Consider random variables (X, Y ) with joint p.d.f.
fX,Y (x, y) = 1/3 x ≥ 0, y ≥ 0, 2x + 3y ≤ 6
0 otherwise.
(a) Let W = X + Y . Compute FW (w) and fw(w).
(b) Compute E[W] and V ar[W].
(c) Let Z = Y − X. What are the minimum and maximum of Z?
(d) Write FZ(z) in terms of double integral on x and y. You want to consider two separate cases for w ≥ 0 and w < 0.
(e) Find fZ(z).
(f) Compute E[Z] and V ar[Z].

Answer:

Ice rental at the local skating rink is $150 for 2h. Skate rental is $3 per person. The Grade 8 class went skating. All students rented skates. The total cost was $231. How many students went skating?

Answer:

Possible expressions for the radius of the can and the depth of the paint in the can are [tex]r = \sqrt{9\cdot x^{2}+24\cdot x+16}[/tex] and [tex]h = 2\cdot x[/tex], respectively.

Step-by-step explanation:

Let be the initial volumes of the initial cans represented by these expressions:

[tex]V_{1} = (8\cdot x^{3}+31\cdot x^{2}+32\cdot x)\cdot \pi[/tex] (1)

[tex]V_{2} = (10\cdot x^{3}+17\cdot x^{2})\cdot \pi[/tex] (2)

The resulting volume of the paint can is the sum of the two functions:

[tex]V_{3} = (18\cdot x^{3}+48\cdot x^{2}+32\cdot x)\cdot \pi[/tex] (3)

Then, we proceed to factor the polynomial:

[tex]V_{3} = 2\cdot (9\cdot x^{2}+24\cdot x +16)\cdot x \cdot \pi[/tex]

[tex]V_{3} = \pi\cdot (9\cdot x^{2}+24\cdot x + 16)\cdot (2\cdot x)[/tex] (3b)

By direct comparison with the volume formula for the cylinder we have the following expressions:

[tex]r^{2} = 9\cdot x^{2}+24\cdot x + 16[/tex]

[tex]r = \sqrt{9\cdot x^{2}+24\cdot x+16}[/tex]

[tex]h = 2\cdot x[/tex]

Possible expressions for the radius of the can and the depth of the paint in the can are [tex]r = \sqrt{9\cdot x^{2}+24\cdot x+16}[/tex] and [tex]h = 2\cdot x[/tex], respectively.

There are 10 ways to pick 4 jelly beans from a jar containing 5 red and 3 purple jelly beans, ensuring at least 2 are red.

To calculate the number of ways, we consider the cases where we choose exactly 2 red jelly beans, 3 red jelly beans, or all 4 red jelly beans.

Case 1: Choosing 2 red jelly beans - There are 5 red jelly beans to choose from, and we need to select 2. This can be done in [tex]5C2 = 10[/tex] ways.

Case 2: Choosing 3 red jelly beans - There are 5 red jelly beans to choose from, and we need to select 3. This can be done in [tex]5C3 = 10[/tex] ways.

Case 3: Choosing all 4 red jelly beans - There are 5 red jelly beans, and we need to select 4. This can be done in [tex]5C4 = 5[/tex] ways.

Adding up the possibilities from all three cases, we get 10 + 10 + 5 = 25 ways. However, we need to subtract the case where we select all 4 purple jelly beans, which is only 1 way. Therefore, the final number of ways is 25 - 1 = 24 ways.

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The inverse Laplace transform of X(s) = [tex]1/s^7[/tex] is f(t) =[tex]t^6.[/tex]

To locate the inverse Laplace rework of X(s) =[tex]1/(s^7[/tex]), we are able to use algebraic manipulation and the inverse Laplace transform assets said in Theorem 7.2.1, which permits us to discover the original characteristic while given its Laplace rework.

Using the assets of the Laplace transform, we will rewrite the given expression as:

X(s) = [tex]x^-1(1/s^7) = (1/s^7)[/tex]

We need to locate the feature f(t) such that its Laplace transform is X(s) = [tex]1/s^7[/tex]. By making use of Theorem 7.2.1, we understand that the inverse Laplace remodels of X(s) will give us f(t).

