(Order statistics and independence) Let X be the minimum and Y the maximum of two independent, nonnegative random variables S and T with common continuous density f. Let Z denote the indicator function of the event (S > 2T). Vhat is the distribution of Z? b) Are X and Z independent? Are Y and Z independent? Are (X, Y) and Z independent? c) Is X independent of Y?

Answer:

The answer is:-

Step-by-step explanation:

x^3+y^3+z^3=k

xyz^9=k

xyz/9=k

xyz=9k

Hence, k=9 and, x, y and z are 1.

The length of the hypotenuse or the value of 'x' is 60.

We know a right-angled triangle has three sides they are -: Hypotenuse,

Opposite and Adjacent.

We can remember SOH CAH TOA which is,

sin = opposite/hypotenuse, cos = adjacent/hypotenuse and

tan = opposite/adjacent.

From the given diagram respect to the reference angle of 37°,

Hypotenuse is 'x' and adjacent is 48.

And we know,  cos = adjacent/hypotenuse.

Therefore, cos37° = 48/x.

x = 48/0.8.

x = 60.

Q. A right-angle triangle is shown, Find the value of x.

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1) The  model is R = 1820 - 19t

2) After six years R = 1706 barrels

3) There would be non more oil in 96 years.

We know that an equation can be used as a model. This implies that we can be able to use an equation to show how a variable is changing over time and that is what the problem that we have here is all about.

If at the current time we have 1820 billion barrels Then they decrease by 19 barrels per year and t is the number of years then the model is;

R = 1820 - 19t

After six years we would have;

R = 1820 - (19 * 6)

R = 1706 barrels

When there are no further barrels;

0 = 1820 - 19t

1820 = 19t

t = 1820/19

t = 96 years.

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The steps regarded essential for a successful simulation study are:

a. Problem formulation

b. Model validation

c. Model verification

d. Experimental design

e. Output analysis

To give any solution to the problem the very first step necessary is to understand the problem . Now through the observations of the process the problem is defined and formulated .

Keeping in mind the real life solutions and necessities a model is developed . Now the further process of verification and validation is carried out . In verification we make sure that the model works as we thought through animation whereas validation makes sure that the model reflects reality .

Experimental design multiple similar models are compared with minute changes and the performance are compared to find the best suited model.

Finally the results of the study are discussed in output analysis through written reports , documents or presentation and the best product is chosen .

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The translation needed to obtain the graph of g(x) = 7^x-4 from the graph of (x) = 7^x is 4 units right

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

g(x) = 7^x-4

f(x) = 7^x

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

g(x) = f(x - 4)

This represents a translation to the right by 4 units

Hence, the translation is 4 units right

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The volume of the cube is 1 cubic meter.

Option B is the correct answer.

It is a polygon having six faces.

The volume of a cube is side³.

Example:

The volume of a cube of 3 cm in length is 27 cm³.

We have,

Side length = 100 cm

Now,

Volume.

= Side³

= 100³

= 1000000 cm³ _____(1)

Now,

1 m = 100 cm

1 m³ = 1000000 cm³ _____(2)

From (1) and (2),

Volume = 1 cubic meter.

Thus,

The volume of the cube is 1 cubic meter.

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a) Probability, P(y = 2) = P(AA) is equals to the 0.81.

b) Probability, P(y = 3), is equals to the 0.162.

c) For P(y=5), 5ᵗʰ battery must be an A. four outcome for which Y= 5 , probability is equals to the 0.00324.

d) General formula for p(y) is P(y)= (y-1)(0.1)⁽ʸ⁻²⁾(0.9)².

A new battery’s voltage may be acceptable (A) or unacceptable (U). Assume that 90% of all batteries have acceptable voltages. So, probability of acceptance = 0.90 and

probability of unacceptance = 1 - 0.9 = 0.1

Let the number of batteries that must be tested be "Y". A particular flashlight requires two batteries, that is batteries will be independently selected.

a) For P(y = 2), both the batteries must be acceptable. Hence probability, P(y = 2)

= P(AA) = 0.90× 0.90 = 0.81

b)For P(y = 3 ) = P(AUA) + P(UAA)

=0.9× 0.1×0.9 + 0.1× 0.9× 0.9 = 0.162

C) For any(Y = n), nᵗʰ batterry should be acceptable, as search for two acceptable batteries will complete once second acceptable battery gets selected. Hence for P(y=5), 5ᵗʰ battery must be an A. four outcome for which Y

= 5 are (AUUUA) ,(UAUUA),(UUAUA),(UUUAA)

P(y=5) = P(AUUUA) + P(UAUUA) + P(UUAUA) + P(UUUAA)

= 4× 0.9² × 0.1³

= 0.00324

d) From above probabilities we conclude, the general formula for p(y) is p(y) =(y- 1)(0.1)⁽ʸ⁻²⁾(0.9)².

