Find the slope for the following line:

Because x-bar is an unbiased estimator of μ, sample averages have the same variance as the individual observations in the population.

The sample mean or x-bar is a statistic that estimates the population mean or μ. When we say that x-bar is an unbiased estimator of μ, it means that on average, the sample mean is equal to the population mean. This property is desirable because it means that our estimates are not systematically too high or too low.

One consequence of the x-bar being an unbiased estimator of μ is that sample averages have the same variance as the individual observations in the population. This means that the spread or variability of the sample mean is equal to the spread or variability of the individual observations. However, the central limit theorem tells us that the distribution of sample averages is more Normal than the distribution of individual observations. This is because as the sample size increases,

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Answer: The best prediction possible for the number of times you roll a three is 1/6 of a roll.

Step-by-step explanation:

When you find the statistics of something like rolling a die, you will need to divide how many times you roll the die by how many sides are on the die.

6 - how many times you roll the die

6 - how many sides are on the die

When you divide 6 by 6,  you will get one. Since both of the statistical numbers are the same, you will get an 1/6 chance of rolling a 1, 2, 3, 4, 5, and 6. So 1/6 is the answer! Hope this helps!!!

P.S (I learned this in 5th grade :))

We can be 95% confidence interval that the true mean reading achievement score for the population of third graders falls within the interval (74.02, 75.98).

The margin of error for a 95% confidence interval for the mean reading achievement score for a population of third graders depends on the sample size, standard deviation, and the level of confidence desired.

Assuming the sample is randomly selected and follows a normal distribution, the margin of error (E) for a 95% confidence interval can be calculated using the following formula:

E = 1.96 * (s / √(n))

where s is the sample standard deviation, n is the sample size, and 1.96 is the z-score associated with a 95% confidence level.

For example, if we have a sample of 100 third graders with a sample standard deviation of 5, the margin of error for a 95% confidence interval would be:

E = 1.96 * (5 / √(100))

= 0.98

Therefore, the 95% confidence interval for the mean reading achievement score for the population of third graders would be the sample mean plus or minus the margin of error:

sample mean ± margin of error

For instance, if the sample mean is 75, the 95% confidence interval for the mean reading achievement score would be:

=75 ± 0.98 or (74.02, 75.98)

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The pairs of triangles that can be proven by SAS similarity is (c)

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

The list of options

The SAS similarity is used to check if two triangles are similar by comparing the lengths of the corresponding sides and the angle between the corresponding sides.

Using the above as a guide, we have the following:

The pair of triangles in (c) can be proved by SAS

This is because

Corresponding sides are similar and the angle between them are equal

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Answer:

it's a triangle

Step-by-step explanation:

Answer:

answer is d

cause central angle is twice the inscribed angle sustended by same arc

Thus, the integral of xe^yds on the circle from (2,0) to (5,4) is equal to 207/8 e^4 - 23/8.

To solve this problem, we will use the line integral formula:
∫(P dx + Q dy) = ∫(P(x,y) dx + Q(x,y) dy)

where P and Q are the x and y components of the vector field F(x,y) = (xe^y, 0).

First, we need to parameterize the given circle. We can do this by using the parametric equations:

x = 2 + 3t
y = 4t

where 0 ≤ t ≤ 1.

Next, we can compute the differential ds:

ds = √(dx^2 + dy^2) = √(9 + 16) dt = 5 dt

Now, we can substitute the parametric equations and ds into the line integral formula:

∫(P dx + Q dy) = ∫(xe^y dx) = ∫(xe^y dx/dt dt) = ∫((2+3t)e^(4t) 3 dt)

Evaluating the integral gives:

∫(xe^yds) = ∫((2+3t)e^(4t) 3 dt) = 207/8 e^4 - 23/8

Therefore, the integral of xe^yds on the circle from (2,0) to (5,4) is equal to 207/8 e^4 - 23/8.

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The surface area of the given prism is 386 square inches.

To find the surface area of a prism, we need to find the area of all its faces and add them up.

