Write an equation of the line passing through the point (8,3) that is parallel to the line y=-3/4+4

(a) The probability that X = n is 2^n-1 / 2^N - 1.

(b) ɸ(t) = E[2^tx] = ∑_{n=1}^N 2^tx P(X=n) = ∑_{n=1}^N 2^tx (2^n-1 / 2^N - 1) = (2^t / 2^N - 1) ∑_{n=1}^N 2^(n-1)t = (2^t / 2^N - 1) (2^Nt - 1) / (2^t - 1) = 2^Nt / (2^N - 1)

(c) The probability that X = n and Y = m is 2^(n-m-1) / 2^N - 1.

(d) This is the same as finding the probability that the difference between the maximum and minimum elements of a subset of {1,...,N} is k. If k = 0, then the only subsets with maximum and minimum elements differing by k are the single-element subsets, so P(W=k) = N / 2^N - 1. If k > 0, then there are (N-k) choices for the minimum element m and 2^(k-1) subsets of {m+1,...,m+k-1}, so P(W=k) = (N-k) 2^(k-1) / 2^N - 1.

To find the probability mass function of X, we need to find the probability that X = n for each n ∈ {1,...,N}. This is the same as finding the probability that the maximum element of a subset of {1,...,N} is n. There are 2^n-1 subsets of {1,...,n-1}, and each of these subsets can be combined with n to form a subset of {1,...,N} with maximum element n. Therefore, the probability that X = n is 2^n-1 / 2^N - 1.

To find the value of ɸ(t) for any t ∈ R, we need to compute the expected value of 2^tx. Using the formula for expected value, we get:

ɸ(t) = E[2^tx] = ∑_{n=1}^N 2^tx P(X=n) = ∑_{n=1}^N 2^tx (2^n-1 / 2^N - 1) = (2^t / 2^N - 1) ∑_{n=1}^N 2^(n-1)t = (2^t / 2^N - 1) (2^Nt - 1) / (2^t - 1) = 2^Nt / (2^N - 1)

To find the joint probability mass function of (X,Y), we need to find the probability that X = n and Y = m for each n,m ∈ {1,...,N}. If n = m, then the only subset of {1,...,N} with maximum element n and minimum element m is {n}, so P(X=n, Y=m) = 1 / 2^N - 1. If n ≠ m, then there are 2^(n-m-1) subsets of {m+1,...,n-1}, and each of these subsets can be combined with m and n to form a subset of {1,...,N} with maximum element n and minimum element m. Therefore, the probability that X = n and Y = m is 2^(n-m-1) / 2^N - 1.

To find the probability mass function of W = X - Y, we need to find the probability that W = k for each k ∈ {0,...,N-1}. This is the same as finding the probability that the difference between the maximum and minimum elements of a subset of {1,...,N} is k. If k = 0, then the only subsets with maximum and minimum elements differing by k are the single-element subsets, so P(W=k) = N / 2^N - 1. If k > 0, then there are (N-k) choices for the minimum element m and 2^(k-1) subsets of {m+1,...,m+k-1}, so P(W=k) = (N-k) 2^(k-1) / 2^N - 1.

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Answer:

the answer is A

So basically it's simple just use some scientific calculator and put the number one by one with the bracket because there's a rule of mathematics which we call BODMAS

According to the information, the expressions that would match the descriptions would be both expressions have the same result, so it doesn't matter which description they are associated with. both are the largest and the smallest value.

To find the correct descriptions that match the expressions we must look at the graph. In this case q and n represent two numbers on the number line. In this case, to relate them to a description we must find the number to which they refer:

In accordance with the above, we could say that the descriptions would look like this:

In this case, both expressions have the same result, so it doesn't matter which description they are associated with. both are the largest and the smallest value.

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airplane A

To compare the rate of speed of Airplane A and Airplane B, we need to calculate the speed of Airplane B first.

The formula to calculate distance is:

distance = rate x time

For Airplane A, we know:

distance = 1662 miles

time = 6 hours

So, we can rearrange the formula to solve for the rate:

rate = distance / time = 1662 / 6 = 277 miles per hour

For Airplane B, we know:

y = 922/3

x = rate of speed over 6 hours

We can use the formula to solve for x:

distance = rate x time

922/3 = x * 6

x = (922/3) / 6 = 153.67 miles per hour

Comparing the rates of the two airplanes, we can see that Airplane A traveled at a faster rate with 277 miles per hour compared to Airplane B's rate of 153.67 miles per hour. Therefore, Airplane A traveled at a faster rate.

