Prove that if X and Y are non-negative independent random variables, then X^2 is independent of Y^2.
*** Please prove using independent random variables or variance or linearity of variance, or binomial variance.

To win the Powerball jackpot, you need to choose the correct five numbers from the integers 1-69 and pick the correct Powerball, which is one number picked from the integers 1-26. The order in which you pick the numbers is not relevant.

To calculate the number of ways to choose the correct five numbers, we use combinations. The expression for combinations is nCk, where n is the number of items available to be chosen from, and k is the number of items chosen. In this case, n = 69 and k = 5. The number of ways to choose five numbers from the integers 1-69 is calculated as:
69C5 = 69! / (5!(69-5)!) The symbol ! is called "factorial." The Factorial of a natural number is the product of the number and all natural numbers below it. For instance, 4! = 4 × 3 × 2 × 1 = 24.  So, the combination can be simplified as:
69C5 = 69! / (5!(69-5)!) = 69 × 68 × 67 × 66 × 65 / (5!) = 11,238,513 Therefore, there are 11,238,513 ways to choose the correct five numbers from the integers 1-69.

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The number of elements in set f is 161, in the union of sets e and f is 194 and the complement of set u has zero elements.

Based on the given information, we can use the formula for calculating the number of elements in a set union:

n(e ∪ f) = n(e) + n(f) - n(e ∩ f)

Using the values given, we can rearrange the formula to solve for n(f):

n(f) = n(e ∪ f) - n(e) + n(e ∩ f)

Plugging in the values, we get:

n(f) = 194 - 106 + 73 = 161

Therefore, the number of elements in set f is 161.

Next, we can use the formula for calculating the number of elements in a set intersection:

n(e ∩ f) = n(e) + n(f) - n(e ∪ f)

Using the values given, we can rearrange the formula to solve for n(e ∪ f):

n(e ∪ f) = n(e) + n(f) - n(e ∩ f)

Plugging in the values, we get:

n(e ∪ f) = 106 + 161 - 73 = 194

Therefore, the number of elements in the union of sets e and f is 194.

Finally, we can use the formula for calculating the complement of a set:

n(U\A) = n(U) - n(A)

Using the values given, we can plug in and solve for the complement of set u:

n(U\U) = n(U) - n(U) = 0

Therefore, the complement of set u has zero elements.

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The power expression (-4 + 3i)^3 in rectangular form is 117 + 44i

To find the cube of the complex number (-4 + 3i), we can expand it using the binomial theorem or by using the properties of complex conjugates.

Here's one way to do it:

(-4 + 3i)^3

= (-4 + 3i)^2 (-4 + 3i)

= (16 - 24i + 9i^2) (-4 + 3i)

= (16 - 24i - 9) (-4 + 3i) (since i^2 = -1)

= (-17 - 24i) (-4 + 3i)

= -4(-17) + 3(-17i) - 24i(-4) - 24i(3i)

= 68 - 51i + 96i + 72

= 117 + 44i

Therefore, (-4 + 3i)^3 is equal to 117 + 44i in rectangular form.

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The perfect square trinomial as a squared binomial. is B  (c - 7)²

A perfect square trinomial refer to a  trinomial (that is it has an expression with  three  perfect terms) which can be factored into a square of a binomial (which is an  expression with two terms). It has the forms below.

+a² + 2ab + b²

where  a and b are  constants.

This is  a perfect square ,  consider the square of the binomial

This indicate  that the trinomial a² + 2ab + b² is the perfect  square of the binomial a + b.

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By using special right triangles to calculate the distance, we get to know that  WX is [tex]7\sqrt{3}[/tex],  XY is equal to 7 and  YZ is equal to 7[tex]\sqrt{2}[/tex]

A triangle is said to be right-angled if one of its inner angles is 90 degrees, or if any one of its angles is a right angle. The right triangle or 90-degree triangle is another name for this triangle.

the matching for the given questions are

            1 - B : (WY-14)

            2-C : (YX-7)

            3-A : (YZ- [tex]7\sqrt{2}[/tex])

Here there are two right-angled triangles, that are WXY & YXZ.  

the length of WX is [tex]7\sqrt{3}[/tex].

here we use the trigonometry principles as we know the angle and one side length.

                    cos 30°=[tex]\frac{7\sqrt{3} }{x}[/tex]

                    [tex]\frac{\sqrt{3} }{2}[/tex]= [tex]\frac{7\sqrt{3} }{x}[/tex]

                 therefore; x=14 ⇒ WY = 14

 for knowing XY⇒

                   sin 30° = [tex]\frac{x}{14}[/tex]

                    [tex]\frac{1}{2}[/tex] = [tex]\frac{x}{14}[/tex]

                    ⇔ x=7

                  therefore, XY is equal to 7.

and finally for YZ,

                  sin 45°= [tex]\frac{7}{y}[/tex]

                    [tex]\frac{1}{\sqrt{2} }[/tex] = [tex]\frac{7}{y}[/tex]

                    therefore, y=7[tex]\sqrt{2}[/tex]

                 YZ is equal to 7[tex]\sqrt{2}[/tex]

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The equation that describes the relationship between the rate of change and the current amount of radioactive material is: dQ/dt = -kQ.

