Let X and Y have joint pdf .
a. Compute P(X < 1/2 Ç Y > 1/4).
b. Derive the marginal pdfs of X and Y.
c. Are X and Y independent? Show some calculations in support of your answer.
d. Derive f(x|y) = {the conditional pdf of X given Y=y}

Answer:

[tex] \sqrt{9} \sqrt{9} = 9[/tex]

Based on the information, we can infer that the mean is 14.28, the median is 15, and there is no unique mode (several repeated values).

To calculate the measures of central tendency, we will use the data provided in the table. The pairs of shoes sold on each day are as follows: 10, 14, 15, 12, 16, 18, 16.

Mean:

The mean is calculated by adding all the values and dividing by the total number of items.

Mean = (10 + 14 + 15 + 12 + 16 + 18 + 16) / 7 = 101 / 7 = 14.28

Median:

The median is the value in the middle of an ordered data set. To calculate it, we first order the values from smallest to largest.

10, 12, 14, 15, 16, 16, 18

Since there are an odd number of values, the median is the value in the middle position, which in this case is 15. Therefore, the median is 15.

Mode:

The mode is the value that occurs most frequently in a data set. In this case, there is no value that is repeated more times than the others. The values 16 and 12 are repeated twice each, but there is no single mode. Therefore, there is no single mode in this data set.

Note: Here is the question in English:

The table shows the number of shoes sold in 1 store during a week, calculates the measures of central tendency, and perform the corresponding analysis.

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Answer:

log(3x⁹y⁴) = log 3 + 9 log x + 4 log y

Answer:

[tex]\log 3+ 9\log x +4 \log y[/tex]

Step-by-step explanation:

Given logarithmic expression:

[tex]\log 3x^9y^4[/tex]

[tex]\textsf{Apply the log product law:} \quad \log_axy=\log_ax + \log_ay[/tex]

[tex]\log 3+\log x^9 +\log y^4[/tex]

[tex]\textsf{Apply the log power law:} \quad \log_ax^n=n\log_ax[/tex]

[tex]\log 3+ 9\log x +4 \log y[/tex]

Only Graph A satisfies the criteria for a density curve, making it the correct answer.

Graph A represents a density curve because the area under the curve equals 1, and the curve is above the horizontal axis. In a density curve, the total area under the curve represents the probability, and it should always equal 1.

This indicates that the curve represents a probability distribution, where the probability of an event occurring within a certain range is given by the area under the curve within that range.

The fact that the curve is above the horizontal axis indicates positive values, which is consistent with a density curve representing a probability distribution.

On the other hand, Graph B does not represent a density curve because the area under the curve equals 2, which is not valid for a probability distribution. The area under a density curve should always equal 1, indicating that the total probability is accounted for.

As a result, only Graph A meets the requirements for a density curve, making it the right response.

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Answer:

The slope-intercept equation is y = four-thirds x + 1

Step-by-step explanation:

slope:  [tex]\frac{4}{3}[/tex]

point: (3, 5)

y = mx + b

[tex]y=\frac{4}{3}x+b[/tex]

[tex]5=\frac{4}{3}(3)+b[/tex]

[tex]5=4+b[/tex]

[tex]b=5-4[/tex]

[tex]b=1[/tex]

Equation: [tex]y=\frac{4}{3}x+1[/tex]

[tex]-2=\frac{4}{3}(0)+1[/tex]

[tex]-2=0+1[/tex]

[tex]-2\neq 1[/tex]

Thus, the second statement is true!

Answer:

-2(3x + 12y - 5 - 17x - 16y + 4)

-6x - 24y + 10 + 34x + 32y - 8

-6x + 34x - 24y + 32y + 10 - 8

28x + 8y + 2

The predicted population of Knox, rounded to the nearest whole number, after 8 years is 32,599.

To predict the population of Knox after 8 years, we can use the given information that the population is currently 42,000 and it is declining at a rate of 3.2% per year.

To calculate the population after 8 years, we need to apply the rate of decline for each year. Let's break down the calculation step by step:

Calculate the population after the first year:

Population after 1 year = 42,000 - (3.2% of 42,000)

= 42,000 - (0.032 * 42,000)

= 42,000 - 1,344

= 40,656

Calculate the population after the second year:

Population after 2 years = 40,656 - (3.2% of 40,656)

= 40,656 - (0.032 * 40,656)

= 40,656 - 1,299.71

= 39,356.29

Continue this process for each year up to 8 years, applying the 3.2% rate of decline each time.

