes placed within the hedge.

The statement "75 ft and 23 m" is false, as the correct dimensions are 75 ft and approximately 22.86 m.

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The terms of the sequence continue to increase without bound, we can say that the sequence is not bounded.

To determine whether the sequence is increasing or decreasing, we need to compare consecutive terms of the sequence.

For n = 1, a1 = 6(1) + 2 = 8

For n = 2, a2 = 6(2) + 2 = 14

For n = 3, a3 = 6(3) + 2 = 20

Since each term of the sequence is greater than the previous one, we can say that the sequence is increasing.

To determine if the sequence is bounded, we need to check if it approaches infinity or if it has a finite upper and lower bound. Since the terms of the sequence continue to increase without bound, we can say that the sequence is not bounded.

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The quadratic function with the given features is defined as follows:

y = 0.86x² - 5.86x + 5.

The standard definition of a quadratic function is given as follows:

y = ax² + bx + c.

When x = 0, y = 5, hence the coefficient c is given as follows:

c = 5.

Hence:

y = ax² + bx + 5.

When x = 1, y = 0, hence:

a + b + 5 = 0

a + b = -5.

The discriminant is given as follows:

D = b² - 4ac.

Hence:

D = b² - 20a

The minimum value is of -4, hence:

-D/4a = -5

(b² - 20a)/4a = -5

b² - 20a = 20a

b² = 40a

Since a = -5 - b, we have that the value of b is obtained as follows:

b² = 40(-5 - b)

b² + 40b + 200 = 0.

b = -5.86.

Hence the value of a is of:

a = -5 + 5.86

a = 0.86.

Then the equation is:

y = 0.86x² - 5.86x + 5.

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You should originally invest $148.

To solve this problem, we need to use the formula for present value of a bond:

[tex]PV = C/(1+r)^t[/tex]

where PV is the present value, C is the annual coupon payment, r is the annual interest rate, and t is the time to maturity in years.

We know that the General Electric bonds had a time to maturity of 87 years and an annual rate of 5.25%. We also know that they paid a total of $6,630 at maturity. Let's call the original investment amount X.

Using the formula, we can set up the following equation:

[tex]6,630 = X/(1+0.0525)^{87[/tex]

Simplifying this equation, we get:

[tex]X = 6,630 * (1+0.0525)^{-87[/tex]

Using a calculator, we get:

X = $147.91

Rounding this to the nearest dollar, the answer is:

$148

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The Answer is True.

The lognormal distribution is commonly used to model data that follows a log-transformed distribution. Taking the logarithm of a variable can often help to transform skewed or highly variable data into a more normal distribution, which can make it easier to analyze statistically.

Therefore, log10 transformation is a common technique used to create a log distribution for data that has a large range of values.

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Answer:

A. There is a strong negative correlation between the number of minutes played and the number of tokens used.

Hope this helps!

Answer:

A

Step-by-step explanation:

sorry im in a rush bye gtg :D

The test results suggest that there is not enough evidence to reject the null hypothesis.

A hypothesis test is a statistical method used to determine whether an assumption about a population parameter can be supported by sample data. In this case, the Small Business Administration hypothesized that 80% of franchisees own only one location. They collected a random sample of 120 franchisees and found that 85 owned only one store. To determine whether this sample result supports their hypothesis, they performed a hypothesis test with a significance level of 0.05.

The critical value for this test was determined based on the desired level of confidence, which in this case was 95%. The calculated test statistic was then compared to this critical value to determine whether the null hypothesis (that 80% of franchisees own only one location) can be rejected.

In this scenario, the calculated test statistic fell within the confidence interval, indicating that the null hypothesis cannot be rejected based on the sample data. This means that there is not enough evidence to support the claim that franchisers favor owners who operate multiple stores, at least not to the extent that it would significantly impact the distribution of franchise ownership.

It's important to that while the sample data may not support the hypothesis, it's possible that the true population parameter could still differ from the hypothesized value. However, based on the available data and the results of the hypothesis test, it appears that there is not enough evidence to support the claim that franchisers favor multi-store owners.

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The resulting matrix using the specified row transformation to change the matrix; - 4 times row 1 added to row 2 is 0 -8

To apply the specified row transformation, we need to subtract 4 times the first row from the second row.