This equation represents the fact that the rate at which a radioactive material decays (dQ/dt) is proportional to the current amount of the material (Q) and is negative because the material is decreasing over time. The proportionality constant is represented by -k, where k is a positive constant.

This equation is a first-order linear differential equation that models exponential decay, which is commonly observed in radioactive materials. The solution to this equation, Q(t) = Q0 * e^(-kt), provides the amount of radioactive material remaining at any time t, given an initial amount Q0.

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aince 24 cookies is exactly half of 48 cookies you will have to half the cups of flour and since half of 3 cups is 1 1/2 cups the answer is 1 1/2 cups of flour

Alvin's total payment to the store if the donations wasn't taxed is $54.18.

Cost of shirts = $12.50

Cost of pants = $27

Cost of socks = $6.25

Donation to charity = $5

Tax = 7.5%

Total = $12.50 + $27 + $6.25

= $45.75

Total payment = $45.75 + (0.075 ×45.75) + 5

= $54.18125

Hence, the total payment Alvin made to the store is $54.18

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Answer:

Step-by-step explanation:

13x+20

The coordinates of B' after the reflection over the line y = -2 are given as follows:

B'(-2, -9).

The original coordinates of B are given as follows:

B(-2,5).

The reflection line is given as follows:

y = -2.

y = -2 is a horizontal line, hence the reflection line is obtained as follows:

Moving 7 units below the line, the coordinates of B' are given as follows:

B'(-2, - 2 - 7) = B'(-2, -9).

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Answer:

Step-by-step explanation:

Answer:

t = 3 meters

Step-by-step explanation:

From the given figure,

Perimeter = 16 meters

Now, the formula for perimeter of a rectangle = 2(l + b)

Where,

'l' is the length of the rectangle

and

'b' is the breadth of the rectangle.

Since length is the longest side of a rectangle, therefore from the given figure

=> l = 5 meters

and

=> b = t meters

Substituting values in the formula,

16 = 2(5 + t)

=> 16/2 = 5 + t

=> 8 = 5 + t

=> 8 - 5 = t

=> t = 3 meters

The transformation that will take parallelogram PQRS onto itself is given as follows:

D. a rotation of 360° counterclockwise about the center of the

parallelogram.

A rotation over a line or over a degree measure is going to change the orientation of the figure.

To keep the same orientation, the rotation must be over the measure of the circumference of a circle, which is of 360º.

Hence option D is the correct option in the context of this problem.

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Answer:

26.4 degrees.

Step-by-step explanation:

Difference = subtraction.

You have to do 21.8 - (-4.6) = 26.4.  

I'm sorry but i can't help with this BUT i can give you a graph calculator

You can use Desmos graphing calculator to plot the inequality (y\ge-x^2-1).

h t t p s : / /w w w . d e s m o s. c o m / c a l c u l a t o r

The area under the curve that lies between z = -0.36 and z = 1.68 is approximately 0.5941.

To find the area under the curve between two z-scores, we need to use a standard normal distribution table or a calculator that can calculate the cumulative distribution function (CDF) of the standard normal distribution. The CDF represents the area under the curve to the left of a given z-score.

Using a standard normal distribution table or calculator, we can find the CDF values for z = -0.36 and z = 1.68. Let's assume that the CDF value for z = -0.36 is 0.3594 and the CDF value for z = 1.68 is 0.9535.

The area under the curve between z = -0.36 and z = 1.68 can be calculated as follows:

Area = CDF(1.68) - CDF(-0.36)

Area = 0.9535 - 0.3594

Area = 0.5941

Therefore, the area under the curve that lies between z = -0.36 and z = 1.68 is approximately 0.5941. This means that the probability of observing a standard normal random variable between these two z-scores is 0.5941 or 59.41%.

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Once you find their values, yp(x) is the particular solution.

To find the particular solution for the given non-homogeneous differential equations, we use the method of undetermined coefficients.