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Answer: The passion fruit will run out first and 420 ounces left.

Step-by-step explanation: If you put the same amount of cranberry and passion fruit in each juice than they will be using an equal amount each time. We know that there are more ounces of cranberry than passion fruit which already tells us that there will be more cranberry when the passion fruit runs out. In order to find how much is left we can simply subtract the amount of cranberry and passion fruit.

1100-680=420

Answer: The equation of a line in the form of y = mx + b can be found using the slope-intercept form, where m is the slope of the line and b is the y-intercept. To find the slope, you can use the two points on the line: (1, 2) and (3, 5). The slope is found by the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the two points.

m = (5 - 2) / (3 - 1) = 3 / 2 = 1.5

The y-intercept can be found by plugging in one of the points and the slope into the equation and solving for b. Let's use the point (1, 2).

2 = 1.5 * 1 + b

b = 2 - 1.5 = 0.5

So the equation of the regression line is:

y = 1.5x + 0.5

Therefore, the correct answer is: yˆ = 1.5x + 0.5

Step-by-step explanation:

The value of sec2(pi/5) - tan2(pi/5) is equals to  1.

What is trigonometric identity ?

A trigonometric identity is a mathematical equation that relates the values of trigonometric functions of an angle. These identities are true for all values of the angle for which both sides of the equation are defined.

We can use the trigonometric identities to simplify this expression. Recall that:

sec(x) = 1/cos(x)

tan(x) = sin(x)/cos(x)

sin2(x) + cos2(x) = 1

Using these identities, we can rewrite the expression as:

sec2(pi/5) - tan2(pi/5)

= (1/cos2(pi/5)) - (sin2(pi/5)/cos2(pi/5))

= (1 - sin2(pi/5))/cos2(pi/5)

= cos2(pi/5)/cos2(pi/5) (using the identity sin2(x) + cos2(x) = 1)

= 1

Therefore, sec2(pi/5) - tan2(pi/5) = 1.

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The height of the wall is 14 feet

Similar triangles are triangles that have the same shape, but their sizes may vary. All equilateral triangles, squares of any side lengths are examples of similar objects. In other words, if two triangles are similar, then their corresponding angles are congruent and corresponding sides are in equal proportion.

Therefore ;

TH/ JS = TV/VS

4/JS = 6/21

21× 4 = 6 JS

JS = 21× 4/6

JS = 84/6

JS = 14 feet

therefore the height of the wall is 14 feet

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F ∩ H = [4, 9)

FUH= ∞

Generally, To find the intersection of F and H, denoted as F ∩ H, we need to identify the values that are in both sets F and H.

From the definitions of F and H, we have:

F = {v | v=4 or v>4}

H = {v | v<9}

The values that are in both F and H are those that satisfy both conditions, i.e., v=4 or v>4 and v<9. This is equivalent to just the condition v=4, since any value that satisfies v<−4 will also satisfy v>4 and v<9.

Therefore, F ∩ H = {v | 4}.

In interval notation, we can write this as:

F ∩ H = [4, 9)

To find the union of F and H, denoted as F ∪ H, we need to identify all the values that are in either set F or set H or both.

From the definitions of F and H, we have:

F = {v | v=4 or v>4}

H = {v | v<9}

this denotes

F=  {v | v=4> x > ∞}

H = {v |-∞ <v<9}

FUH= ∞

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9. a)The required value of h is 4

9.b) The required value of h is 4

10.a)The required value of h is -6

10.b) They are linearly dependent for all values of h.

Here the given vectors are:

v1 = [tex]\left[\begin{array}{ccc}1\\-3\\2&\end{array}\right][/tex]

v2 = [tex]\left[\begin{array}{ccc}-3\\10\\-6&\end{array}\right][/tex]

v3 = [tex]\left[\begin{array}{ccc}2\\-7\\h&\end{array}\right][/tex]

The vectors v1 and v2 are not scalar multiples of one another so they are not linearly dependent.

[tex]\left[\begin{array}{ccc}1&-3&2\\-3&10&-7\\2&-6&h\end{array}\right][/tex]= 0

1(10h -42) -3(-14 +3h) +2 (18-20) =0

10h -9h-4 =0

h = 4

Hence the value of h is 4 so that the vector v3 is in span{v1,v2}

b) The given vectors will be linearly dependent if h =4

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Answer:

Step-by-step explanation:

24. Gravetter/Wallnau/Forzano, Essentials - Chapter 3 - End-of-chapter question 24

A sample yielded the following scores:

2, 3, 4, 4, 5, 5, 5, 6, 6, 7

n= 10

Assume that the scores are measurements of a discrete variable and find the median.

