30 points to whoever solves

a) f(x) is a PDF. b) the marginal of X and Y is (y/2 + 1) / 2 c) the covariance of X and Y is: -1/18

What is meant by PDF?

In probability theory, a probability density function (PDF) is a function that describes the relative likelihood for a continuous random variable to take on a given value.

What is covariance?

Covariance is a statistical measure that quantifies the degree to which two random variables are linearly associated.

(a) To show that f(x) is a probability density function (PDF), we need to show that it satisfies the following two conditions:

Non-negativity: f(x) is non-negative for all x in its domain.

Normalization: The integral of f(x) over its domain is equal to 1.

The domain of f(x) is given by the inequality 0 ≤ x + 2y ≤ 2. To find the integral of f(x) over its domain, we need to integrate it with respect to y from (0-x/2) to (2-x/2), and then integrate the result with respect to x from 0 to 2:

∫(0 to 2) ∫(0-x/2 to 2-x/2) (x+y) dy dx

Solving the inner integral with respect to y, we get:

∫(0 to 2) [xy + [tex]y^2[/tex]/2] |_0-x/[tex]2^{(2-x/2)[/tex] dx

= ∫(0 to 2) ([tex]x^2[/tex]/4 - [tex]x^3[/tex]/12 + 1) dx

= [[tex]x^3[/tex]/12 - [tex]x^4[/tex]/48 + x] |_[tex]0^2[/tex]

= 2 - 2/3 + 2 = 8/3

Since the integral is finite and positive, the first condition of non-negativity is satisfied. To satisfy the normalization condition, we divide the function by the integral:

f(x) = (x+y) / (8/3)

Therefore, f(x) is a PDF.

(b) To find the marginal of X, we integrate f(x,y) over the range of y:

f(x) = ∫(0-x/2 to 2-x/2) (x+y) / (8/3) dy

= (x/2 + 1) / 2

Similarly, to find the marginal of Y, we integrate f(x,y) over the range of x:

f(y) = ∫(0 to 2) (x+y) / (8/3) dx

= (y/2 + 1) / 2

(c) To find the covariance of X and Y, we use the formula:

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

To find E[XY], we integrate xy*f(x,y) over the range of x and y:

E[XY] = ∫(0 to 2) ∫(0-x/2 to 2-x/2) xy*(x+y)/(8/3) dy dx

= ∫(0 to 2) [[tex]x^3[/tex]/6 - [tex]x^4[/tex]/24 + [tex]x^2[/tex]/4] dx

= 2/3

To find E[X] and E[Y], we integrate xf(x) and yf(y) over their respective ranges:

E[X] = ∫(0 to 2) x*(x/2+1)/2 dx

= 7/3

E[Y] = ∫(0 to 2) y*(y/2+1)/2 dy

= 7/6

Therefore, the covariance of X and Y is:

Cov(X, Y) = E[XY] - E[X]E[Y] = 2/3 - (7/3)*(7/6) = -1/18

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The measures of the angles in the unit circles are 53 degrees, 138 degrees, 300 degrees and 214 degrees

The question is an illustration of unit circles and the angles would be solved using

(x, y) = (cos(∅), sin(∅))

Figure (a)

Here, we have

(x, y) = (3, 4)

This means that

tan(∅) = y/x

So, we have

tan(∅) = 4/3

Take the arc tan of both sides

∅ = 53 degrees

Figure (b)

Here, we have

(x, y) = (-√5, 2)

This means that

tan(∅) = y/x

So, we have

tan(∅) = 2/-√5

Take the arc tan of both sides

∅ = 138 degrees

Figure (c)

Here, we have

(x, y) = (1, -√3)

This means that

tan(∅) = y/x

So, we have

tan(∅) = -√3/1

Take the arc tan of both sides

∅ = 300 degrees

Figure (d)

Here, we have

(x, y) = (-3, -2)

This means that

tan(∅) = y/x

So, we have

tan(∅) = -2/-3

Take the arc tan of both sides

∅ = 214 degrees

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The probability of event A, the sum being greater than 5, is 11/36. The probability of event B, the sum being an odd number, is 1/2.

To calculate the probability of each event, we need to determine the total number of possible outcomes and the number of favorable outcomes for each event.