1) y'' - 2y' + y = 7e^t*cos(t)

For this equation, we assume a particular solution of the form:
yp(t) = (Ae^t)*cos(t) + (Be^t)*sin(t)

Plug this into the given equation and solve for A and B. Once you find their values, yp(t) is the particular solution.

2) y'' - y' + 9y = 3sin(3x)

For this equation, we assume a particular solution of the form:
yp(x) = C*cos(3x) + D*sin(3x)

Plug this into the given equation and solve for C and D. Once you find their values, yp(x) is the particular solution.

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Step-by-step explanation:

1st we want to find the measure of <1 so we use cos

cos( 1 ) =18/30

cos( 1 ) =18/30 cos (1) = 0.6

cos( 1 ) =18/30 cos (1) = 0.61= cos^-1(0.6)

cos( 1 ) =18/30 cos (1) = 0.61= cos^-1(0.6)1° = 53.13° so the angle of 1 is 53.13°

2nd we can solve angle 2 by using sin

sin(2) = 18/30

sin(2) = 18/30 sin(2) = 0.6

sin(2) = 18/30 sin(2) = 0.62 = sin^-1(0.6)

2 = 36.869° round to 36.87°

so the angle 2 is 36.87°

The sum of the series is 1/2. The sequence An= (6n) / (4n+3) converges to 3/2.

The first series can be written as:

∑n=1 to [infinity] (3^n-1) / (8^n) = ∑n=1 to [infinity] [(3/8)^n - (1/8)^n]

We can simplify the series as:

∑n=1 to [infinity] (3^n-1) / (8^n) = [(3/8)^1 - (1/8)^1] + [(3/8)^2 - (1/8)^2] + [(3/8)^3 - (1/8)^3] + ...

= (3/8 - 1/8) + (9/64 - 1/64) + (27/512 - 1/512) + ...

= 2/8 + 8/64 + 26/512 + ...

=(1/4) + (1/8) + (1/32) + ...

This is a geometric series with first term a = 1/4 and common ratio r = 1/2. Since the absolute value of r is less than 1, the series converges. The sum of the series is:

sum = a / (1 - r) = (1/4) / (1 - 1/2) = (1/4) / (1/2) = 1/2

For the second sequence:

The sequence is given by An = (6n) / (4n+3).

Taking the limit as n approaches infinity, we have:

lim n→∞ An = lim n→∞ (6n) / (4n+3) = lim n→∞ (6/4 + 9/(4n+3))

As n approaches infinity, the second term goes to zero, and we are left with:

lim n→∞ An = 3/2

Thus, the sequence converges to 3/2.

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Answer:

Step-by-step explanation:

15/4 n+5.4

The abbreviated chemical reaction that summarizes RNA polymerase-directed transcription is: RNAp + DNA → mRNA + DNA'.

A chemical reaction is a process that involves the rearrangement of the molecular or ionic structure of a substance, resulting in a change in its chemical properties. During a chemical reaction, atoms are either rearranged within molecules or combined with other molecules to form new products. Chemical reactions are essential for many biological processes and are also used to produce a variety of products, such as medicines, plastics, and food additives. The reactants of a chemical reaction are the original molecules or ions before the reaction takes place, and the products are the molecules or ions formed after the reaction has occurred.

This reaction represents the process by which RNA polymerase binds to the DNA double helix, reads the genetic code, and produces a complementary mRNA molecule. The DNA molecule is then released in its original form (DNA'), allowing for the mRNA molecule to be used in translation.

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The probability that the mean run time for the 40 runners is between 141 and 143 minutes is approximately 0.4394

Probability is the measure of the likelihood of an event occurring, expressed as a number between 0 and 1.

Mean is a measure of central tendency that represents the average value of a set of numbers.

According to the given information :

Using the Central Limit Theorem, we know that the sample mean follows a normal distribution with a mean of 142 and standard deviation of 8/√40 = 1.2649. To find the probability that X is between 141 and 143, we standardize the values:

z1 = (141 - 142) / 1.2649 = -0.7925

z2 = (143 - 142) / 1.2649 = 0.7925

Using a standard normal table or calculator, we can find the area between -0.7925 and 0.7925 to be 0.4394. Therefore, the probability that the mean run time for the 40 runners is between 141 and 143 minutes is approximately 0.4394.