Now, we need to find a characteristic f(t) that has a Laplace transform [tex]1/s^7[/tex]. By examining the Laplace transform a desk or the usage of regarded formulas, we will decide that the Laplace remodel of [tex]t^n[/tex](wherein n is a high-quality integer) is given by means of[tex]n!/s^(n+1).[/tex]

In our case, we're looking for a function whose Laplace remodel is[tex]1/s^7.[/tex]Comparing this with the Laplace transform formulation cited earlier, we see that the exponent within the denominator of sought to be [tex]8 ^(7+1).[/tex]

Hence, f(t) must be t^6 (given that 6+1 = 7), and its Laplace remodel maybe [tex]6!/s^7 = 720/s^7.[/tex]

Therefore, the inverse Laplace transform of X(s) = [tex]1/s^7 is f(t) = t^6.[/tex]

In precis, by applying algebraic manipulation and making use of the inverse Laplace rework assets, we determined that the inverse Laplace transform of [tex]1/s^7 is f(t) = t^6[/tex]. This approach that a unique feature corresponding to the given Laplace rework is [tex]t^6.[/tex].

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Answer

3.7 or approximately 4 gallons

Step-by-step explanation:

7x2=14 total liters.

3.78 liters=1 gallon

14/3.78= 3.7 gallons

In Part A: Standard error = 3.5 / √100 = 3.5 / 10 = 0.35 years and in Part B: the mean of the sample means is 13.2 years.

Part A: The standard error of the sample means can be calculated using the formula: standard deviation / square root of the sample size. In this case, the standard deviation is 3.5 years and the sample size is 100. So, the standard error is given by:

Standard error = 3.5 / √100 = 3.5 / 10 = 0.35 years

Part B: The mean of the sample means is equal to the population mean, which is 13.2 years. When we take multiple samples from a population, the mean of those sample means is expected to be equal to the population mean. In this case, the mean of the sample means is 13.2 years. The standard error of the sample means is 0.35 years, indicating the average deviation of the sample means from the population mean. The mean of the sample means is 13.2 years.

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The orthonormal basis for (2, 1), (2, 5) is therefore u1, u2 = (2/5, 1/5), (2/5, -1/5) because u2 = v2_orth/||v2_orth|| = (2/5, -1/5).

In R2, the internal result of the two vectors u and v is as follows: The Gram-Schmidt procedure can be used to request the transformation of (2, 1), (2, 5) into an orthonormal premise. u, v = 2u1v1 + u2v2. An orthonormal premise is made by changing over a bunch of directly free vectors utilizing the Gram-Schmidt process. Our set's principal vector, v1 = (2, 1), should serve as our starting point.

We standardize v1 to obtain our first orthonormal premise vector: We must locate the second vector in our set, v2 = (-2, -5), and we can orthogonalize v2 by deducting its projection from u1: u1 = v1/||v1|| = (2/5, 1/5) proj_u1(v2) = (v2 u1)u1 = (- 8/5, - 4/5)v2_orth = v2 - proj_u1(v2) = (6, - 21/5)Our second orthonormal premise vector is acquired by normalizing v2_orth: The orthonormal reason for (2, 1), (2, 5) is subsequently u1, u2 = (2/5, 1/5), (2/5, - 1/5) in light of the fact that u2 = v2_orth/||v2_orth|| = (2/5, - 1/5).

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a) the corresponding standardized value (z) for x = 4.1 is approximately 1.39.

b) The area under the standard normal probability density function from negative infinity to z is interpreted as the probability that the random variable is less than or equal to z.

a) To calculate the standardized value (z) for the point x = 4.1, we can use the formula:

z = (x - μ) / σ

where x is the value, μ is the mean, and σ is the standard deviation.

In this case, x = 4.1, μ = 0.33, and σ = 2.69. Plugging these values into the formula:

z = (4.1 - 0.33) / 2.69

z ≈ 1.39

So, the corresponding standardized value (z) for x = 4.1 is approximately 1.39.

b) The area under the standard normal probability density function from negative infinity to z is interpreted as the probability that the random variable is less than or equal to z.

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Answer:

[tex]y=-\frac{1}{3}[/tex]

Step-by-step explanation:

- 3 + 9y = - 6

- 3 + 3 + 9y = - 6 + 3

9y = - 3

9y ÷ 9 = - 3 ÷ 9

[tex]y=- \frac{1}{3}[/tex]

The account balance after 10 years will be approximately $5,259.99.