Event A: The sum is greater than 5. There are 11 favorable outcomes (6,5), (6,4), (6,3), (6,2), (6,1), (5,6), (4,6), (3,6), (2,6), (1,6), and (5,5). The total number of possible outcomes is 36. So, the probability of event A is 11/36.

Event B: The sum is an odd number. There are 18 favorable outcomes (1,3), (1,5), (1,5), (2,1), (2,3), (2,5), (2,3), (3,1), (3,3), (3,5), (4,1), (4,3), (4,5), (5,1), (5,3), (5,5), (6,1), (6,3), and (6,5). The probability of event B is 18/36 or 1/2.

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Answer: y = 18000(0.89)^x, where x represents the number of years in the future.

Step-by-step explanation:

Answer:

Y= 18,000 X (0.11x)

Step-by-step explanation:

x being years y being car

an organized list representing the sample space of drawing one letter tile and one number tile: {(A, 1), (A, 2), (A, 3), (B, 1), (B, 2), (B, 3), (C, 1), (C, 2), (C, 3)}

In probability theory, sample space refers to the set of all possible outcomes of a random experiment.

For example, if we toss a coin, the sample space would consist of two possible outcomes, heads or tails. Similarly, if we roll a die, the sample space would consist of six possible outcomes, the numbers 1 through 6.

The sample space is an important concept because it allows us to define the probabilities of events. In order to calculate the probability of an event, we divide the number of outcomes that are favorable to the event by the total number of possible outcomes in the sample space.

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The complete question is;

The bag contains tiles labeled with the letters A, B,

and C. The box contains tiles labeled with the

numbers 1, 2, and 3. June draws one letter tile and

one number tile. Represent the sample space using

either a table or an organized list.

B

B

A

2

3

Answer:

b

Step-by-step explanation:

The expression that is equivalent to 5[4+3(x−6)] is 15x - 70

To simplify the expression  5[4+3(x−6)]  , we can first simplify the expression inside the square brackets, which is (x-6) multiplied by 3 and then added to 4. This gives us:4 + 3(x - 6) = 4 + 3x - 18 = 3x -

this expression back into the original equation, we get:5[4+3(x−6)] = 5(4 + 3x - 18) = 5(3x - 14) = 15x - 70

Therefore, the expression 15x - 70 is equivalent to 5[4+3(x−6)] as seen here.

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Missing parts;

Which expression is the equivalent to  5[4+3(x−6)]

By Probability , You expect Stella to remove a apple chew from the bag

 For apple ,  Pₐ = 7/21

    For lime ,  Pi = 2/21

What is probability with and example?

Probability is a measure of how likely or likely-possible something is to happen. For instance, there is a 1 in 2 chance of receiving a head when flipping a coin, and there are 2 potential outcomes overall (a head or tail). P(heads) = 12 is what we write.

Out of 21 times that the experiment was performed

A orange chew was selected 12 times

A orange chew was selected 7 times

A lime chew was selected 2 times

then for the orange ones we have a relative frequency (that can be thought as the probability ) of

          P₀ = 12/21

    For apple ,  Pₐ = 7/21

    For lime ,  Pi = 2/21

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after 5 years with different compounding frequencies: a. Monthly: $13,952.19 b. Daily: $14,059.89 c. Quarterly: $13,784.98 d. Weekly: $14,001.84

How to solve the question ?

To calculate the balance after 5 years with compound interest, we can use the formula:

A = P (1 + r/n) in power nt

where A is the balance after t years, P is the principal balance, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years.

For this problem, P = $10,000 and r = 6.92%. To find the balance after 5 years with different compounding frequencies, we can use the following values for n:

a. Monthly: n = 12 A = $10,000 (1 + 0.0692/12)¹²*⁵ = $13,952.19

b. Daily: n = 365 A = $10,000 (1 + 0.0692/365)³⁶⁵*⁵ = $14,059.89

c. Quarterly: n = 4 A = $10,000 (1 + 0.0692/4)⁴*⁵= $13,784.98

d. Weekly: n = 52 A = $10,000 (1 + 0.0692/52)⁵²*⁵ = $14,001.84

Therefore, after 5 years with different compounding frequencies: a. Monthly: $13,952.19 b. Daily: $14,059.89 c. Quarterly: $13,784.98 d. Weekly: $14,001.84

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Step-by-step explanation:

Area of a circle = pi r^2

   44  cm^2      = pi r^2

     r  =    sqrt (44/pi)

     r  = 3.74 cm

Answer:

≈ 3,74 cm

Step-by-step explanation:

Given:

A (area) = 44 cm^2

Find: r (radius) - ?

