(a) Explain why a gamma random variable with parameters (n, λ) has an approximately normal distribution when n is large.
(b) Then use the result in part (a) to solve Problem 9.20, page 395.
(d) What does the central limit theorem say with continuity correction? (e) Find the exact probability. steps, find the probability that the walk is within 500 steps from the origin calculations, explain why X ︽.Norm(a/λ, a/λ2). 9.18 Consider a random walk as described in Example 9.13. After one million 9.19 Let X ~ Gamma(a,A), where a is a large integer. Without doing any 9.20 Show that lim Hint: Consider an independent sum of n Exponential() random variables and apply the central limit theorem. 9.21 A random variable Y is said to have a lognormal distribution if log Y has a normal distribution. Equivalently, we can write Y -eX, where X has a normal distribution. (a) If X1, X2,... is an independent sequence of uniform (0,1) variables, show that the product Y =「L-i X, has an approximate lognormal distribution. Show that the mean and variance of log Y are, respectively, -n and n (b) If Y = ex, with X ~ Norm(μ, σ2), it can be shown that

The perimeter of triangle XYZ with the vertices, X(2, 5), Y(10 9), and Z(6, 1), is approximately 24.07 units.

To find the perimeter of the triangle, we need to add up the lengths of all three sides. To do this, we can use the distance formula, which is:

d = √[(x2 - x1)² + (y2 - y1)²]

Using this formula, we can find the length of each side of triangle XYZ:

Side XY:

d = √[(10 - 2)² + (9 - 5)²] = √64 + 16 = √80

Side YZ:

d = √[(6 - 10)² + (1 - 9)²] = √16 + 64 = √80

Side ZX:

d = √[(6 - 2)² + (1 - 5)²] = √16 + 16 = √32

Therefore, the perimeter of triangle XYZ = XY + YZ + ZX = √80 + √80 + √32

Perimeter ≈ 24.07 (rounded to two decimal places)

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The answer is (C) newspapers. Convenience products are low-cost consumer goods and services that are readily available and easily accessible to consumers.

These products are typically purchased frequently, with minimal effort and little thought or research. Examples of convenience products include bread, gasoline, newspapers, and other everyday items that consumers purchase without much consideration.

Cars, furniture, large appliances, and a cruise are not typically considered convenience products. These are generally higher-cost items that require more thought and research before purchase.

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Answer:

No solutions

Step-by-step explanation:

\(y=-\dfrac{1}{2}x+4 \\\\x+2y=-8\)

Substitute:

\(x+2(-\dfrac{1}{2}x+4)=-8\\\\x-x+8=-8\\\\8=-8\)

This system has no solutions.

Hope this helps!

Answer:1683

Step-by-step explanation:33 ft x 51 ft = 1683

Answer:

-1

Step-by-step explanation:

Derivative of cos(x)=-sin(x)

Derivative of e^x=e^x

Derivative of cos(x)-e^x= - sin(x)-e^x

plug in 0 to the spots of x and you will get 1 because sin(0) is 0 e^0 is 1

0-1=-1

Answer:

P = 20x + 6

Step-by-step explanation:

Perimeter formula:

P = 2(l + w)

Given:

l = 8x - 1

w = 2x + 4

Work:

P = 2(l + w)

P = 2(8x - 1 + 2x + 4)

P = 2(10x + 3)

P = 20x + 6

Answer:

1. 2(8x-1)  + 2(2x +4)

2. 16x -2 + 4x + 8

3. (20x + 6)

Step-by-step explanation:

Ed22

Answer:

8% annual

Step-by-step explanation:

first find the rate of 1 year. 960/3 = 320.

then find what % of 4,000, 320 is.

8%

Answer:

x = 24

Step-by-step explanation:

The lights on or off in the daytime is the independent variable. (option c).

The variable that you manipulate or control in your study is called the independent variable. In your case, the independent variable is whether the drivers have their headlights on or off during the daytime.

The dependent variable, on the other hand, is the variable that you observe or measure in your study. It is dependent on the independent variable, as it is expected to be affected by it.

A confounding variable is a variable that is not controlled for in a study and can affect the dependent variable. It is a potential source of bias that can make it difficult to determine whether the independent variable truly has an effect on the dependent variable.

Finally, the fact that the independent variable in your study is a random variable means that it is not under your control, but is determined by chance.