Assuming this is a simple interest situation:

The 8.75% account will earn (0.0875)(5000) each year.
     $437.5 earned per year

The 9.5 % account will earn (0.095)(7000) each year

     $665 earned per year

Together, these accounts earn $1102.5 per year.

If x is the number of years until the desired interest is earned, then you need to solve 1102.5x = 8820.  Dividing by 1102.5 on both sides, you find x=8.

Again, assuming this is simple interest and not compound interest, it will take 8 years.

The possible rational roots of f(x) are ±1, ±2, ±4, ±8, ±1/2, and ±1/4.

The Rational Root Theorem states that if a polynomial [tex]f(x) = anxn + an-1xn-1 + ... + a1x + a0[/tex] has a rational root, then it must be of the form p/q, where p is a factor of the constant term a0 and q is a factor of the leading coefficient an.

For the given polynomial [tex]f(x) = -x + 14x3 - 18x2 - 4x5 - 26x4 + 8[/tex], the constant term is 8 and the leading coefficient is -4.

The factors of 8 are ±1, ±2, ±4, and ±8. The factors of -4 are ±1, ±2, and ±4.

Therefore, the possible rational roots of f(x) are:

p/q = ±1/1, ±2/1, ±4/1, ±8/1, ±1/2, ±2/2, ±4/2, ±8/2, ±1/4, ±2/4, ±4/4, ±8/4

Simplifying gives us the possible rational roots:

±1, ±2, ±4, ±8, ±1/2, ±4/2, ±8/2, ±1/4, ±2/4, ±8/4

Simplifying further gives us the final list of possible rational roots:

±1, ±2, ±4, ±8, ±1/2, ±2, ±4, ±1/4, ±1/2, ±2

Removing duplicates, the final list of possible rational roots is:

±1, ±2, ±4, ±8, ±1/2, ±1/4

Therefore, the possible rational roots of f(x) are ±1, ±2, ±4, ±8, ±1/2, and ±1/4.

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Answer:

Step-by-step explanation:

Im not really sure what the heck you doing because im only in 8th grade but right now im in central and i'm kind of good at math but i'll try. Lets see........ I think they would all just be $11.50, but like i said im only in 8th grade and not that good at this kind of math so before you answer it please check with someone else to see if i got it right for you, cause i don't want to give you the wrong answer and you fail it.

Class B has higher percentage of girls than class A that is 26.6 % more.

A percentage is a number or ratio that can be expressed as a fraction of 100. If you need to calculate the percentage of a number, divide the number by the whole number and multiply by 100. Percentages therefore mean 1 in 100.

Given,

Total number of students in class A = 60

Number of girls student in class A = 28

percentage of girls in class A

= (28/60)×100

= 0.467 × 100

= 46.7%

Total number of students in class B = 60

Number of girls student in class B = 44

percentage of girls in class A

= (44/60)×100

= 0.733 × 100

= 73.3 %

Difference in percentage of girls in class A and class B.

= 73.3% - 46.7%

= 26.6%

Hence, class B has 26.6 percent more girls than class A.

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Answer: 10,210$

Step-by-step explanation:
Its simple really. The formula is 800 x 2.65 x 1. 10,210.

Use this next time, good luck :)

Answer: $879.50

Step-by-step explanation:

The simple interest formula is I = prt

Plug values in:

(Percent move decimal over 2 and time has to be in years, so 3 years and 9 months is 3.75 years)

I = (800)(0.0265)(3.75)

I = (21.2)(3.75)

I = 79.5

Add the interest to the principle:

800 + 79.5 = $879.50

Hope this helps!

The Complete value for the Table is

x               f(x)                   g(x)

1                8                         1.1

5               20                  1.61051

10              35                  2.5937

20             65                   6.7274

The function f(x) grows with faster rate.

A function is an expression, rule, or law in mathematics that describes a relationship between one variable (the independent variable) and another variable (the dependent variable).