Median =5Correct

Points:

1 / 1

Median = 5

To find the median for a set of numbers with an even quantity the two middle numbers are added together and divided by 2.

n=102+(n = 102+1)2

= 5th score + 6th score2

= 5 + 52

= 102=5

Assume that the scores are measurements of a continuous variable and find the median by locating the precise midpoint of the distribution. (Use two decimal places.)

Median = 4.83Correct

The median for the discrete variable is 5 and for continuous variable is 5.

Median in statistics is the middle value of a list of data when arranged in order. It is a measure of central tendency that separates the data into two halves: the upper half and the lower half.

Given that: scores = 2, 3, 4, 4, 5, 5, 5, 6, 6, 7

a) The scores are in ascending order; then the median which is the middle number will be:

median = (5th term + 6 term)/2

= (5+5)/2

= 10/2

= 5

b) Use the cumulative percentage table ;

X     frequency    cumulative frequency     cumulative percentage

2         1                     1                                           10%

3         1                     2                                          20%

4        2                     4                                          40%

5        3                     7                                           70%

6        2                     9                                           90%

7        1                      10                                          100%

Fraction = the no. required to reach 50%/ no. in that interval

= 1/3

Since we have a total of 4 boxes when we reach the value of 4.5

The median of the continuous variable = 4.5 + (1/3)

= 4.833

≈ 5

Hence, the median for the discrete variable is 5 and for continuous variable is 5.

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Answer:

1/6

Step-by-step explanation:

1/4x2/3 = 2/12 = 1/6

The error which Jake made is likely to be a result of confusion or oversight in identifying the variable y as having a coefficient of 1.

The coefficients of an expression are the numerical factors that multiply the variables, while the constants are the numerical values that do not contain any variables.

In the expression 9m + 4 + y, the coefficients are 9 and 1 (which is the implied coefficient for y since there is no visible numerical factor), and the constant is 4.

So Jake correctly identified 9 as a coefficient and 4 as a constant, but missed the implied coefficient of 1 for the variable y. The error that Jake made is likely to be a result of confusion or oversight in identifying the variable y as having a coefficient of 1, rather than assuming it is a constant or neglecting it entirely.

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Answer:(A) 1.314.08

Step-by-step explanation:

Inflation

The creation of the bin.matrix using cbind() allows us to organize and manipulate our data in a way that is conducive to statistical analysis.

A binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, with a constant probability of success.

First, let's define what a binomial distribution is. For example, if we toss a fair coin 10 times, the number of heads that come up is described by a binomial distribution.

Now, let's talk about the cbind() function. cbind() is short for "column bind". It is used to combine vectors or matrices into a single matrix, where each input vector or matrix becomes a column in the final matrix.

In this exercise, we have created 7 vectors, each with a different success probability for a binomial distribution, ranging from 0.2 to 0.8. We will use cbind() to combine these vectors into a matrix called bin.matrix.

The dimension of the bin.matrix is

=> 300 x 7.

This means that the matrix has 300 rows and 7 columns. Each row represents a trial, and each column represents a different success probability. For example, the first column might represent a trial with a success probability of 0.2, while the second column might represent a trial with a success probability of 0.3, and so on.

By creating this matrix, we can easily perform statistical analyses on the data contained within it. We can calculate summary statistics, such as the mean and variance, for each column, or we can compare the distributions of the different columns to see if there are any significant differences between them.

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Step-by-step explanation:

reflect the shape, will give you:

A = (-1,0)

B= (3, -5)

C= (-6, -7)

D= (-4, -1)

If a point A(x, y) is dilated by a factor P, the new point would be at

A'(Px, Py).

A = -1/2.5 = -0.4             0/2.5= 0

= ( - 0.4, 0)

B = 3/2.5 =  1.2              -5/2.5 = -2

= (1.2, -2)

C = -6/2.5= -2.4       -7/2.5 = -2.8

= ( -2,4, -2.8)

D = -4/2.5 = -1.6.     -1/2.5 = -0.4

= ( -1.6, -0.4)

You're Welcome!

Answer:

Step-by-step explanation:

( -1.6, -0.4)

The value of x is 60.

We can use the fact that the opposite angles of a parallelogram are equal to find the value of x. Angle BCD is therefore equal to angle ABD, and angle CBD is equal to angle CAD.