So the resulting matrix will be:

[  2              3

8 - 4 ( 2 )   4 - 4 ( 3 ) ]

which simplifies to:

[ 2   3

0   -8 ]

Therefore the resulting matrix for the specified row transformation is 0 and - 8.

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The repeated measures test, also known as a within-subjects or paired design, decreases the influence of individual differences compared to an independent t-test. The correct answer is c. paired/independent.

A repeated measures test decreases variability between subjects because it is a within-subjects design, meaning that each participant is measured multiple times under different conditions. This reduces the variability between participants and increases the power of the test. In contrast, an independent t-test is a between-subjects design and compares the means of two independent groups, resulting in more variability between subjects. The type of test statistic used depends on the design of the study - a within design uses a paired test statistic, while a between design uses an independent test statistic. Therefore, the answer is c. paired/independent.

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Answer:

Tooo mny

Step-by-step explanation:

I think, you need to add al the sides then subtract by 180

Hi! The dy/dx using implicit differentiation for the given equation is (4x + y*sin(x)) / (cos(x) - 8y).

The given equation is y cos(x) = 2x^2 + 4y^2

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Answer:

see explanation

Step-by-step explanation:

(a)

given y is directly proportional to [tex]\sqrt{x}[/tex] , then the equation relating them is

y = k[tex]\sqrt{x}[/tex] ← k is the constant of proportion

to find k use the condition y = 4 when x = 1

4 = k[tex]\sqrt{1}[/tex] = k

y = 4[tex]\sqrt{x}[/tex] ← equation of proportion

(b)

when x = 36 , then

y = 4 × [tex]\sqrt{36}[/tex] = 4 × 6 = 24

(c)

when y = 36 , then

36 = 4[tex]\sqrt{x}[/tex] ( divide both sides by 4 )

9 = [tex]\sqrt{x}[/tex] ( square both sides to clear the radical )

9² = ([tex]\sqrt{x}[/tex] )² , then

81 = x

The equation y has two outputs for each input of x, which proves that y²= 4x+16 is not a function.

A function is a relation between two sets of values such that each element of the first set is associated with a unique element of the second set.

In this case, y²= 4x+16 is an equation that is not a function as it does not satisfy the definition of a function.

It does not meet the criteria of having a unique output for each input. For example, when x = 0, the equation yields y²= 16.

Since y can be both positive and negative, there are two outputs for the same input. This violates the definition of a function and therefore this equation is not a function.

This can be proven mathematically by rearranging the equation to solve for y.

y²= 4x+16

y² -4x= 16

y² -4x+4= 16+4

(y-2)²= 20

y= ±√20 + 2

This equation shows that y has two outputs for each input of x, which proves that y²= 4x+16 is not a function.

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a. The null hypothesis for this statement would be that the number of children the younger sister has is not dependent on the number of children the older sister has.

b. The null hypothesis for this statement would be that the distribution of family sizes for older and younger sisters is the same.

For a, The alternative hypothesis would be that there is a dependency between the two variables. This hypothesis can be tested using a chi-squared test for independence.

For b,The alternative hypothesis would be that the distributions are different. This hypothesis can be tested using a two-sample t-test for comparing means or a chi-squared test for comparing proportions.

Both hypotheses can be true or false independently. It is possible that the number of children the younger sister has is independent of the number of children the older sister has, but the distribution of family sizes could be different for older and younger sisters. Conversely, it is also possible that the number of children the younger sister has is dependent on the number of children the older sister has, but the distribution of family sizes is the same for both.

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Complete Question:  

A sociologist is studying influences on family size. He finds pairs of sisters, both of whom are married, and determines for each sister whether she has 0, 1, or 2 or more children. He wants to compare older and younger sisters. Explain what the following hypotheses mean and how to test them.

   a. The number of children the younger sister has is independent of the number of children the older sister has.

     b. The distribution of family sizes is the same for older and younger sisters. Could one hypothesis be true and the other false? Explain.

Answer:

The answer is 0.