The given prism has a rectangular base with dimensions of 7 inches by 15 inches, and a height of 4 inches.

The two rectangular faces (front and back) have dimensions of 7 inches by 4 inches,

so each has an area of 7 x 4 = 28 square inches.

The two rectangular faces (sides) have dimensions of 15 inches by 4 inches,

so each has an area of 15 x 4 = 60 square inches.

The top and bottom faces are both rectangles with dimensions of 7 inches by 15 inches,

so each has an area of 7 x 15 = 105 square inches.

Therefore, the total surface area of the prism is:

2(28 sq in) + 2(60 sq in) + 2(105 sq in) = 56 sq in + 120 sq in + 210 sq in

                                                               = 386 sq in.

So, the surface area of the given prism is 386 square inches.

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The given information f(x)dx = 7 can be used to find the values of various definite integrals as follows:

(a) f(u)du = (Type exact answers, using radicals as needed.)

We cannot determine the value of this definite integral with the given information. We need to know the function f(x) to compute f(u)du.

(b) f(z)dz = (Type exact answers, using radicals as needed.)

Since the definite integral of f(x)dx is 7, we know that ∫ f(x)dx = 7. We can now use the substitution u = f(x) to rewrite the integral as ∫ 1/u du = ln|u| + C, where C is a constant of integration. Substituting back u = f(x), we get f(x)dz = ln|f(x)| + C. Therefore, f(z)dz = ln|z| + C.

(c) f(t)dt= (Type exact answers, using radicals as needed.)

Again, we cannot determine the value of this definite integral with the given information. We need to know the function f(x) to compute f(t)dt.

(d) [-f(x)]dx= (Type exact answers, using radicals as needed.)

Since the definite integral of f(x)dx is 7, we know that ∫ f(x)dx = 7. Therefore, the definite integral of [-f(x)]dx can be found by multiplying the integral of f(x)dx by -1, giving [-f(x)]dx = -∫ f(x)dx = -7.

In summary, we can find the value of f(z)dz and [-f(x)]dx using the given information. However, we need to know the function f(x) to compute the values of f(u)du and f(t)dt.

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The side which must have the same length as BC is RS.

The correct answer choice is option C.

Given the figure ABCD with a similar figure QRST

If AD = 12 and QT = 12

AB = 8 and QR = 8

Then,

BC = RS

and

CD = ST

Hence, the side corresponding to BC is RS

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For  Sky View School mean is 7.933, median is 6, mode is 5

For Riverside School  mean is 8, median is 6.5, mode is 5

To calculate the measures of center, we can find the mean, median, and mode of each set of data.

For Sky View School:

Mean=119/15 = 7.933

Median: To find the median, we need to put the class sizes in order from smallest to largest.

0, 0, 1, 2, 3, 4, 5, 5, 5, 6, 6, 7, 8, 9, 9

The median is the middle value, which is 6.

Mode: The mode is the most common class size. In this case, the mode is 5.

For Riverside School:

Mean =120/15

= 8

Median:

0, 0, 1, 2, 2, 3, 4, 5, 5, 6, 7, 7, 8, 8, 10

The median is the average of the two middle values, which is 6.5

Mode: The mode is 5.

Part B:

To calculate the measures of variability, we can find the range and interquartile range (IQR) for each set of data.

For Sky View School:

Range: The range is the difference between the largest and smallest values.

$Range = 9 - 0 = 9$

IQR: To find the IQR, we first need to find the first quartile (Q1) and third quartile (Q3)

0, 0, 1, 2, 3, 4, 5, 5, 5 | 6, 6, 7, 8, 9, 9

Q1 is the median of the lower half, which is 3.

Q3 is the median of the upper half, which is 8.

IQR = Q3 - Q1 = 8 - 3 = 5

For Riverside School:

Range = 10 - 2 = 8$

IQR

Q1 is the median of the lower half, which is 2.

Q3 is the median of the upper half, which is 8.