This is because you will randomly assign the drivers to the two groups, with some having their headlights on and others having their headlights off.

So, the correct option is (c).

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The Maclaurin series for f(x) converges absolutely for x within the interval (-2/3, 2/3).

To find the Maclaurin series for the function f(x) = (3cos(x))ln(1+x), we can use the standard formulas for the Maclaurin series expansion of elementary functions.

First, let's find the derivatives of f(x) up to the third order:

f(x) = (3cos(x))ln(1+x)

f'(x) = -3sin(x)ln(1+x) + (3cos(x))/(1+x)

f''(x) = -3cos(x)ln(1+x) - (6sin(x))/(1+x) + (3sin(x))/(1+x)² - (3cos(x))/(1+x)²

f'''(x) = 3sin(x)ln(1+x) - (9cos(x))/(1+x) + (18sin(x))/(1+x)² - (12sin(x))/(1+x)³ + (12cos(x))/(1+x)² - (3cos(x))/(1+x)³

Next, we evaluate these derivatives at x = 0 to find the coefficients of the Maclaurin series:

f(0) = (3cos(0))ln(1+0) = 0

f'(0) = -3sin(0)ln(1+0) + (3cos(0))/(1+0) = 3

f''(0) = -3cos(0)ln(1+0) - (6sin(0))/(1+0) + (3sin(0))/(1+0)² - (3cos(0))/(1+0)² = -3

f'''(0) = 3sin(0)ln(1+0) - (9cos(0))/(1+0) + (18sin(0))/(1+0)² - (12sin(0))/(1+0)³ + (12cos(0))/(1+0)² - (3cos(0))/(1+0)³ = -9

Now we can write the first three nonzero terms of the Maclaurin series:

f(x) = f(0) + f'(0)x + (f''(0)/2!)x² + (f'''(0)/3!)x³ + ...

f(x) = 0 + 3x - (3/2)x² - (9/6)x³ + ...

Simplifying, we have:

f(x) = 3x - (3/2)x² - (3/2)x³ + ...

To determine the values of x for which the series converges absolutely, we need to find the interval of convergence. In this case, we can use the ratio test:

Let aₙ be the nth term of the series.

|r| = lim(n->infinity) |a_(n+1)/aₙ|

= lim(n->infinity) |(3/2)(xⁿ+1)/(xⁿ)|

= lim(n->infinity) |(3/2)x|

For the series to converge absolutely, we need |r| < 1:

|(3/2)x| < 1

|x| < 2/3

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Answer:

cashier

Step-by-step explanation:

the discount is $7.50

Answer:

\($1,800\)

Answer:

\(\dfrac{1}{2}\) day

Step-by-step explanation:

Given that:

Owen who is an alien has only 4/5 of his socks clean.

and he wears 2/5 socks a day.

Out of his total 4/5 clean socks.

It will take him (4/5 ÷ 2/5) days to wash all his socks

i.e

the numbers of days since he washed all his socked is:

= \(\dfrac{4}{5} \div \dfrac{2}{5}\)

= \(\dfrac{4}{5} \times \dfrac{5}{2}\)

= \(\dfrac{1}{2}\)

The probability of picking blue ball by Liz and Mike are equal =1/3

Probability is a branch of mathematics that deals with the study of random events or experiments. It is concerned with quantifying the likelihood or chance of a particular event occurring, given certain assumptions or conditions.

The probability of an event is typically represented as a number between 0 and 1, with 0 indicating that the event is impossible, and 1 indicating that the event is certain. For example, if you flip a fair coin, the probability of getting heads is 0.5, and the probability of getting tails is also 0.5.

There are different methods for calculating probabilities, depending on the nature of the event or experiment being considered. Some of the most commonly used methods include the classical probability method, the empirical probability method, and the subjective probability method.

Probability has many applications in various fields, including statistics, finance, engineering, and science. It is used to model and analyze complex systems and to make predictions and decisions based on uncertain information.

Given : X contains 9 blue balls and 18 red balls.

Y contains 7 blue balls and 14 red balls.

Probability of picking blue ball is 9/9+18

=9/27=1/3

Now Mike pick ball from bag Y one ball not affect these pick ball

so, 7/7+14=7/21=1/3

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Answer:

graph

Step-by-step explanation:

graph shown

intersected large curve

Answer:

360 cubes

Step-by-step explanation:

Given:

In a right triangle, the measure of one acute angle is 12 more than twice the measure of the other acute angle.