Given:

We have f(x) = 3x+ 5 and g(x) = [tex](1.1)^x[/tex]

The Complete value for the Table is

x                   f(x)                                           g(x)

1               3x+ 5 = 3(1) +5= 8                         1.1

5              3x+ 5 = 3(5) +5= 20                      [tex](1.1)^5[/tex]= 1.61051

10             3x+ 5 = 3(10) +5= 35                    [tex](1.1)^{10}[/tex]= 2.5937

20            3x+ 5 = 3(20) +5= 65                    [tex](1.1)^{20}[/tex]= 6.7274

So, the function f(x) grows with faster rate.

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Answer: (0, 2)

Step-by-step explanation:

The solution of a system of equations is where the lines intersect at.

We will use substitution to solve the system.

Equations:

y = 1/2x + 2

y = -1/5x + 2

Set both equations to equal each other:

1/2x + 2 = -1/5x + 2

Simplify:

7/10x = 0

x = 0

Plug 0 back in:

y = 1/2(0) + 2

y = 0 + 2

y = 2

The solution is (0, 2)

(This can also be seen by looking at the graph)

Hope this helps!

Answer:

The volume of a sphere with a diameter of 8.6 m = 333.0 m^3

Step-by-step explanation:

Answer:

Step-by-step explanation:

x + 4 + 2x + 23 = 90

3x + 27 = 90

3x = 63

x = 21

x + 4 = 21 + 4 = 25 for <1

2(21) + 23  = 42 + 23 = 65 for <2

The final expression after collecting terms is: 19zx^(2) - 3z^(2)x + 2z^(2)y

Explanation: To collect terms in an expression, we need to combine like terms. Like terms are terms that have the same variables with the same exponents. In the given expression, the like terms are 13zx^(2) and 6zx^(2). We can combine these terms by adding their coefficients:

13zx^(2) + 6zx^(2) = 19zx^(2)

The other terms, 2z^(2)y and -3z^(2)x, do not have any like terms, so we cannot combine them with any other terms. Therefore, the final expression after collecting terms is:

19zx^(2) - 3z^(2)x + 2z^(2)y

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The total number of months he made deposits and multiply that by the monthly deposit amount.

Chase started an IRA at the age of 26 and deposited $400 each month until he retired at the age of 65. To determine the amount of money Chase deposited over the length of the investment, we need to calculate the total number of months he made deposits and multiply that by the monthly deposit amount.

Number of months = (Retirement age - Starting age) * 12
Number of months = (65 - 26) * 12
Number of months = 468

Total deposits = Number of months * Monthly deposit amount
Total deposits = 468 * 400
Total deposits = $187,200

To determine how much Chase made in interest upon retirement, we need to subtract the total deposits from the retirement savings.

Interest = Retirement savings - Total deposits
Interest = $838,879.58 - $187,200
Interest = $651,679.58

Therefore, Chase deposited a total of $187,200 over the length of the investment and made $651,679.58 in interest upon retirement.

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The simplified expression is u^(3) - v^(3).

To perform the indicated operation on the algebraic expressions, we need to multiply each term in the first expression by each term in the second expression and then simplify the resulting expression.

Step 1: Multiply each term in the first expression by each term in the second expression:

(u)(u^(2)) + (u)(uv) + (u)(v^(2)) - (v)(u^(2)) - (v)(uv) - (v)(v^(2))

Step 2: Simplify the resulting expression by combining like terms:

u^(3) + u^(2)v + uv^(2) - u^(2)v - uv^(2) - v^(3)

Step 3: Simplify further by canceling out terms that are equal but opposite in sign:

u^(3) - v^(3)

Therefore, the simplified expression is u^(3) - v^(3).

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The amount of Kingston's birth weight that was much bigger than Karmichael would be = 1 pound 4 ounces

The weight of Kingston at birth = 9 pounds 2 ounces

The weight of Karmichael at birth = 7 pounds 11 ounces.

Next, the weights are converted to ounces such as:

1 pound = 16 ounces

For 9 pounds = 16×9 = 144 ounces+ 2 ounces = 146 ounces

For 7 pounds = 16×7 = 112 ounce + 11 ounces = 123 ounce

The difference between the birth weight of Kingston and Karmichael = 146-123 = 23 ounce

Convert 23 ounce to pounds = 23/16 = 1 pound 4 ounces

Therefore, amount of Kingston's birth weight that was much bigger than Karmichael would be = 1 pound 4 ounces.