IQR = Q3 - Q1 = 8 - 2 = 6

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The antenna is approximately 104.6 feet tall and the distance between anchor 2 and the base of the antenna is approximately 66.3 feet.

Let's denote the height of the antenna as h and the distance between anchor 2 and the base of the antenna as x. We can use trigonometry to create two equations based on the angles α and β:

tan(α) = h / (54 - x)

tan(β) = h / x

We can rearrange the first equation to get h = (54 - x)tan(α), and we can rearrange the second equation to get h = xtan(β). We can then set these two expressions for h equal to each other and solve for x:

(54 - x)tan(α) = xtan(β)

54tan(α) - xtan(α) = xtan(β)

54tan(α) = xtan(α) + xtan(β)

54tan(α) = x(tan(α) + tan(β))

x = 54tan(α) / (tan(α) + tan(β))

Now that we have the value of x, we can substitute it back into one of the equations to find the height of the antenna:

h = (54 - x)tan(α) ≈ 104.6 feet

We can also substitute x into the equation for the distance between anchor 2 and the base of the antenna:

x = 54tan(α) / (tan(α) + tan(β)) ≈ 66.3 feet

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Answer:

To transform the graph of f(x)=x√ into g(x)=x+9, we need to apply a horizontal shift to the right by 9 units. This can be done by replacing x in f(x) with x-9 to get g(x)=(x-9)√. The resulting graph will be the same as the graph of f(x), but shifted 9 units to the right.

Answer:

Sydney is 28

Step-by-step explanation:

Let's call Daisy D, and call Sydney S.

Daisy is 10 years older.

So D = S + 10

S = D - 10

Sum of their ages = 66.

D + S = 66

D + (D - 10) = 66

2D - 10 = 66

2D = 76

D = 38

If Daisy is 10 years older, Sydney must be 38 - 10 = 28.

to check if we're correct, 38 + 28 = 66.

Answer:

[tex]\textsf{A.} \quad y=2^x[/tex]

Step-by-step explanation:

From inspection of the given table, we can see that the number of new infections is twice the number of infections recorded for the previous day. Therefore, we can use the exponential function formula to write a function that models the given data.

[tex]\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function}\\\\$y=ab^x$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $b$ is the base (growth/decay factor) in decimal form.\\\end{minipage}}[/tex]

The initial value is the number of new infections on day 0:

The growth factor is 2, since the number of new infections doubles each day:

Substitute the values of a and b into the formula:

[tex]y=1 \cdot 2^x[/tex]

[tex]y=2^x[/tex]

Therefore, the exponential function that models the data is:

[tex]\boxed{y=2^x}[/tex]

The expected frequency is 50 hence, the answer is (d) 50.

Expected frequency is the frequency we would expect to see in a particular category or group if the null hypothesis is true. The null hypothesis assumes a specific distribution or pattern in the data, and the expected frequency is calculated based on that assumption.

Here we have

Individuals in a random sample of 150 were asked whether they supported capital punishment.

If the opinions of the individuals are uniformly distributed, then the expected frequency for each group would be the same.

Since there are three groups (yes, no, and no opinion), the expected frequency for each group is:

Expected frequency = Total frequency / Number of groups

= 150/3 = 50

Therefore,

The expected frequency is 50 hence, the answer is (d) 50.

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The values of the equation are x=2.71 or x=1.29

The given quadratic equation is 2x²-8x+7=0

We solve by using the formula x = -b±√b²-4ac/2a

From the equation, a =2, b=-8 and c=7

x=8±√64-4(2)(7)/2(2)

x=8±√64-56/4

x=8±√64-56/4

x=8±√8/4

x=8+√8/4  or x=8-√8/4

x=2.70 or x=1.29

Hence, the values of the equation are x=2.71 or x=1.29

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It is highly likely that Jamie's family stopped to rest during their 4.5 hour trip to the campsite in the mountains. As their average speed was 1.78 miles per hour, which is slower than the average walking speed, it is likely that Jamie's family stopped to rest during their trip to the mountain campsite.