After performing these calculations for each year, we arrive at the population after 8 years:

Population after 8 years ≈ 32,599

Therefore, the predicted population of Knox, rounded to the nearest whole number, after 8 years is 32,599.

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Answer:

it 10:579

Step-by-step explanation:

it is the anser

Answer:

violence figer in the past two years

The examples match with the following properties:

1. C. Distributive Property

2. A. Commutative Property

3. B. Associative Property

4. B. Associative Property

The correct matching of the examples on the left with the corresponding properties on the right is as follows:

1. 3(x + 3) = 3x + 9 : C. Distributive Property

  This example demonstrates the distributive property, where we distribute the factor 3 to both terms inside the parentheses, resulting in 3 times x and 3 times 3, which simplifies to 3x + 9.

2. 2 + 3 + 4 = 4 + 3 + 2 : A. Commutative Property

  This example illustrates the commutative property of addition, which states that the order of adding numbers does not affect the sum. Here, the numbers are rearranged on both sides of the equation, but the sum remains the same.

3. 4(2 x 3) = (4 x 2)3 : B. Associative Property

  This example showcases the associative property of multiplication, where the grouping of factors does not affect the product. The expression on the left side is calculated as 4 times the product of 2 and 3, while the expression on the right side is calculated as the product of 4 and 2, followed by multiplying the result by 3. Both calculations yield the same result.

4. 6 + (7 + x) = (6 + 7) + x : B. Associative Property

  This example also demonstrates the associative property, but in the context of addition. The expression on the left side associates the addition of 7 and x first, while the expression on the right side associates the addition of 6 and 7 first. Both expressions ultimately add up to the same sum.

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The question probable may be:

Match the example on the left with the corresponding property on the right.

1. 3(x + 3) = 3x + 9

2. 2 + 3 + 4 = 4 + 3 +2.

3. 4(2 x 3) = (4 x 2)3

4. 6 + (7 + x) = (6 + 7) + x

A. Commutative Property

B. Associative Property

C. Distributive Property

Todd should expect approximately 624 people at his gym during the week of July 15.

A: To find the quadratic equation that models the data, we can use the vertex form of a quadratic equation:

[tex]y = a(x - h)^2 + k[/tex] where (h, k) represents the vertex of the parabola.

Let's analyze the data to determine the vertex. We observe that the number of people is highest during the first week and gradually decreases over the following weeks.

This suggests a downward-opening parabola.

From the data, the highest point occurs during the week of 6/3 to 6/9 with 624 people.

Therefore, the vertex is located at (6/3 to 6/9, 624).

Using the vertex form, we have:

[tex]y = a(x - 6/3 to 6/9)^2 + 624[/tex]

Now, we need to find the value of 'a.'

To do this, we can substitute any other point and solve for 'a.' Let's use the data from the week of 5/27 to 6/2:

[tex]618 = a(5/27 to 6/2 - 6/3 to 6/9)^2 + 624[/tex]

Simplifying the equation and solving for 'a,' we find:

[tex]618 - 624 = a(-6/3)^2[/tex]

-6 = 4a

a = -3/2

Therefore, the quadratic equation in vertex form that models the data is:

[tex]y = (-3/2)(x - 6/3 to 6/9)^2 + 624[/tex]

B: To predict the number of people Todd should expect during the week of July 15, we substitute x = 7/15 into the equation and solve for y:

[tex]y = (-3/2)(7/15 - 6/3 to 6/9)^2 + 624[/tex]

Simplifying the equation, we find:

[tex]y = (-3/2)(1/15)^2 + 624[/tex]

y = (-3/2)(1/225) + 624

y = -3/450 + 624

y = -1/150 + 624

y = 623.993

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Answer:

The table shows that there are a total of 142 books on Diana's bookshelf, with 77 books written by female authors and the rest by male authors. However, the number of non-fiction books written by female authors is missing from the table, so it is impossible to determine the exact number without more information.

Step-by-step explanation:

Answer:

104 books

Step-by-step explanation:

                 Fiction         Non-Fiction      Total

Male              36                     38

Female                                    x

Total              77                     142

----------------------------------------------------------------------------

x = 142 - 38 = 104

The diameter of the engine cylinder can be within the range of 4.995 cm to 5.005 cm (option e).

Given that the diameter of the engine cylinder needs to be 5 cm wide with a tolerance of ± 0.005 cm.

To determine the permissible range of the diameter, we need to consider both the upper and lower limits.

Upper limit: Add the tolerance to the desired diameter.

  Upper limit = 5 cm + 0.005 cm = 5.005 cm.

Lower limit: Subtract the tolerance from the desired diameter.