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Answer:

Step-by-step explanation:

when we find the volume of a cylinder we must know

the radius

the length of the cylinder-L

L^2=11^2+4^2

L^2= 144+16

L=√160

L=4√10 INCHES

                       = 3.14*16*4√10

                       = 3.14*64*2.23*1.41

                       = 632.192528

                       ≈632 CUBIC INCHES

The eigenvalues of the given matrix are -1 and 2. The eigenspace corresponding to the eigenvalue -1 is spanned by the vector [1, 2, 0], and the eigenspace corresponding to the eigenvalue 2 is spanned by the vector [1, 0, 1].

To find the eigenvalues of the given matrix, we need to solve the characteristic equation. The characteristic polynomial is given as:

|A - λI| = 0

where A is the given matrix, λ is the eigenvalue, and I is the identity matrix.

Substituting the given matrix into the characteristic equation, we get:

|[-1, 8, -4; -4, 7, -1; 8, -4, 6] - λ[1, 0, 0; 0, 1, 0; 0, 0, 1]| = 0

which simplifies to:

|[-1-λ, 8, -4; -4, 7-λ, -1; 8, -4, 6-λ]| = 0

Expanding the determinant, we get:

(-1-λ)[(7-λ)(6-λ) - (-1)(-4)] - 8[-4(6-λ) - (-1)(8)] + (-4)[-4(-4) - 8(8)] = 0

Simplifying further, we get:

(λ+1)(λ^2 - 2λ - 15) + 8(λ-2) + 4(4 - 4λ - 64) = 0

This is a cubic equation in λ. Solving for λ, we find that the eigenvalues are λ = -1, λ = 2, and λ = -3.

Next, we need to find the eigenvectors corresponding to each eigenvalue. For λ = -1, substituting λ = -1 into the matrix equation (A - λI)v = 0, where v is the eigenvector, we get:

|[-2, 8, -4; -4, 8, -1; 8, -4, 7]|v = 0

Row reducing the augmented matrix, we get:

[-2, 8, -4; -4, 8, -1; 8, -4, 7] --> [1, -4, 2; 0, 0, 1; 0, 0, 0]

The reduced row-echelon form shows that the eigenvector corresponding to λ = -1 is [1, 2, 0].

For λ = 2, substituting λ = 2 into the matrix equation (A - λI)v = 0, we get:

|[-3, 8, -4; -4, 5, -1; 8, -4, 4]|v = 0

Row reducing the augmented matrix, we get:

[-3, 8, -4; -4, 5, -1; 8, -4, 4] --> [1, -8/3, 4/3; 0, 1, -5/3; 0, 0, 0]

The reduced row-echelon form shows that the eigenvector corresponding to λ = 2 is [1, 0, 1].

Therefore, the eigenvalues of the given matrix are -1 and 2, and the corresponding eigenvectors are [1, 2,].

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Answer: it will take 7 years for the lake to have no more trout, considering the fishermen's impact and the natural population oscillation.

Step-by-step explanation:

Let's analyze the problem step by step:

1. The average trout population is 20,000.

2. The population naturally oscillates above and below average by 2,000 every year. This means the population goes up by 2,000 and then goes down by 2,000, making it a net change of 0 in the population every two years.

3. Fishermen catch 3,000 fish every year.

We want to find out how long it will take for the lake to have no more trout. Since the fishermen are catching 3,000 fish per year and the natural population oscillation has a net change of 0 over a two-year period, the main factor affecting the trout population is the number of fish caught by fishermen.

Let's denote the number of years it takes for the lake to have no more trout as "x". We can represent this situation with the following equation:

20,000 - 3,000x = 0

Now we can solve for x:

3,000x = 20,000

x = 20,000 / 3,000

x ≈ 6.67

Since we cannot have a fraction of a year, we need to round up to the next whole number to ensure that there will be no more trout left in the lake.

T satisfies property 1, and is thus a linear transformation.

T also satisfies property 2, and is a linear transformation.

To show that T is a linear transformation, we need to verify two properties:

1. T(a*u + b*v) = a*T(u) + b*T(v) for any vectors u, v in R3 and any scalars a, b.

2. T(c*u) = c*T(u) for any vector u in R3 and any scalar c.

Let's start with property 1. Suppose u = (u1, u2, u3) and v = (v1, v2, v3) are two arbitrary vectors in R3, and let a, b be scalars. Then, we have:

T(a*u + b*v) = T(a*u1 + b*v1, a*u2 + b*v2, a*u3 + b*v3)  [by definition of vector addition and scalar multiplication]

= (a*u1 + b*v1, a*u2 + b*v2, -(a*u3 + b*v3))  [by definition of T]

= (a*u1, a*u2, -a*u3) + (b*v1, b*v2, -b*v3)  [by distributivity of scalar multiplication over vector addition]

= a*(u1, u2, -u3) + b*(v1, v2, -v3)  [by definition of T]

= a*T(u) + b*T(v)  [by definition of T]

Therefore, T satisfies property 1, and is thus a linear transformation.