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Answer:

D

Step-by-step explanation:

Mengxi invested $4,200 in the term deposit paying 5% /year and $5,800 in Canada Savings Bonds paying 3.5% /year. The total interest earned is $413.

Let's assume that Mengxi invests x dollars in the term deposit paying 5% /year, and the remaining (10000 - x) dollars in Canada Savings Bonds paying 3.5% /year.

At the end of the year, Mengxi earns a total of $413 in simple interest. The interest earned from the investment in the term deposit is calculated as 0.05x, while the interest earned from the investment in Canada Savings Bonds is 0.035(10000 - x).

Thus, we can write the following equation to represent the total interest earned:

0.05x + 0.035(10000 - x) = 413

Simplifying the equation, we get:

0.015x + 350 = 413

0.015x = 63

x = 4200

Therefore, Mengxi invested $4,200 in the term deposit paying 5% /year, and $5,800 (10000 - 4200) in Canada Savings Bonds paying 3.5% /year.

To check the answer, we can calculate the interest earned from each investment and add them up:

0.05(4200) + 0.035(5800) = 210 + 203 = 413

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In answering the question above, the solution is If you know the lengths Pythagorean theorem of the other two sides of the right triangle, you may use these formulae to determine the length of the missing side.

The fundamental Euclidean geometry relationship between the three sides of a right triangle is the Pythagorean Theorem, sometimes referred to as the Pythagorean Theorem. This rule states that the areas of squares with the other two sides added together equal the area of the square with the hypotenuse side. According to the Pythagorean Theorem, the square that spans the hypotenuse (the side that is opposite the right angle) of a right triangle equals the sum of the squares that span its sides. It may also be expressed using the standard algebraic notation, a2 + b2 = c2.

[tex]a^2 + b^2 = c^2[/tex]

where a and b are the lengths of the other two sides, and c is the length of the hypotenuse.

c = √[tex](a^2 + b^2)[/tex]

You may rewrite the formula as follows to get the length of one of the other sides:

a = √[tex](c^2 - b^2)[/tex]

b = √[tex](c^2 - a^2)[/tex]

If you know the lengths of the other two sides of the right triangle, you may use these formulae to determine the length of the missing side.

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Answer: 1

Step-by-step explanation:

3(x-2) = 4x+2

3x-6 = 4x+2

To move 3x, deduct it on both sides of the equation.

3x - 6 - (3x) = 4x + 2 - (3x)

0 - 6 = [1]x +2

Given:-

[tex] \: [/tex]

Solution:-

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

━━━━━━━━━━━━━━━━━━━━━━━

hope it helps ⸙

a) Null hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are not related variables. Alternative hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are related variables. So the option E is correct.

b) There is insufficient information to draw the conclusion that the variables are connected even though the value is greater than the 0.05 threshold for significance.

c) The difference between these percentage are fairly similar which suggests that sex and risk of lower-extremity Injury in  interscholastic soccer are not related.

a) The null and alternative hypotheses about sex and the risk of a lower-extremity Injury while playing interscholastic soccer is:

Null hypothesis: Sex and lower-extremity injury risk in interscholastic soccer are unrelated factors. Alternative hypothesis: sex and the likelihood of sustaining a lower-extremity injury in collegiate soccer are associated variables.

So the option E is correct.

b) The conclusion is that there is no significant relationship between the gender of the athlete and the risk of lower-extremity injuries. This conclusion is based on the fact that the chi-square statistic (1.63) and the corresponding p-value (0.202) are both greater than the 1.63 X standard for significance. This indicates that there is not enough evidence to show that the variables are related.

c) Of 997 participants in girls' soccer, 79 experienced lower-extremity injuries. Of 1,664 participants in boys' soccer, 156 experienced lower-extremity injuries.

Girls = 79/997 × 100 = 7.92%

Boys = 156/1664 × 100 = 9.38%

The difference between the two percentages is consistent with the conclusion that there is no significant relationship between the gender of the athlete and the risk of lower-extremity injuries because the difference between the two percentages is small.

This indicates that the risk of lower-extremity injuries is relatively similar for both boys and girls, which is consistent with the conclusion that there is no significant relationship between gender and the risk of lower-extremity injuries.