Based on the given angles, we can construct two equations:

Angle ABD + angle CBD = 100 (because angle BCD = angle ABD) Angle ABD + angle ACD = 80 (because angle CAD = angle CBD)

When we substitute the given values, we get:

x + 40 = 100 x + 20 = 80

When we solve for x in both equations, we get:

x = 60 x = 60.

We can be confident that our answer is correct because both equations give us the same value for x. As a result, the value of x is 60.

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The first derivative of the composite function r(x) = √[x + √(x + √x)] is equal to r' (x) = [1 / [2 · √[x + √(x + √x)]]] · [1 + [1 / [2 · √(x + √x)]] · [1 + 1 / (2 · √x)]].

In this problem we find the case of a composite function of the form:

f ° g ° h (x) = f [g [h (x)]]

Whose first derivative must be found by means of the following differentiation rules:

Chain rule

d [f [u (x)]] / dx = (df / du) · (du / dx)

Derivative of a square root

d (√x) / dx = 1 / (2√x)

And the first order derivative of the function r(x) = √[x + √(x + √x)] is:

r' (x) = [1 / [2 · √[x + √(x + √x)]]] · [1 + [1 / [2 · √(x + √x)]] · [1 + 1 / (2 · √x)]]

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The transformation from ln ( x ) to ln ( x + 2 ) is horizontal transformation or shift.

What kinds of functions can be transformed?

Transformations can be divided into three categories: translations, reflections, and dilations. There are two types of transformations that can be done to a function: vertical (which affects the y-values) and horizontal (affects the x-values).

                               The four transformation variables are included in the equation of a function that is shown below (a, b, h, and k).

f(x) = ln(x)

if g(x) = ln(x + 2)

Then,

The transformation from ln(x) to ln(x+2) is horizontal transformation or shift. As 2 units are added in g(x) the function is horizontally shifted by 2 units in left-hand direction.

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Both items have different rate of change, m₁ = -2, m₂ = -1/2, Item 1 is steeper.

Correct option is C.

The slope of a line is defined as the change in y coordinate with respect to the change in x coordinate of that line. The net change in y coordinate is Δy, while the net change in the x coordinate is Δx. So the change in y coordinate with respect to the change in x coordinate can be written as,

m = Δy/Δx

where,m is the slope

Slope of line denotes rate of change and steepness.

Given,

A) a line with y-intercept (0,0) and passes through point (-2 , 4).

slope of the line

= (y₂ - y₁)/(x₂ - x₁)

= (4 - 0)/(-2 - 0)

= 4/-2

m₁ = -2

B) In the graph, line passes through (2, -1) and (-2 , 1 )

slope of the line

= (y₂ - y₁)/(x₂ - x₁)

= (1 - -1)/(-2 - 2)

= 2/-4

m₂ = -1/2

Hence, by the calculation

Items have different rate of change, Item 1 is steeper.

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The proportion of selling times in the sample that are greater than or equal to 10 days is 0.85.

As per the data given:

HOME Realty claims that it can sell a detached, residential house faster than any other realty company.

With the aim of examining HOME's claim, you sample 20 customers who sold a detached.

The data has been summarized in the histogram(attached at the end of solution)

Here we have to determine the proportion of selling times in the sample that are greater than or equal to 10 days.

Selling Time Frequency

0 - 10                    3

10 - 20                  4

20 - 30                 7

30 - 40                 3

40 - 50                 3

Frequencies for which selling times are 10 or greater

= 4 + 7 + 3 + 3

= 17

Total frequencies = 3 + 4 + 7 + 3 + 3

= 20

Proportion for which selling times are 10 or greater

= [tex]$\frac{17}{20}[/tex]

= 0.85

Therefore the proportion is 0.85.

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HOME Realty claims that it can sell a detached, residential house faster than any other realty company. With the aim of examining HOME's claim, you sample 20 customers who sold a detached, residential house through HOME and record the selling times (in days) of the houses. Your data are summarized in the following histogram(attached at the end of solution):

Based on the histogram, find the proportion of selling times in the sample that are greater than or equal to 10 days. Write your answer as a decimal, and do not round your answer.

Answer:

  4.5 mm

Step-by-step explanation:

You want the length of AB in similar triangles ACE and BCD, given BD is parallel to AE, and AC=36 mm, ED=9 mm, EC=72 mm.

Corresponding segments in the two triangles are proportional, so we have ...

  AB/ED = AC/EC

  AB/(9 mm) = (36 mm)/(72 mm) = 1/2

  AB = (9 mm)(1/2) = 4.5 mm . . . . . multiply by 9 mm

The length of segment AB is 4.5 mm.