Now, let's check property 2. Suppose u = (u1, u2, u3) is an arbitrary vector in R3, and let c be a scalar. Then, we have:

T(c*u) = T(c*u1, c*u2, c*u3)  [by definition of scalar multiplication]

= (c*u1, c*u2, -c*u3)  [by definition of T]

= c*(u1, u2, -u3)  [by distributivity of scalar multiplication over vector addition]

= c*T(u)  [by definition of T]

Therefore, T also satisfies property 2, and is a linear transformation.

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The derivative of P with respect to a is 828/[tex]a^{11}[/tex].

Starting with the given expression using chain rule:

P = ln(92) - 9 ln(a)

We can rewrite this using the properties of logarithms:

P = ln(92) - ln([tex]a^9[/tex])

P = ln(92/[tex]a^9[/tex])

Now, we can find the derivative of P using the chain rule:

dP/da = d/dx [ln(92/[tex]a^9[/tex])] * d/dx [92/[tex]a^9[/tex]]

Using the chain rule:

dP/da = [-9/a] * [-92/[tex]a^{10}[/tex]]

Simplifying:

dP/da = 828/[tex]a^{11}[/tex]

Therefore, the derivative of P with respect to a is 828/[tex]a^{11}[/tex].

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The correct option is C)Yes, using the result from the 2014 survey only slightly increases the sample size. This means that using the result from the 2014 survey, which estimated the proportion of e-cigarette users, only slightly increases the sample size needed for the current survey.

Proportion refers to the relative or fractional amount or share of a particular characteristic or attribute within a population or sample. It is commonly expressed as a percentage or a decimal, representing the ratio of the number of individuals or items exhibiting the characteristic of interest to the total number of individuals or items in the population or sample.

According to the given information:

C. Yes, using the result from the 2014 survey only slightly increases the sample size.

The sample size required for a survey depends on several factors, including the desired confidence level, margin of error, and the estimated proportion of the population with the characteristic of interest (in this case, the current e-cigarette usage rate).

In this scenario, the confidence level is given as 99% and the margin of error as 2 percentage points. The estimated proportion of the population with the characteristic of interest is 3.74% based on the 2014 survey. Using these parameters, a sample size can be calculated using a sample size formula for estimating proportions.

The formula for calculating the sample size for estimating proportions is:

n = (Z^2 * p * (1-p)) / (E^2)

Where:

n = sample size

Z = z-score corresponding to the desired confidence level

p = estimated proportion of the population with the characteristic of interest

E = margin of error

Plugging in the given values:

Confidence level = 99% => Z = 2.62 (corresponding z-score for a 99% confidence level)

Margin of error = 2 percentage points => E = 0.02

Estimated proportion of e-cigarette users from the 2014 survey = 3.74% => p = 0.0374

Using these values in the sample size formula, we get:

n = (2.62^2 * 0.0374 * (1-0.0374)) / (0.02^2)

n ≈ 1022.8

So, the sample size required for the current survey is approximately 1023. This means that using the result from the 2014 survey, which estimated the proportion of e-cigarette users, only slightly increases the sample size needed for the current survey.

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The five-number summary of the given data set is 2, 16.5, 54.5, 64.5, 70

To find the five-number summary of the given data set, we first need to order the data in ascending order:

2, 6, 7, 27, 46, 54, 55, 58, 61, 66, 69, 70

The five-number summary includes the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum of the data set.

Minimum: The smallest value in the data set is 2.

Q1 (First quartile): The median of the lower half of the data set, which includes the values up to and including the median. To find Q1, we take the median of the first half of the data set, which is:

2, 6, 7, 27, 46, 54

The median of this set is 16.5, which is the first quartile.

Q2 (Median): The median of the entire data set is:

2, 6, 7, 27, 46, 54, 55, 58, 61, 66, 69, 70

The median of this set is the average of the two middle values, which are 54 and 55. Therefore, the median is (54 + 55) / 2 = 54.5.

Q3 (Third quartile): The median of the upper half of the data set, which includes the values from the median to the maximum. To find Q3, we take the median of the second half of the data set, which is:

55, 58, 61, 66, 69, 70

The median of this set is 64.5, which is the third quartile.

Maximum: The largest value in the data set is 70.

Therefore, the five-number summary of the given data set is:

2, 16.5, 54.5, 64.5, 70

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Answer:

Step-by-step explanation:

6000/100=60

60*5=300

300*4=1200