[tex]a = \pi {r}^{2} [/tex]

[tex]\pi {r}^{2} = 44[/tex]

[tex] {r}^{2} = \frac{44}{\pi} [/tex]

[tex]r = \sqrt{ \frac{44}{\pi} } ≈3.74[/tex]

The total mass in tones of the 2500 books is 1.75 tones

A proportion is defined as an equation having two ratios that are set equal to each other.

From the information given, we have that;

There are 2500 books

But then, a book weighs 700 grams

So, we have;

1 book = 700 grams

then, 2500 books = x grams

cross multiply

x = 1, 750, 000 grams

Let's convert the grams to tones , we get;

I gram = 10⁻⁶

Then 1. 75 ×  10⁻⁶ = x

cross multiply

x = 1. 75 tones

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If a 25 bag-sample has mean of 415 grams and variance of 400, then the p-value of test statistic is 0.0169.

To test whether the bag filling machine is working correctly at the 424 gram setting, we use one-sample t-test.

The null hypothesis is that mean weight of the bags filled by machine is 424 grams, and

The Alternative hypothesis is that the mean weight is less than 424 grams.

The test statistic "t" :

⇒ t = (415 - 424)/√(400/25)) = -2.25,

The degrees of freedom for this test is = 25-1=24.

To find the p-value of the test statistic, we need to determine the probability of observing a t-value less than or equal to -2.25,

Using a t-distribution table with 24 degrees of freedom, the p-value corresponding to t = -2.25 is approximately 0.0169.

Therefore, the p-value of the test statistic is 0.0169.

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Answer: 20+12=44-12

Step-by-step explanation:

You can distribute the multiplication on both sides of the equation.

Whenever you see a number before parentheses, it means to multiply that number to all terms in the parentheses.

so the new equation will be: (4 * 5) +(4 *3) = (2 *22) - (2*6)

Next, we simplify: 20 + 12 = 44 - 12

If the above equation is what you are looking for then use that but you can simplify further and add the numbers together so it will be

32=32  

The area of the square ABCD is approximately 420/π cm².

To find the area of the square, follow these steps:
1. Calculate the radius of the circle:
Since the area of the circle is 105 cm², you can use the formula for the area of a circle,

which is A = πr²,

where A is the area and r is the radius.
105 = πr²
Divide both sides by π:
r² = 105/π
Take the square root of both sides:
r = √(105/π)
2. Calculate the side length of the square:
Since the circle touches all 4 sides of the square, the diameter of the circle is equal to the side length of the square. The diameter is twice the radius (d = 2r), so:
d = 2√(105/π)
3. Calculate the area of the square:
Now, use the formula for the area of a square, which is A = s²,

where A is the area and s is the side length. In this case, s = d:
A = (2√(105/π))²
A = 4 * (105/π)
4. Multiply to find the area:
A = 420/π cm².

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The true statements about the image △A'B'C' include the following:

A. AB is parallel to A'B'.

B. [tex]D_{O, \frac{1}{2} }[/tex] (x, y) = (1/2x, 1/2y)

C. The distance from A' to the origin is half the distance from A to the origin.

In Mathematics and Geometry, a dilation simply refers to a type of transformation which typically changes the size of a geometric object, but not its shape.

This ultimately implies that, the size of the geometric shape would be increased (stretched or enlarged) or decreased (compressed or reduced) based on the scale factor applied.

Next, we would apply a dilation to the coordinates of the pre-image by using a scale factor of 0.5 centered at the origin as follows:

Ordered pair A (-4, 3) → Ordered pair A' (-4 × 1/2, 3 × 1/2) = Ordered pair B' (-2, 1.5).

Ordered pair B (4, 4) → Ordered pair B' (4 × 1/2, 4 × 1/2) = Ordered pair B' (2, 2).

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A fifth-degree polynomial can have at most 4 critical numbers, and hence 4 relative extrema. It can have at most 3 points of inflection.