Walking 8 miles is not an easy feat, especially in a mountainous area where the terrain can be challenging and the altitude can make it harder to breathe. It is natural to feel fatigued and need to take breaks during such a trip.

camping trips often involve carrying heavy backpacks and other gear, which can make the journey even more strenuous. Jamie's family would likely need to take breaks to catch their breath and rest their muscles along the way. It is also possible that they stopped to enjoy the scenery or have a snack.

Therefore, it is highly probable that Jamie's family stopped to rest during their 4.5 hour trip to the campsite in the mountains. Walking 8 miles in a mountainous area is a challenging task, and breaks are necessary to ensure that the family arrives at the campsite safely and without any injuries or exhaustion.

During their summer break, Jamie's family took a trip to go camping in the mountains. They had to hike 8 miles to reach their campsite, as there were no roads available. The journey took them about 4.5 hours to complete on foot. Based on the given data, it is possible that Jamie's family stopped to rest during the trip.

calculate their speed, divide the distance (8 miles) by the time taken (4.5 hours):

8 miles / 4.5 hours = 1.78 miles per hour.

As their average speed was 1.78 miles per hour, which is slower than the average walking speed, it is likely that Jamie's family stopped to rest during their trip to the mountain campsite.

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To find the mean salary for the 50 employees of the new company, we need to combine the salaries of all the workers from both companies and divide by the total number of employees. So, the mean salary for the 50 employees of the merged company is $90 per week.

The mean salary for the 20 workers in Company A is $93 per week, and for the 30 workers in Company B, it's $88 per week. To find the mean salary for the 50 employees of the merged company, you'll need to calculate the total combined salary for all workers and divide it by the total number of employees.

finding the total salary of the 20 workers in company A. We know that the mean salary is $93 per week, so we can multiply that by the number of workers to get the total salary:

Total salary in company A = 20 x $93 = $1860

Next, we'll do the same thing for company B. The mean salary is $88 per week, and there are 30 workers:

Total salary in company B = 30 x $88 = $2640

Now we can add the two total salaries together to get the total salary for all 50 employees:

Total salary for all 50 employees = $1860 + $2640 = $4500

Finally, we can divide the total salary by the total number of employees (50) to find the mean salary for the new company:

Mean salary for the new company = $4500 / 50 = $90 per week

Therefore, the mean salary for the 50 employees of the new company is $90 per week.

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Using the given values, we can evaluate sin(a+b) to be approximately 0.656 and cos(a-b) to be approximately 0.308.

First, we can use the identity sin^2θ + cos^2θ = 1 to find sin a and sin b:

sin a = √(1 - cos^2a) ≈ 0.534

sin b = √(1 - cos^2b) ≈ 0.615

Next, we can use the sum and difference identities to find sin(a+b) and cos(a-b):

sin(a+b) = sin a cos b + cos a sin b = 0.656

cos(a-b) = cos a cos b + sin a sin b =0.308

Finally, we can use the identity cos^2θ + sin^2θ = 1 to find cos a and cos b:

cos a = √(1 - sin^2a) =0.846

cos b = √(1 - sin^2b) =0.785

Therefore, using the given values, we have found that sin(a+b) is approximately 0.656 and cos(a-b) is approximately 0.308.

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(a) ab l 1 bc l = ab + bc

(b) cd l 1 db l = cd - db

(c) db l 2 ab l = 2ab + db

(d) dc l 1 ca l 1 ab = -ab + ca + dc

In each of the given combinations of vectors, we need to add or subtract the given vectors.

In case (a), the vectors are parallel and have the same direction, so we can simply add them to get the resultant vector: ab + bc = ac.

In case (b), the vectors are also parallel and have opposite directions, so we can subtract them to get the resultant vector: cd - db = cb.

In case (c), we need to double the vector db and subtract it from vector ab to get the resultant vector: ab - 2db = ab - db - db = ac - db.

In case (d), we need to add vector dc and subtract vectors ca and ab to get the resultant vector: dc - ca - ab = db. Therefore, the resultant vectors are: (a) ac, (b) cb, (c) ac - db, (d) db.