The balance in Kelly's account after 10 years can be determined using the formula for compound interest which is given as:

A = [tex]P(1 + r/n)^{nt}[/tex]

Where, A is the balance after time t, P is the principal amount (initial deposit), r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time in years.

Using this formula and substituting the given values, we get:

A = [tex]3000(1 + 0.052/4)^{4 \times 10}[/tex]

= [tex]3000(1.0125)^{40}[/tex] ≈ $5,259.99

Therefore, the account balance after 10 years will be approximately $5,259.99.

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Answer:

The total would be 36,600.

Step-by-step explanation:

First you will convert 33 months into years which is 2 years and 10 months.

You will then add the 2 years and the 11 years together which gets you 13 so now you have a total of 13 years and 10 months. (this information will be used later)

Now we will start by multiplying the 20000 by 16% and we should get a total of 1200. That means the interest is 1200 per year so lets first take our 13 years and multiply it by the 1200 (13x1200). You should have 15600 as an answer (save this number for later) since we have 10 months were going to take that 1200 and divide it into 12 month meaning the monthly interest should be 100 per month since we have 10 months we will multiply 100 by 10 and get 1000. Now lets bring back that 15600 and add the additional 1000 to it, our answer should be 16,600, and then we add the original 20000 to the 16600 and your final answer should be 36600.

Answer:

9°C

Step-by-step explanation:

Group the terms and get [tex]24+8-1-10-12[/tex]

Then simplify to get [tex]32-23[/tex]

Subtract and you get 9

So the answer is 9°C.

Hope this helps!

Given the function `y = (x-6) / (x^2 + 5x + 6)`, let's identify the vertical asymptotes and holes: Factoring the denominator, we get`(x^2 + 5x + 6) = (x+2)(x+3)`So, `y = (x-6) / (x+2)(x+3)`

The vertical asymptotes of the function are the roots of the denominator. Thus, the vertical asymptotes of the function are `x = -2` and `x = -3`.Now, we'll look for the holes in the function. A hole is a point where the function is undefined but can be simplified by canceling common factors.

In the given function, we notice that the numerator `(x-6)` and the denominator `(x+2)(x+3)` have a common factor of `(x-6)`. Thus, there is a hole at `x = 6`.We can cancel `(x-6)` from both numerator and denominator to obtain the simplified function `y = 1 / (x+3)`.Therefore, the vertical asymptotes are `x = -2` and `x = -3`, and there is a hole at `x = 6`.

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Answer:

B) They estimate that 2.3% of those surveyed answered incorrectly

Step-by-step explanation:

hope it helps you dude

An test for divisibility that a 5-digit number "abcde" will be divisible by 11 if and only if "a - b + c - d + e" is divisible by 11.

To show that a 5-digit number "abcde" is divisible by 11 if and only if "a - b + c - d + e" is divisible by 11, the concept of modular arithmetic.

The 5-digit number "abcde" as a sum of its digits multiplied by their respective place values:

"abcde" = a × 10000 + b × 1000 + c × 100 + d × 10 + e

Then express "a - b + c - d + e" in terms of the digits:

a - b + c - d + e = a × (10,000 mod 11) - b × (1,000 mod 11) + c × (100 mod 11) - d × (10 mod 11) + e

examine the patterns of the modulos:

10,000 mod 11 = 1

1,000 mod 11 = 10

100 mod 11 = 1

10 mod 11 = 10

Substituting these values back into the expression,

a - b + c - d + e = a × 1 - b ×10 + c × 1 - d × 10 + e

Simplifying further:

a - b + c - d + e = a - b + c - d + e

observe that the expression "a - b + c - d + e" is equivalent to the original 5-digit number "abcde." This means that if "abcde" is divisible by 11, then "a - b + c - d + e" will also be divisible by 11.

Conversely, if "a - b + c - d + e" is divisible by 11, it implies that the expression and the 5-digit number "abcde" have the same remainder when divided by 11. Since they are equivalent, "abcde" must also be divisible by 11.