  Lower limit = 5 cm - 0.005 cm = 4.995 cm.

Therefore, the permissible range for the diameter of the engine cylinder is between 4.995 cm and 5.005 cm.

Hence, the final answer is that the diameter of the engine cylinder can be within the range of 4.995 cm to 5.005 cm.

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The number line that represents the solution set for the inequality 3(8 – 4x) < 6(x – 5) is option C: A number line from negative 5 to 5 in increments of 1. An open circle is at negative 3, and a bold line starts at negative 3 and is pointing to the left.

To determine the solution set for the inequality 3(8 – 4x) < 6(x – 5), we need to solve it step by step:

Simplify the inequality:

24 - 12x < 6x - 30

Combine like terms:

-12x - 6x < -30 - 24

-18x < -54

Divide both sides of the inequality by -18, remembering to flip the inequality sign:

x > (-54) / (-18)

x > 3

The inequality tells us that x must be greater than 3. To represent this on a number line, we place an open circle at the value 3 and draw a bold line pointing to the right to indicate that the solution set includes all values greater than 3.

Therefore, option C accurately represent the solution set for the inequality 3(8 – 4x) < 6(x – 5).

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Answer:

2147483648

Step-by-step explanation:

Write the geometric sequence as an explicit formula

[tex]8,\,32,\,128\rightarrow8(4)^0,8(4)^1,8(4)^2\rightarrow a_n=a_1r^{n-1}\rightarrow a_n=8(4)^{n-1}[/tex]

Find the n=15th term

[tex]a_{15}=8(4)^{15-1}=8(4)^{14}=8(268435456)=2147483648[/tex]

The exponential equation a¹⁰ = 200 is equivalent to the logarithmic equation Log = 200 a.

The rule of logarithms that we should use here is given below:

[tex]\log \text{x} = \text{a} \iff 10^{\text{a}} = \text{x}[/tex]

The given expression is Log = 200 a.

We can follow the rule log x = a ⇔ 10^a = x to convert this logarithmic equation to an exponential one.

Log = 200 a can be rewritten as a¹⁰ = 200

Therefore, we have found that the exponential equation a¹⁰ = 200 is equivalent to the logarithmic equation Log = 200 a.

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The mass of the sea glass in the box is 1265 grams.

To find the mass of the sea glass in grams, we need to subtract the mass of the empty box from the total mass of the box with the sea glass. Let's convert all the units to grams for consistency.

Mass of the empty box = 235 grams

Total mass of the box with sea glass = 1 1/2 kilograms = 1.5 kilograms = 1500 grams

To determine the mass of the sea glass, we subtract the mass of the empty box from the total mass:

Mass of the sea glass = Total mass of the box with sea glass - Mass of the empty box

Mass of the sea glass = 1500 grams - 235 grams

Performing the subtraction:

Mass of the sea glass = 1265 grams

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Answer:

Isaac Newton (late 17th century) - Newton made significant contributions to calculus and mathematical physics. His book "Philosophiæ Naturalis Principia Mathematica" laid the groundwork for classical mechanics.

Annette Creighton opened Creighton Consulting and rented a small office space. She also hired a part-time worker to handle phone calls and make deliveries.

1. Annette Creighton decided to start her own consulting firm and named it Creighton Consulting.

2. As part of setting up the business, she began by renting a small office space. This would serve as the headquarters for her consulting operations.

3. To ensure smooth communication and timely deliveries, Annette realized the need for assistance. She hired a part-time worker to handle incoming phone calls and make necessary deliveries on behalf of the company.

4. The part-time worker would be responsible for answering the phone and addressing client inquiries or forwarding them to Annette if necessary. This step was crucial to maintaining good customer service and professionalism.

5. Additionally, the worker would also handle deliveries, ensuring that any documents, packages, or materials were transported promptly to clients or other relevant destinations.

6. By hiring a part-time worker, Annette aimed to streamline her operations and focus on core consulting tasks, such as client meetings, project management, and providing expert advice to clients.

7. The worker's role would involve effective communication skills, organization, and attention to detail to represent Creighton Consulting in a professional manner.

8. Annette understood that investing in administrative support would free up her time and allow the business to operate more efficiently, ultimately benefiting both her and her clients.

9. With the office space secured and the part-time worker in place, Annette was ready to embark on her consulting journey, confident in her ability to handle client inquiries and maintain smooth business operations.

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Answer:

Step-by-step explanation:

12 14 33 35========1,234,000,124,581.

The  system of equation: -x + 7 = -2x³ + 5x² + x - 2 has three solutions

The correct answer is option D.