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The candidate needs to answer the remaining questions with at least 75% rate to get a 70% rate on the entire test.

Let's denote the number of remaining questions that the candidate needs to answer correctly as x.

The total number of questions the candidate needs to answer correctly to get a 70% rate is 42 questions, so the candidate needs to answer (42 - 12) = 30 more questions correctly.

We can set up an equation to solve for x:

x / 40 = 30 / 40

Simplifying this equation, we get:

x = 0.75 x 40

x = 30

So the candidate needs to answer 30 out of the remaining 40 questions correctly, which is a rate of 75%.

Therefore, the candidate needs to answer the remaining questions with at least 75% rate to get a 70% rate on the entire test.

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The parameterization of the line from (-1, 0) to (3, -2) is P(t) = (-1 + 4t, -2t), where t varies from 0 to 1.

Direction vector = (3, -2) - (-1, 0) = (3 + 1, -2 - 0) = (4, -2)

In order to get a parameterization of the line, we can use the equation:

P(t) = P0 + t * v

where P(t) is a point on the line, P0 is a known point on the line (in this case, (-1, 0)), v is the direction vector of the line, and t is a parameter that varies along the line.

Substituting the values we have, we get:

P(t) = (-1, 0) + t * (4, -2)

Simplifying, we get:

P(t) = (-1 + 4t, -2t)

So the parameterization of the line from (-1, 0) to (3, -2) is P(t) = (-1 + 4t, -2t), where t varies from 0 to 1.

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The Taylor polynomial T3(x) for the function f centered at the number a is 1, -1.

To find the slope of the tangent line to the curve at a given point, we need to find the derivative of the curve and evaluate it at that point. So, let's find the derivative of the curve x(t) = cos^3(4t), y(t) = sin^3(4t):

x'(t) = 3cos^2(4t) * (-sin(4t)) * 4 = -12cos^2(4t)sin(4t)

y'(t) = 3sin^2(4t) * cos(4t) * 4 = 12sin^2(4t)cos(4t)

Now, let's evaluate these derivatives at t = pi/6:

x'(pi/6) = -12cos^2(2pi/3)sin(2pi/3) = -6sqrt(3)

y'(pi/6) = 12sin^2(2pi/3)cos(2pi/3) = 6sqrt(3)

So, the slope of the tangent line at t = pi/6 is:

y'(pi/6) / x'(pi/6) = (6sqrt(3)) / (-6sqrt(3)) = -1

Therefore, the answer is option 1, -1.

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The missing values in the triangle are:

∠B = 90°

AB =  20.98

CB = 16.99

First, remember that the sum of the interior angles of any triangle is always equal to 180°, then we can write:

51 + 39 + ∠B = 180

∠B = 180 - 51 - 39 = 90

So we have a right triangle.

Now, to find the values of AB and CB, we can use trigonometric relations, we know that teh hypotenuse is 27 units, then we can use:

cos(51°) = CB/27

27*cos(51°) = CB = 16.99

And:

cos(39°) = AB/27

27*cos(39°) = AB = 20.98

These are the missing values.

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The equation that represents the proportional relationship is y = 16x.

To find the equation that represents the proportional relationship, we need to determine the constant ratio between the values of x and y.

Let's observe the given values:

x: 1 2 3

y: 16 32 48

We can see that when x increases by 1, y increases by 16. This means that the constant ratio between x and y is 16.

To write the equation representing this proportional relationship, we can use the formula:

y = kx

Where:

y represents the dependent variable (in this case, the y-values)

x represents the independent variable (in this case, the x-values)

k represents the constant of proportionality (the ratio between x and y)

Substituting the values into the equation:

y = 16x

Therefore, the equation that represents the proportional relationship is y = 16x.

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If IQ scores are normally distributed with a mean of 100 and a standard deviation of 15, we can use the normal distribution properties to find the percentage of IQ scores between 85 and 130.