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The complete question is:

Researchers studied a random sample of high school students who participated in interscholastic athletics to learn about the risk of lower-extremity injuries (anywhere between hip and toe) for interscholastic athletes. Of 997 participants in girls' soccer, 79 experienced lower-extremity injuries. Of 1,664 participants in boys' soccer, 156 experienced lower-extremity injuries.

(a) Write null and alternative hypotheses about sex and the risk of a lower-extremity Injury while playing interscholastic soccer.

A. Null hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are related variables. Alternative hypothesis: Sex and risk of a lower-extremity Injury in interscholastic soccer are not related variables.

B. Null hypothesis: Sex and risk of a lower extremity injury in interscholastic soccer are explanatory variables. Alternative hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are not explanatory variables.

C. Null hypothesis: Sex and risk of a lower extremity injury in interscholastic soccer are not explanatory variables. Alternative hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are explanatory variables.

D. Null hypothesis: There is a weak relationship between sex and risk of a lower-extremity injury in interscholastic soccer. Alternative hypothesis: There is a strong relationship between sex and risk of a lower-extremity injury in interscholastic soccer

E. Null hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are not related variables. Alternative hypothesis: Sex and risk of a lower-extremity injury in interscholastic soccer are related variables.

(b) For these data, the value of the chi-square statistic is 1. 63, and the p-value for the chi-square test is 0. 202. Based on these results, state a conclusion about the two variables in this situation and explain how you came to this conclusion. The p-value is greater than the 1. 63 X standard for significance, so there is not sufficient evidence to be able to conclude that the variables are related.

(c) For each sex separately, calculate the percent of participants who had a lower-extremity injury. (Round your answers to one decimal place. ) girls 7.9% boys 9.4%. Explain how the difference between these percentages is consistent with the conclusion you stated in part (b). These percentages are fairly similar , which suggests that sex and risk of a lower-extremity injury in interscholastic soccer are not related

Answer:

x=36

y=6

system of equations

The solution of the inequaltiy is v <= 6 or v >= 2.

To solve the compound inequality 4v+3<=27 or 2v-4>=0, we need to solve each inequality separately and then combine the solutions.

For the first inequality, 4v+3<=27, we can isolate the variable on one side of the inequality by subtracting 3 from both sides:

4v <= 24

Next, we can divide both sides by 4 to get:

v <= 6

For the second inequality, 2v-4>=0, we can isolate the variable on one side of the inequality by adding 4 to both sides:

2v >= 4

Next, we can divide both sides by 2 to get:

v >= 2

Now, we can combine the solutions to get the final solution for the compound inequality:

v <= 6 or v >= 2

This means that the solution set includes all values of v that are less than or equal to 6 or greater than or equal to 2.

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I chose the Quadratic Formula and Graphing because they are both straightforward methods that can give us the exact solutions.

A quadratic equation in the form ax^2 + bx + c = 0 where a > 1 is 2x^2 + 5x + 3 = 0.

To find the number of solutions based on the discriminant, we can use the formula D = b^2 - 4ac. In this case, D = (5)^2 - 4(2)(3) = 25 - 24 = 1. Since D > 0, the quadratic equation has two distinct real solutions.

Two ways to find the specific solutions or show that there is no solution are the Quadratic Formula and Graphing.

The Quadratic Formula is x = (-b ± √D)/2a. Plugging in the values from the equation, we get x = (-5 ± √1)/4 = (-5 ± 1)/4. This gives us the two solutions x = -1 and x = -2.

Graphing is another way to find the solutions. We can graph the equation y = 2x^2 + 5x + 3 and find the x-intercepts, which are the solutions to the equation. The graph shows that the x-intercepts are -1 and -2, which are the same solutions we found using the Quadratic Formula.

I chose the Quadratic Formula and Graphing because they are both straightforward methods that can give us the exact solutions. The Quadratic Formula is a formula that can be applied to any quadratic equation, and Graphing allows us to visually see the solutions. The other methods, such as Factoring, Square Root Property, and Completing the Square, may not always be applicable or may require more steps to find the solutions.

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The angle opposite to the side length c is given as follows:

θ = 53º.

The Law of Cosines is a trigonometric formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is also known as the Cosine Rule.