The area of each circle, to the nearest tenth is:

1. 19.6 cm²

2. 379.9 in.²

3. 28.3 mm²

4. 78.5 in.²

5. 132.7 cm²

6. 162.8 yd²

The area of a circle is calculated as:

A = πr², where, r is the radius of the circle, and π is a constant.

1. r = 5/2 = 2.5 cm

Area = πr² = 3.14 * 2.5²

= 19.6 cm²

2. r = 11 in.

Area = πr² = 3.14 * 11²

= 379.9 in.²

3. r = 3 mm

Area = πr² = 3.14 * 3²

= 28.3 mm²

4. r = 10/2 = 5 in.

Area = πr² = 3.14 * 5²

= 78.5 in.²

5. r = 13/2 = 6.5 cm

Area = πr² = 3.14 * 6.5²

= 132.7 cm²

6. r = 7.2 yd

Area = πr² = 3.14 * 7.2²

= 162.8 yd²

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The random number x satisfies Poisson distribution of probability and the probability that there are 8 occurrences in ten minutes is 0.0241 and The probability that there are less than 3 occurrences is 0.0659.

Given,

The random variable x is the number of occurrences of an event over an interval of ten minutes. It can be assumed that the probability of an occurrence is the same in any two time periods of an equal length. It is known that the mean number of occurrences in ten minutes is 5.3.

1) A Poisson distribution characterizes the random variable x that is provided.

2) Having 8 occurrences in 10 minutes is likely based on the following formula:

P(X = 8) = (e^(-5.3) * 5.3^8) / 8! = 0.0241 (rounded to four decimal places) (rounded to four decimal places)

Consequently,

(a) 0.0241 is the correct response.

3) The likelihood that there are fewer than 3 occurrences is as follows:

P(X 3) = P(X = 0), P(X = 1), and P(X = 2) = (e(-5.3) * 5.30), (e(-5.3) * 5.31), and (e(-5.3) * 5.32), respectively, / 0!, / 1/1, and / 2/ respectively, / 2! = 0.0659 (rounded to four decimal places)

Accordingly,

(a) 0.0659 is the correct response.

The random number x satisfies Poisson distribution of probability and the probability that there are 8 occurrences in ten minutes is 0.0241 and The probability that there are less than 3 occurrences is 0.0659.

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The random number x satisfies Poisson distribution of probability and the probability that there are 8 occurrences in ten minutes is 0.0241 and The probability that there are less than 3 occurrences is 0.0659.

Given,

The random variable x is the number of occurrences of an event over an interval of ten minutes. It can be assumed that the probability of an occurrence is the same in any two time periods of equal length. It is known that the mean number of occurrences in ten minutes is 5.3.

1) A Poisson distribution characterizes the random variable x that is provided.

2) Having 8 occurrences in 10 minutes is likely based on the following formula:

P(X = 8) = ([tex]e^(-5.3)[/tex] * [tex]5.3^8[/tex]) / 8! = 0.0241 (rounded to four decimal places) (rounded to four decimal places)

Consequently,

(a) 0.0241 is the correct response.

3) The likelihood that there are fewer than 3 occurrences is as follows:

P(X 3) = P(X = 0), P(X = 1), and

P(X = 2) = ([tex]e(-5.3)[/tex]* 5.30), ([tex]e(-5.3)[/tex]* 5.31), and ([tex]e(-5.3)[/tex] * 5.32),

respectively, / 0!, / 1/1, and / 2/ respectively, / 2! = 0.0659 (rounded to four decimal places)

Accordingly,

(a) 0.0659 is the correct response.

The random number x satisfies Poisson distribution of probability and the probability that there are 8 occurrences in ten minutes is 0.0241 and The probability that there are less than 3 occurrences is 0.0659.

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The total duration of the phone call was 33 minutes.

Given that, by using Company A's calling plan, the cost of an overseas phone call is a $0.95 connection fee plus 33 cents per minute. If the total cost of the call is $11.84 and we have to calculate the time how long was the phone call.

Let the number of minutes a phone call held be x.

Now, we will be representing the above given information in terms of linear equation and then solve for the value of x which will give us the number of minutes the phone call held.

[tex]Total[/tex] [tex]cost=[/tex] $[tex]0.95[/tex] [tex]+[/tex] $[tex]0.33x[/tex]

$[tex]11.84=[/tex] $[tex]0.95[/tex] [tex]+[/tex] $[tex]0.33x[/tex]

[tex]11.84=0.95+0.33x[/tex]

[tex]11.84-0.95=0.33x[/tex]

[tex]0.3=0.33x[/tex]

[tex]\frac{0.3}{0.33}=x[/tex]

[tex]33=x[/tex]

Therefore, the call will go 33 minutes long.

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