Specifically, we can standardize the scores and use a normal distribution table or calculator to find the area under the curve between the z-scores corresponding to 85 and 130.To find the percentage of IQ scores between 85 and 130, we first need to standardize the scores by subtracting the mean and dividing by the standard deviation. This gives:

z1 = (85 - 100) / 15 = -1.00

z2 = (130 - 100) / 15 = 2.00

We can then use a normal distribution table or calculator to find the area under the curve between these two z-scores. For example, using a standard normal distribution table, we can find that the area to the left of z = -1.00 is 0.1587 and the area to the left of z = 2.00 is 0.9772. Therefore, the area between these two z-scores is:

0.9772 - 0.1587 = 0.8185

This means that approximately 81.85% of IQ scores are between 85 and 130. Alternatively, we can use a normal distribution calculator to find the same result. For example, using an online calculator, we can input the mean (100), standard deviation (15), and the lower and upper limits (85 and 130) and obtain a probability of 0.8185, or 81.85%.

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The total number of ways Emma can color the flag, based on the different colored pens would be 24 ways.

For the first stripe, Emma has four options to choose from since she can select any of the four colored pens.

For the second stripe, she needs to ensure that it is a different color from the first stripe. Therefore, she has three options remaining to choose from.

Similarly, for the third stripe, she needs to select a color different from both the first and second stripes. Thus, she is left with two options:

= 4 (options for the first stripe) × 3 (options for the second stripe) × 2 (options for the third stripe)

= 24 ways

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Tables and graphs to show Equivalent ratios makes it easy to see that different pairs of numbers can represent the same ratio. This helps to develop a deeper understanding of ratios and proportions.

Tables and graphs are great tools to show equivalent ratios because they provide a visual representation of the data that makes it easy to compare different values.

To use a table to show equivalent ratios, we can create a table with two columns: one for the numerator and one for the denominator. Then, we can list different values of the numerator and denominator that have the same ratio in each row. For example, we can create a table to show equivalent ratios for 2:3 by listing different pairs of numbers that have the same ratio, such as 4:6, 8:12, and 10:15.

To use a graph to show equivalent ratios, we can create a coordinate plane with the horizontal axis representing the numerator and the vertical axis representing the denominator. Then, we can plot different points that represent pairs of numbers with the same ratio. For example, we can plot points for 2:3, 4:6, 8:12, and 10:15 on the graph, and we will see that they all fall on the same line.

Using tables and graphs to show equivalent ratios makes it easy to see that different pairs of numbers can represent the same ratio. This helps to develop a deeper understanding of ratios and proportions.

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To find the marginal probability mass function of x and y, we sum f(x, y) over the values of y and x, respectively. For x = 1, we have f(1,0) = 0 and f(1,1) = 1/8, so the marginal probability mass function of x is given by:

f(x) = Σ f(x,y) = f(1,0) + f(1,1) + f(2,0) + f(2,1) = 0 + 1/8 + 0 + 2/8 = 3/8

Similarly, for y = 0, we have f(1,0) = 0 and f(2,0) = 2/8, so the marginal probability mass function of y is given by:

f(y) = Σ f(x,y) = f(1,0) + f(1,1) + f(2,0) + f(2,1) = 0 + 1/8 + 2/8 + 1/8 = 1/2

The expected value of x is given by:

E[X] = Σ x f(x) = 1(3/8) + 2(5/8) = 1.625

The expected value of y is given by:

E[Y] = Σ y f(y) = 0(1/2) + 1(1/2) = 0.5

The covariance between x and y is given by:

Cov[X,Y] = E[XY] - E[X]E[Y]

We can find E[XY] by summing xyf(x,y) over all values of x and y:

E[XY] = Σ Σ xy f(x,y) = 1(0) + 1(1/8) + 2(0) + 2(2/8) = 1/2

Substituting in the values we have found, we get:

Cov[X,Y] = E[XY] - E[X]E[Y] = (1/2) - (1.625)(0.5) = -0.25

Therefore, the covariance between x and y is -0.25.

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