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The interior of S consists of all the points in S that are not in the boundary of S. These points are:

The rational numbers strictly between 13 and 16

The rational numbers strictly between 1 and 5

The rational numbers strictly between 5 and 7

The natural numbers strictly between 1 and 18, excluding 20

The set S consists of the rational numbers between 13 and 16 (inclusive), the open interval between 1 and 5, the open interval between 5 and 7, the singleton set {20}, and the set of natural numbers between 0 and 18.

To find the interior of S, we need to find all the points in S that have a neighborhood entirely contained in S. In other words, we need to find all the points in S that are not on the boundary of S.

The boundary of S includes the endpoints of the closed interval [13, 16] and the endpoints of the open intervals (1, 5) and (5, 7), as well as the points 20, 0, and 18.

Therefore, the interior of S consists of all the points in S that are not in the boundary of S. These points are:

The rational numbers strictly between 13 and 16

The rational numbers strictly between 1 and 5

The rational numbers strictly between 5 and 7

The natural numbers strictly between 1 and 18, excluding 20

Note that the point 20 is not in the interior of S because it is on the boundary of S. Similarly, the points 0 and 18 are not in the interior of S because they are in the boundary of S.

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Answer:

fixed - monthly rent, car payment, savings for new guitar

variable - movies, video games, snacks

Step-by-step explanation:

Fixed costs are costs that do not vary with output.

the amount of rent paid is fixed.  

Variable costs are costs that vary with production

the amount paid at the movies depend on the number of movies watched

Answer:

0.42846931021

Step-by-step explanation:

Answer:

x = 50.... 25/50 = 15/30 or 1/2 = 1/2

Step-by-step explanation:

25/x = 15/30

consider the cross multiply

25 * 30 = 15 * x

750 = 15x

divide both sides with 15 to make the co-efficient of x, 1.

x = 50

Answer:

5×15/6

5×5/2

25/2

12.2 is your answer ☺️☺️☺️. If I'm right so,

Please mark me as brainliest. thanks!!!

Answer:

Down below

Step-by-step explanation:

[tex]A=\pi r^2\\A=3.14*(15)^2\\A=3.14*225\\A=706.5[/tex]

Answer:

150*3= 450 miles

Hope this helped :)

Answer:

(x - 2)(3x^2 + 5)

Step-by-step explanation:

All four terms here have 3 as a factor.  Factor out 3:

3x^(3)-6x^(2)+15x-30 =>  3(x^3 - 2x^2 + 5x - 10)

The last two terms can be rewritten as 5(x - 2).  The first two terms can be rewritten as 3x^2(x - 2).  So (x - 2) is a factor of 3(x^3 - 2x^2 + 5x - 10).  We get:        

3x^2(x - 2) + 5(x - 2) = (x - 2)(3x^2 + 5)

The margin of error for a 95% confidence level is 8.2.

The margin of error (ME) is determined using the formula:

ME= z ∗ σ/√n

where:

z is the z-score for the desired confidence level

σ is the population standard deviation

n is the sample size

For a 95% confidence level, the z-score is 1.96. Thus, we have:

z = 1.96

σ  = 13.3

n = 10

Substituting these values into the formula, we have:

ME =  1.96 ∗ 13.3/√10

ME = 8.2

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The function h(z) = 1 4iz + 2 defined on the extended complex plane,

(a) The function h(z) = 1/(4iz + 2) can be expressed as a composition of the linear function g(z) and reciprocal function f(z).

(b) The image of the line y = 2 under w = h(z) is the point z = -3/(8i).

(c) The image of the circle |z - i| = 1/2 under w = h(z) is the two points z = i + √7/2 and z = i - √7/2.

(a) To express the function h(z) = 1/(4iz + 2) as a composition of a linear function g(z) and reciprocal function f(z), we can rewrite h(z) as follows:

h(z) = 1/(4iz + 2)

= 1/(4i(g(z)) + 2)

= 1/f(g(z))

Here, g(z) represents the linear function and f(z) represents the reciprocal function. To determine g(z), we set g(z) = 4iz + 2.