Step-by-step explanation:

[tex] log_{2}(32) = 5[/tex]

[tex] log_{5}(5) = 1[/tex]

[tex] log_{3}(1) = 0[/tex]

Let's consider the statement P: "For all x, y ∈ Z, if xy = 0, then x = 0 or y = 0."

(a) The negation of P is: "There exist x, y ∈ Z such that xy = 0 and x ≠ 0 and y ≠ 0."

(b) The contrapositive of P is: "For all x, y ∈ Z, if x ≠ 0 and y ≠ 0, then xy ≠ 0."

(c) To prove P, consider the original statement. If xy = 0 and either x or y is nonzero, then the product of the nonzero integer with the zero integer must be zero. Since the product of any integer and zero is always zero, the statement P holds true.

(d) The converse of P is: "For all x, y ∈ Z, if x = 0 or y = 0, then xy = 0." To prove the converse, consider the two cases where either x or y is zero. If x = 0, then xy = 0 * y = 0. If y = 0, then xy = x * 0 = 0. In both cases, the product xy is zero, proving the converse to be true.

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Substituting in our expression for Σ, we get:

[tex]√n (r1)( . )( . )( rk) ~ Nk(0, (1/n) σ^2 Ik)[/tex]

This is the desired result.

If [tex]Xt[/tex]is an ARMA(p, q) process with [tex]p = q = 0,[/tex] then Xt is just white noise. In this case, Bartlett's formula gives us the asymptotic distribution of the sample autocorrelation coefficients, which can be written as:

[tex]√n (r1)( . )( . )( rk) ~ Nk(0, Ik)[/tex]

where [tex]r1, ..., rk[/tex] are the sample autocorrelation coefficients at lags 1 through [tex]k, √n[/tex] is the scaling factor, and Nk(0, Ik) is the multivariate normal distribution with mean 0 and identity covariance matrix.

To show this result, we can use the properties of white noise to derive the mean and covariance of the sample autocorrelation coefficients. For white noise, the sample mean is zero and the sample variance is constant. Therefore, we have:

E[tex](rj) = 0 for j = 1, ..., k[/tex]

Var [tex](rj) = 1/n for j = 1, ..., k[/tex]

To find the covariance between rj and rk, we use the fact that white noise has no autocorrelation at non-zero lags. Therefore, we have:

Cov [tex](rj, rk) = E[rjrk] - E[rj]E[rk][/tex]

Since Xt is white noise, we have:

E[tex][Xt] = 0[/tex] for all t

Cov [tex](Xt, Xs) = 0 for t ≠ s[/tex]

Therefore, we can write:

E[tex][rjrk] = E[(1/n) ∑(t=1)^(n-j) Xt Xt+j (1/n) ∑(t=1)^(n-k) Xt Xt+k]= (1/n^2) ∑(t=1)^(n-j) ∑(s=1)^(n-k) E[XtXt+jXsXs+k]= (1/n^2) ∑(t=1)^(n-j) E[XtXt+jXt+j+tXt+j+k]= (1/n^2) ∑(t=1)^(n-j) E[XtXt+j]E[Xt+j+tXt+j+k]= (1/n^2) ∑(t=1)^(n-j) Var(Xt)δ(t+k-j)[/tex]

where δ(i) is the Kronecker delta function, which is equal to [tex]1 if i = 0[/tex] and 0 otherwise. Using the fact that Var[tex](Xt) = σ^2[/tex] for all t, we can simplify this expression to:

E[tex][rjrk] = (1/n) σ^2 δ(k-j)[/tex]

Therefore, we have:

[tex]Cov(rj, rk) = E[rjrk] - E[rj]E[rk] = (1/n) σ^2 δ(k-j)[/tex]

Putting this together, we can write the covariance matrix of the sample autocorrelation coefficients as:

[tex]Σ = (1/n) σ^2 Ik[/tex]

where Ik is the k x k identity matrix. Therefore, the asymptotic distribution of the sample autocorrelation coefficients is:

[tex]√n (r1)( . )( . )( rk) ~ Nk(0, Σ)[/tex]

Substituting in our expression for Σ, we get:

[tex]√n (r1)( . )( . )[/tex]

[tex]( rk) ~ Nk(0, (1/n) σ^2 Ik)[/tex]

This is the desired result.

To learn more about autocorrelation visit:

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