To solve the system of equations:

-x + 7 = -2x³ + 5x² + x - 2

We need to simplify and rearrange the equation to find its solutions. Let's start by combining like terms:

-x + 7 = -2x³ + 5x² + x - 2

Simplifying the left side:

7 = -2x³ + 5x² - x - 2

Next, let's arrange the equation in descending order of the variable's exponent:

-2x³ + 5x² - x - 2 = 7

Now, let's move all terms to one side of the equation to set it equal to zero:

-2x³ + 5x² - x - 2 - 7 = 0

Simplifying further:

-2x³ + 5x² - x - 9 = 0

To determine the number of solutions, we can analyze the degree of the equation. Since it is a cubic equation, it can have a maximum of three real solutions.

The given system of equations is:

-x + 7 = -2x³ + 5x² + x - 2

To find the solutions, we need to set the equation equal to zero. Let's rearrange the terms:

2. -2x³ + 5x² - x - 9 = 0

Now, we can try to factor or use numerical methods to solve the equation. However, factoring a cubic equation can be complex and time-consuming. In this case, we'll use numerical methods to approximate the solutions.

One common numerical method is the Newton-Raphson method, which involves making an initial guess for the solutions and iterating to converge on a more accurate solution.

Using numerical software or calculators, we can find the approximate solutions of the equation as follows:

x ≈ -1.607, x ≈ 1.279, and x ≈ 3.328

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The list price of each camera is approximately $26.59.

Let's assume the list price of each camera is represented by 'x'.

According to the given information, the merchant sold a certain number of cameras at the list price and received a total of $432. This can be expressed as:

Number of cameras * List price = Total revenue

x * Number of cameras = $432

Now, if the merchant had three more cameras and sold each one for $12 less, the new selling price would be (x - $12). The total revenue would still be $432. This can be expressed as:

(New number of cameras) * (New selling price) = Total revenue

(x + 3) * (x - $12) = $432

Expanding the equation:

x^2 - $12x + 3x - $36 = $432

x^2 - 9x - $468 = 0

To solve this quadratic equation, we can factor it or use the quadratic formula. Factoring this equation may not yield integer solutions, so we'll use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = -9, and c = -468. Plugging these values into the quadratic formula:

x = (-(-9) ± √((-9)^2 - 4 * 1 * (-468))) / (2 * 1)

x = (9 ± √(81 + 1872)) / 2

x = (9 ± √(1953)) / 2

Calculating the square root of 1953 gives us approximately 44.17. Therefore, the two possible values for x are:

x1 = (9 + 44.17) / 2 ≈ 26.59

x2 = (9 - 44.17) / 2 ≈ -17.59

Since the price of a camera cannot be negative, we discard x2. Therefore, the list price of each camera is approximately $26.59.

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The measure of the outside angle F indicated in the figure is 61 degrees,

The external angle theorem states that "the measure of an angle formed by two secant lines, two tangent lines, or a secant line and a tangent line from a point outside the circle is half the difference of the measures of the intercepted arcs.

Expressed as:

Outside angle = 1/2 × ( major arc - minor arc )

From the figure:

Major arc = 195 degrees

Minor arc = 73 degrees

Outside angle F = ?

Plug the value of the minor and major arc into the above formula and solve for the outside angle F:

Outside angle = 1/2 × ( major arc - minor arc )

Outside angle = 1/2 × ( 195 - 73 )

Outside angle = 1/2 × ( 122 )

Outside angle = 122/2

Outside angle = 61°

Therefore, the outside angle measures 61 degrees.

Option C) 61° is the correct answer.

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To determine the appropriate verb form, consider the time of the action, its relation to the present or past events, and whether it is a general fact, completed action, ongoing situation, or action preceding another. Choose the verb tense that accurately conveys the intended meaning in the paragraph.

To determine which form of verb is appropriate to use in a paragraph, you need to consider the context and the intended meaning of the sentence or paragraph. Here are some general guidelines for using different tenses:

Present Simple Tense:

Use the present simple tense to talk about general facts, habits, routines, and permanent situations.

Example: "The sun rises in the east."

Past Simple Tense:

Use the past simple tense to talk about completed actions or events in the past.

Example: "She studied abroad last year."

Present Perfect Tense:

Use the present perfect tense to talk about past actions or events that have a connection to the present or when the exact time of the action is not specified.

Example: "I have visited Paris several times."

Past Perfect Tense:

Use the past perfect tense to talk about an action or event that happened before another past action or event.

Example: "She had already eaten dinner when I arrived."