To find:

The measures of the 2 acute angles of the triangle.

Solution:

Let x be the measure of one acute angle. Then the measure of another acute is (2x+12).

According to the angle sum property, the sum of all interior angles of a triangle is 180 degrees. So,

\(x+(2x+12)+90=180\)

\(3x+102=180\)

\(3x=180-102\)

\(3x=78\)

Divide both sides by 3.

\(x=\dfrac{78}{3}\)

\(x=26\)

The measure of one acute angle is 26 degrees. So, the measure of another acute angle is:

\(2x+12=2(26)+12\)

\(2x+12=52+12\)

\(2x+12=64\)

Therefore, the measures of two acute angles are 26° and 64° respectively.

Answer:

x = ±2i

Step-by-step explanation:

√-1 is imaginary number i

Step 1: Write equation

3x² = -12

Step 2: Solve for x

The equation is solved for x and the solutions of x = 2i or x = -2i.

Given data ,

To solve the equation 3x^ = -12, we can divide both sides of the equation by 3 to isolate x²:

(3x²)/3 = (-12)/3

Simplifying further, we get:

x² = -4

To find the value of x, we can take the square root of both sides of the equation:

√(x²) = √(-4)

Since we're dealing with a square root of a negative number, we need to introduce the concept of complex numbers.

The square root of -4 can be written as ±2i, where i is the imaginary unit (√(-1)).

So, the solutions to the equation 3x² = -12 are:

x = ±√(-4) = ±2i

Hence , the equation is solved and x can take on the values 2i or -2i.

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Answer:

Step-by-step explanation:

Let's call the angle of elevation of the sun "θ". We want to find the value of θ.

First, we can find the length of the part of the flagpole that is casting the shadow by using trigonometry. We can define:

h = height of the flagpole above the ground

d = length of the shadow

α = angle of inclination of the flagpole (which is the same as the angle between the ground and the sun's rays, since the flagpole is vertical)

Then, we can use the tangent function to relate these variables:

tan(α) = h/d

Rearranging, we get:

h = d tan(α)

We know that the height of the flagpole is 25 feet, and the length of the shadow is 12 feet. So:

25 = 12 tan(α)

Solving for tan(α), we get:

tan(α) = 25/12

Now we can use the inverse tangent function (also called arctangent) to find α:

α = tan^(-1)(25/12)

α ≈ 65.14 degrees

This is the angle between the ground and the sun's rays, as seen from the flagpole. To find the angle of elevation of the sun, we need to subtract α from 90 degrees (since the sun's rays are perpendicular to the ground). So:

θ = 90 - α

θ ≈ 24.86 degrees

Therefore, the angle of elevation of the sun is approximately 24.86 degrees.

a.

= (5x)² - 3 = 25x² - 3

- 5f(x) = 5(x² - 3) = 5x² - 15

b.

- f(x - 2) = (x - 2)² - 3 = x² - 4x + 1

- f(x) - f(2) = (x² - 3) - (2² - 3) = x² - 3 - 1 = x² - 4

a. To calculate f(5x), we substitute 5x into the function f(x) and simplify the expression.

  f(5x) = (5x)² - 3 = 25x² - 3

  To calculate 5f(x), we multiply the function f(x) by 5.

  5f(x) = 5(x² - 3) = 5x² - 15

b. To calculate f(x - 2), we substitute (x - 2) into the function f(x) and simplify the expression.

  f(x - 2) = (x - 2)² - 3 = x² - 4x + 4 - 3 = x² - 4x + 1

  To calculate f(x) - f(2), we evaluate f(x) and f(2) separately and then find their difference.

  f(x) = x² - 3

  f(2) = 2² - 3 = 4 - 3 = 1

  f(x) - f(2) = (x² - 3) - (2² - 3) = x² - 3 - 1 = x² - 4

For the difference quotient of f(x) = -7x² - 5x + 9, we can calculate it as follows:

Difference quotient = [f(x + h) - f(x)] / h

Expanding the function and substituting into the difference quotient formula, we have:

[f(x + h) - f(x)] / h = [-7(x + h)² - 5(x + h) + 9 - (-7x² - 5x + 9)] / h

Simplifying and expanding further:

= [-7(x² + 2hx + h²) - 5x - 5h + 9 + 7x² + 5x - 9] / h

= [-7x² - 14hx - 7h² - 5x - 5h + 9 + 7x² + 5x - 9] / h

= [-14hx - 7h² - 5h] / h

= -14x - 7h - 5

The difference quotient of f(x) = -7x² - 5x + 9 is -14x - 7h - 5.