Explanation:
1. A polynomial function of degree n has (n-1) critical numbers, which are points where the derivative is either zero or undefined. Since a fifth-degree polynomial has a degree of 5, it can have at most (5-1)=4 critical numbers.

2. Relative extrema are local maximum or minimum points of the function. A relative extrema occurs at a critical number where there is a change in the sign of the first derivative (going from positive to negative or negative to positive). Since we have at most 4 critical numbers, there can be at most 4 relative extrema for a fifth-degree polynomial.
3. Points of inflection are points on the graph where the function changes its concavity (from concave up to concave down or vice versa). This occurs when the second derivative of the function changes sign. To find the points of inflection, we need to find the critical numbers of the first derivative, which is a fourth-degree polynomial (one degree less than the original polynomial). A fourth-degree polynomial has at most (4-1)=3 critical numbers.
4. Therefore, a fifth-degree polynomial can have at most 4 relative extrema and 3 points of inflection.

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According to the puzzle the specified values are;

1. TU = 156.15m    

2. Area of rectangle = 12335.85 m²

3. GH = 182.33m

4. Area of triangle = 8934.17 m²

Three line segments meet at three non-collinear points to form a triangle, a geometric shape. The three points where the line segments meet are referred to as the vertices of the triangle, while the three line segments themselves are referred to as the sides of the triangle.

1. Given triangle TUV apply Pythagoras theorem,

175² = 79² + TU²   after simplification,

TU = 156.15m                   (option A is correct)

2. Area of rectangle = length × width

Area of rectangle = TU × UV  =  156.15 × 79  

Area of rectangle = 12335.85 m² (option G is correct)

3. Given triangle GHI apply Pythagoras theorem,

207² = 98² + GH²   after simplification,

GH = 182.33m    (option I is correct)

4. Area of triangle = [tex]\frac{1}{2}[/tex]  × base × height

Area of triangle =  [tex]\frac{1}{2}[/tex]  × HI × GH

Area of triangle =  [tex]\frac{1}{2}[/tex]  × 98m × 182.33m

Area of triangle = 8934.17 m²

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6 Students can arrange themselves in 720 different ways to sit in the front row of an auditorium.

To determine the number of ways 6 students can arrange themselves to sit in the 6 seats in the front row, you can use

the concept of permutations.

There are 6 seats, so for the first seat, there are 6 choices of students who can sit there.

For the second seat, 5 students remain, so there are 5 choices.

For the third seat, 4 students remain, so there are 4 choices.

For the fourth seat, 3 students remain, so there are 3 choices.

For the fifth seat, 2 students remain, so there are 2 choices.

For the last seat, only 1 student remains, so there is 1 choice.

Now, multiply the number of choices for each seat together to find the total number of arrangements:

6 × 5 × 4 × 3 × 2 × 1 = 720 ways.

So, 6 students can arrange themselves in 720 different ways to sit in the front row of an auditorium.

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Surface area of rectangular prism is 280/9yd²

A rectangular prism, also known as a rectangular cuboid, is a three-dimensional geometric shape that has six rectangular faces, where each face is perpendicular to the adjacent faces. It is a special type of cuboid where all the faces are rectangles.

The rectangular prism has eight vertices, 12 edges, and six faces, where each face has a pair of congruent and parallel rectangular sides. The opposite faces of the rectangular prism are congruent and parallel, which means that it has a uniform cross-section throughout its length.

Sides of rectangular prism are

l=3yd

b=1 1/3yd=4/3yd

h=2 2/3yd=8/3yd

Surface area of rectangular prism=2(lb+bh+lh)

=2(3×4/3+4/3×8/3+8/3×3)

=2(4+8+32/9)

=280/9yd²

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Answer:

Volume= 66.7π

Step-by-step explanation:

Volume = 1/3 ×π×h×r^2

V = 1/3 ×π ×8× 5 ^2

V = 66.7πinches^ 3

Answer: 90

Step-by-step explanation:

To find the median speed of the drives, list the speeds in order from lowest to highest and find the middle value. If there's an even number of speeds, find the average of the two in the middle.

To find the median speed of the drives in miles per hour, you would first arrange the speeds in order from smallest to largest and then identify the speed that sits exactly in the middle of this list. If there is an even number of observations, the median is the average of the two middle speeds.