The Law of Cosines states that for any triangle with sides a, b, and c and angle C opposite to side c, the following equation holds true:

c^2 = a^2 + b^2 - 2ab cos(C)

The parameters for this problem are given as follows:

a = 122.5, b = 58.2, c = 164.5.

Hence the measure of angle θ is given as follows:

c^2 = a^2 + b^2 - 2ab cos(θ)

164.5² = 122.5² + 58.2² - 2 x 122.5 x 58.2 x cos(θ)

14259cos(θ) = 8666.76

cos(θ) = 8666.76/14259

θ = arccos(8666.76/14259)

θ = 53º.

The angle is opposite to the side length C.

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Answer:

990 applicants

Step-by-step explanation:

We know

A university law school accepts 4 out of every 11 applicants.

If the school accepted 360 ​students, find how many applications they received.

To get from 4 to 360, we time 90

We take 11 times 90 = 990 applicants

So, they received 990 applicants.

To prove that EVPI > 0 for the simple decision sapling, we need to understand what EVPI is. EVPI stands for Expected Value of Perfect Information, and it is the difference between the expected value of the decision with perfect information and the expected value of the decision without perfect information.

For the simple decision sapling, we have two options: a sure thing of value r, and a risky gamble with a random payoff G. The expected value of the decision without perfect information is simply the maximum of the two options:
EV = max(r, EG)
The expected value of the decision with perfect information is the maximum of the two options, knowing the outcome of the gamble:
EVPI = max(r, G) - max(r, EG)
Since we know that G is a random variable, the expected value of G is simply the average of all possible outcomes. Therefore, the expected value of the decision with perfect information is simply the maximum of the two options, knowing the average outcome of the gamble:
EVPI = max(r, EG) - max(r, EG) = 0
Therefore, EVPI > 0 for the simple decision sapling.

To prove that the EVPI will be larger if the random payoff of the gamble is replaced by a noisy version of the original gamble, we need to understand how the expected value of the decision changes with the addition of the noise variable Y. The expected value of the decision without perfect information is now:
EV = max(r, EG + EY) = max(r, EG)
Since EY = 0, the expected value of the decision without perfect information does not change. However, the expected value of the decision with perfect information now becomes:
EVPI = max(r, G + Y) - max(r, EG)
Since Y is a random variable with mean 0, the expected value of Y is 0. However, the variance of Y is not necessarily 0, which means that the addition of Y adds variability to the decision. This means that the expected value of the decision with perfect information is now larger, because we have more information about the possible outcomes of the gamble. Therefore, the EVPI will be larger when the random payoff of the gamble is replaced by a noisy version of the original gamble.

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The two linearly independent solutions are y₁(x) = 1 - (x²/3) + (x⁴/81) - (x⁶/2187) + … and y₂(x) = x - (x³/9) + (x⁵/405) - (x⁷/21870) + ...

We can use the power series method to find the two solutions of the given differential equation:

Let y = ∑anxn be a power series solution. Then, we have:

y' = ∑nanxn-1

y'' = ∑nan(n-1)xn-2

Substituting these into the differential equation and equating coefficients of like powers of x, we get:

3x∑nan(n-1)xn-1 + ∑anxn = 0

Simplifying, we obtain:

∑[3nan(n-1) + an-1]xn = 0

Since x ≠ 0, the coefficients of like powers of x must be zero. This gives us the recurrence relation:

an = -an-1/(3n(n-1)), n ≥ 1

Using this recurrence relation and the initial condition y(0) = 1, we can find the power series solution:

y₁(x) = 1 - (x²/3) + (x⁴/81) - (x⁶/2187) + ...

To find a second linearly independent solution, we can use the method of reduction of order. Let y₂(x) = v(x)y₁(x). Then, we have:

y₂' = v'y₁ + vy₁'

y₂'' = v''y₁ + 2v'y₁' + vy₁''

Substituting these into the differential equation and simplifying, we get:

v''y1x + (2v'y1' + vy1'')x + 3vy1 = 0

Since y₁(x) is a solution, we have:

y₁'' + (1/3x)y₁ = 0

Multiplying by y1 and integrating, we obtain:

(y₁')² + y₁²/3 = C

where C is a constant of integration. Using the initial condition y₂(0) = 0, we can find the second solution:

y₂(x) = x - (x³/9) + (x⁵/405) - (x⁷/21870) + ...

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