Therefore, the composition of the linear function g(z) and the reciprocal function f(z) is:

h(z) = 1/f(g(z)) = 1/(4iz + 2)

(b) To find the image of the line y = 2 under w = h(z), we substitute y = 2 into the function h(z) and solve for z.

y = 2

1/(4iz + 2) = 2

To simplify the equation, we multiply both sides by (4iz + 2):

1 = 2(4iz + 2)

1 = 8iz + 4

8iz = -3

z = -3/(8i)

Therefore, the image of the line y = 2 under w = h(z) is the point z = -3/(8i).

(c) To determine the image of the circle |z - i| = 1/2 under w = h(z), we substitute z - i = 1/2 into the function h(z) and solve for w.

|z - i| = 1/2

|z - i|² = (1/2)²

(z - i)(z - i*) = 1/4

z² - iz - iz + i² = 1/4

z² - 2iz + 1 = 1/4

z² - 2iz + 3/4 = 0

Now we solve this quadratic equation using the quadratic formula:

z = (-(-2i) ± √((-2i)² - 4(1)(3/4))) / (2(1))

z = (2i ± √(-4i² - 3)) / 2

z = (2i ± √(4 + 3)) / 2

z = (2i ± √7) / 2

z = i ± √7/2

So, the image of the circle |z - i| = 1/2 under w = h(z) is the two points z = i + √7/2 and z = i - √7/2.

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Answer:

A 3-inch slice of the food item contains 255 calories.

Step-by-step explanation:

Set calories per inch slice as variable x.

2x = 170

x = 85 cal

Since each inch slice of the food item contains 85 calories, a 3-inch slice would contain:

3x = 3(85) = 255 cal

Answer:

same

Step-by-step explanation:

The equation of the sphere passing through the point (7, 3, -1) with center (5, 8, 5) is:

(x - 5)^2 + (y - 8)^2 + (z - 5)^2 = 65.

To find the equation of the sphere passing through the point (7, 3, -1) with center (5, 8, 5), we can use the general equation for a sphere in three-dimensional space. The equation of a sphere with center (h, k, l) and radius r is given by:

(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2.

Using the given center (5, 8, 5), we have:

(x - 5)^2 + (y - 8)^2 + (z - 5)^2 = r^2.

Since the sphere passes through the point (7, 3, -1), we can substitute these values into the equation:

(7 - 5)^2 + (3 - 8)^2 + (-1 - 5)^2 = r^2.

Simplifying the equation gives us:

4 + 25 + 36 = r^2.

65 = r^2.

Therefore, the equation of the sphere passing through the point (7, 3, -1) with center (5, 8, 5) is:

(x - 5)^2 + (y - 8)^2 + (z - 5)^2 = 65.

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For both scenarios, a sample is selected without replacement from a batch of items. In the first scenario, the batch contains 3 defective items ('d') and 10 good items ('g'). The ordered sample space consists of all possible ordered pairs of items: {dd, dg, gd, gg}. In the second scenario, the batch contains the items {a, b, c, d}. The ordered sample space also consists of all possible ordered pairs of items: {aa, ab, ac, ad, ba, bb, bc, bd, ca, cb, cc, cd, da, db, dc, dd}.

In the first scenario, the ordered sample space is derived by considering all possible combinations of the two items selected from the batch. Since the selection is done without replacement, the first item can be either defective ('d') or good ('g'). For each case, the second item can also be defective or good, depending on what was chosen as the first item. Therefore, the ordered sample space consists of four possibilities: dd, dg, gd, and gg.

In the second scenario, the batch consists of four distinct items: a, b, c, and d. Again, the ordered sample space is obtained by considering all possible combinations of the two items selected without replacement. Since there are four items, there are 16 possible combinations. Each combination is represented by an ordered pair of the selected items, such as aa, ab, ac, and so on.

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The correct answer is b. close to 1.00.  Ratio close to 1.00 indicates that the between-group variation is similar to the within-group variation.

In an ANOVA (Analysis of Variance) model, MSTR refers to the mean square treatment (or between-group variation), while MSE refers to the mean square error (or within-group variation).

If the true means of the k populations are equal, it means that the between-group variation is similar to the within-group variation, and there is no significant difference between the group means.

In this scenario, we would expect the MSTR/MSE ratio to be close to 1.00 (answer b). A ratio close to 1.00 indicates that the between-group variation is similar to the within-group variation, supporting the assumption that the true means of the populations are equal.

Therefore, the correct answer is b. close to 1.00.

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