For instance, if the speed of the drives listed in the table are 50 mph, 60 mph, 70 mph, 80 mph, and 90 mph, the median would be 70 mph. If the speeds were 50 mph, 70 mph, 80 mph, and 90 mph, the median would be the average of 70 and 80 mph, which is 75 mph.

The frequency table is useful in identifying how often each speed occurred, but it won't directly affect your calculation of the median. It's important to understand this concept as part of a broader understanding of statistics and data analysis.

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Answer:

see explanation

Step-by-step explanation:

(a)

the sum of the interior angles of a polygon is

sum = 180° (n - 2) ← n is the number of sides

here n = 7 , then

sum = 180° × (7 - 2) = 180° × 5 = 900°

(b)

since the polygon is regular then the 7 interior angles are congruent

each interior angle = 900° ÷ 7 ≈ 128.6° ( to 1 decimal place )

(c)

the sum of the exterior angles of a polygon is 360°

since the polygon is regular then the 7 exterior angles are congruent

each exterior angle = 360° ÷ 7 ≈ 51.4° ( to 1 decimal place )

Answer:

a) 900°

b) ≈ 128,6°

c) ≈ 51,4°

Step-by-step explanation:

a) s = (n - 2) × 180°

n - the number of sides (in this case, it's 7)

s - the sum of interior angles

s = (7 - 2) × 180° = 5 × 180° = 900°

.

b) In order to find the size of each angle, we have to divide the sum by the number of sides:

900° / 7 ≈ 128,6°

.

c) Since the sum of exterior angles of a polygon is 360°, we can find the size of one exterior angle by dividing this sum by the number of sides:

360° / 7 ≈ 51,4°

Step-by-step explanation:

Top floor is x     next floor is 2 x   then the bottom is 4 x

= 7x

7x = 245    x = 35

then  35 ft    70 ft   140 ft   = 245 ft total

Step-by-step explanation:

no calculator at hand ?

-1963 - x = -9512

-x = -9512 + 1963 = -7549

x = 7549

so, 7549 has to be subtracted.

y = (46.5 - (0.5b / 58))x + b This equation allows us to plug in any value for x (the number of miles driven) and calculate Kristen's total pay for that month.

To write an equation in slope intercept form to find Kristen's total pay, we need to first determine the variable values for the equation. In this case, we know that Kristen is paid the same salary each month, so we can assign a constant value to represent her base pay. Let's call this value "b" for simplicity.

Next, we need to determine the relationship between the number of miles Kristen drives and the extra pay she receives. We can do this by calculating the rate at which she is paid per mile. To do this, we can use the formula:

Pay per mile = (Total pay - Base pay) / Number of miles driven

Using the values given in the problem, we can calculate the pay per mile for March and April as follows:

March:

Pay per mile = ($7,748.00 - b) / 58 miles

April:

Pay per mile = ($8,532.00 - b) / 72 miles

To find the overall slope of the equation, we can calculate the difference in pay per mile between March and April:

Slope = (Pay per mile for April - Pay per mile for March) / (Number of miles driven in April - Number of miles driven in March)

Slope = (($8,532.00 - b) / 72 miles) - (($7,748.00 - b) / 58 miles)) / (72 miles - 58 miles)

Simplify the equation and solve for the slope:

Slope = 46.5 - (0.5b / 58)

Now that we have the value of the slope, we can write the equation in slope intercept form:

y = mx + b

y represents Kristen's total pay, m represents the slope we just calculated, x represents the number of miles she drives, and b represents her base pay.

Putting it all together, the final equation for Kristen's total pay in a month given the number of miles she drives is:

y = (46.5 - (0.5b / 58))x + b

This equation allows us to plug in any value for x (the number of miles driven) and calculate Kristen's total pay for that month.

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If 4 vertices of a "regular-tetrahedron" are snipped off which leaves a "triangular-face" in place of each corner, then there will be 18 edges in the new polyhedron.

If we snip off the 4 vertices of a "regular-tetrahedron", leaving a triangular face in place of each corner and a hexagonal-face in place of each original face of the tetrahedron,

We get a truncated tetrahedron.

we know that a truncated-tetrahedron has ⇒ four regular "hexagonal-faces", four equilateral "triangular-faces", 12 vertices and 18 edges.

Therefore, the new polyhedron will have 18 edges.

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