To determine which tense to use, consider the timeline of events and the relationship between them. If you are referring to a specific time in the past, the past simple tense might be appropriate. If you want to emphasize the connection to the present, the present perfect tense might be suitable. If you need to establish a sequence of events in the past, the past perfect tense could be used.

However, it's important to note that these guidelines are not absolute, and there can be variations based on specific contexts and writing styles. It's always best to consult grammar rules and consider the meaning and context of your sentences to choose the most appropriate verb tense for your paragraph.

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a) The distribution of X, the student loan debt of a randomly selected college graduate, is normal with a mean of $25,200 and a standard deviation of $11,200. b) The probability is approximately 7.28%.

c) The middle lies between approximately $22,164 and $28,536.

a. The distribution of X, the student loan debt of a randomly selected college graduate, is a normal distribution (bell-shaped curve) with a mean (μ) of $25,200 and a standard deviation (σ) of $11,200. We can represent this as X ~ N(25200, 11200).

b. To find the probability that the college graduate has between $27,250 and $43,650 in student loan debt, we need to calculate the z-scores for these two values and then find the area under the normal curve between those z-scores.

First, we calculate the z-score for $27,250:

z1 = (X1 - μ) / σ = (27250 - 25200) / 11200 ≈ 1.8304

Next, we calculate the z-score for $43,650:

z2 = (X2 - μ) / σ = (43650 - 25200) / 11200 ≈ 1.6518

Now, we need to find the area under the normal curve between these two z-scores. We can use a standard normal distribution table or a calculator to find this area.

Using a standard normal distribution table or a calculator, the probability is approximately P(1.6518 ≤ Z ≤ 1.8304) ≈ 0.0728.

c. To find the middle 20% of college graduates' loan debt, we need to find the range of values that contain the central 20% of the distribution. This range corresponds to the values between the lower and upper percentiles.

The lower percentile is the 40th percentile (50% - 20%/2 = 40%) and the upper percentile is the 60th percentile (50% + 20%/2 = 60%).

Using a standard normal distribution table or a calculator, we can find the z-scores corresponding to these percentiles:

For the lower percentile (40th percentile):

z_lower = invNorm(0.40) ≈ -0.2533

For the upper percentile (60th percentile):

z_upper = invNorm(0.60) ≈ 0.2533

Now, we can convert these z-scores back to the corresponding loan debt values:

Lower debt value:

X_lower = μ + z_lower * σ = 25200 + (-0.2533) * 11200 ≈ $22,164

Upper debt value:

X_upper = μ + z_upper * σ = 25200 + 0.2533 * 11200 ≈ $28,536

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A sequence of transformations that maps quadrilateral MATH onto quadrilateral M"A"T"H" is a rotation of 180° about the origin and a translation by 1 unit left and 1 unit up.

In Mathematics and Geometry, the rotation of a point 180° about the origin in a clockwise or counterclockwise direction would produce a point that has these coordinates (-x, -y).

Additionally, the mapping rule for the rotation of a geometric figure 180° counterclockwise about the origin is given by this mathematical expression:

(x, y)                                            →            (-x, -y)

Coordinates of point M (2, 4)  →  Coordinates of point M' = (-2, -4)

By applying a translation to the image (M') vertically upward by 1 unit and horizontally left by 1 unit, the new coordinate M" of quadrilateral M"A"T"H" include the following:

(x, y)                               →                  (x - 1, y + 1)

M' (-2, -4)                        →                  (-2 - 1, -4 + 1) = M" (-3, -3)

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The most appropriate name for the function that takes the length of a race and returns the time needed to complete it would be "time(length)".

When choosing a name for a function, it is important to consider clarity and readability. The name should accurately describe the purpose of the function and provide a clear indication of what it does.

In this case, the function is expected to take the length of a race as input and return the time needed to complete it as output. Among the given options, "time(length)" is the most suitable choice.

a. length(time): This name suggests that the function takes time as input and returns the length. However, in this scenario, we are interested in finding the time needed to complete the race based on its length, so this option is not the best fit.

b. Time(race): This name implies that the function takes a race as input and returns the time. While it conveys the idea of finding the time, it doesn't explicitly mention that the input is the length of the race, making it less clear.

c. time(length): This option accurately describes the purpose of the function, indicating that it takes the length of the race as input and returns the corresponding time. It is concise, clear, and aligns with the conventional naming conventions for functions.

d. cost(time): This name suggests that the function calculates the cost based on time, which is not relevant to the scenario of finding the time needed to complete a race.

Therefore, "time(length)" is the most suitable and appropriate name for the function.

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Answer:

7(20) + (1/2)(20)(9) = 140 + 90 = 230 cm²