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In linear equation, 3437.5  feet - lbs is the work done to haul the anchor .

What in mathematics is a linear equation?

An algebraic equation with simply a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation.

                          Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.

Anchor  weighing = 100 lbs

   given 3  lb/ft

  the combined weight  = 3( 25 -y ) + 100

                                 = 175 - 3y

    the workdone on small solution is = (175 - 3y)Δy

                      w = ∫₀²⁵ (175 - 3y) dy

                        = 175[y]²⁵₀- 3/2[y²]²⁵₀

                       = 175 [ 25 - 0 ] - 3/2 [ 25²- 0²]

                       = 3437.5  feet - lbs

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A = -3.the particular solution is given by yp= ae⁽⁻ˣ⁾, so substituting the values of a and x, we have:yp= -3e⁽⁻ˣ⁾

so, the particular solution of the given differential equation, satisfying the initial conditions, is yp= -3e⁽⁻ˣ⁾.

to find the particular solution of the differential equation, we'll first assume that the particular solution takes the form of a function of the same type as the right-hand side of the equation. in this case, the right-hand side is e⁽⁻ˣ⁾, so we'll assume the particular solution is of the form yp= ae⁽⁻ˣ⁾.

taking the first derivative of ypwith respect to x, we get:y'p= -ae⁽⁻ˣ⁾

now, substitute the particular solution and its derivative back into the original differential equation:

m(-2x + 5yp = e⁽⁻ˣ⁾

simplify the equation:-2mx + 5myp= e⁽⁻ˣ⁾

substitute yp= ae⁽⁻ˣ⁾:

-2mx + 5mae⁽⁻ˣ⁾ = e⁽⁻ˣ⁾

cancel out the common factor of e⁽⁻ˣ⁾:-2mx + 5ma = 1

now, we'll use the initial condition y(0) = 0 to find the value of a:

0 = a

substituting a = 0 back into the equation, we get:-2mx = 1

solving for x, we find:

x = -1 / (2m)

finally, we'll find the derivative of ypat x = 0 using y'(0) = 3:y'p= -ae⁽⁻ˣ⁾

y'p0) = -ae⁽⁰⁾3 = -a

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Dodie's triangular plot with 160 rose plants will have 12 rows.

To solve it,

Assigning a variable to the number of rows we want to find. Assume that this variable is "n".

Since there are 5 more plants in each row than in the previous row,

We can use an arithmetic sequence to represent the number of plants in each row.

Specifically, the first row will have 1 plant, and the second row will have

1 + 5 = 6 plants, the third row will have 1 + 5 + 5 = 11 plants, and so on.

The formula for an arithmetic sequence is,

an = a1 + (n-1)d

Where an is the nth term of the sequence,

a1 is the first term,

And d is a common difference.

Using this formula, we can write an expression for the total number of plants in all n rows,

160 = n/2 (2 + (n-1)5)

Simplifying this equation, we get,

160 = 2.5n² + 2.5n - 5

Now we can solve for n using the quadratic formula,

n = (-2.5 ± √(2.5²+ 4(2.5)(165)))/(2(2.5))

After simplifying this equation, we get two solutions,

n = -13.2 and n = 12.2.

Since we can't have a negative number of rows, we'll take the positive solution, n = 12.2.

Now, since we can't have a fraction of a row, we'll round down to the nearest integer.

Therefore, Dodie's plot will have 12 rows.

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Answer:

Tomorrow: 0.075

The day after tomorrow: 0.0375

Step-by-step explanation:

There are only two results to what is going to happen.  Rain or not rain.  So all we have to do is divide each day by 2.

Answer:

y=- -5 (negative 5)

Step-by-step explanation:

The Alternating Series Test (AST) is used to determine if a series is convergent or divergent. It assumes that the terms alternate in sign and are monotonically decreasing in magnitude, and if lim_(n)a_n = 0, then the series is convergent. The series is given in the general formula for the AST, and the absolute value of each term is equal to the corresponding term.

The series for convergence or divergence using the alternating series test is given below:

[infinity] (−1)n 2nn n! n = 1

The general formula for the alternating series test is as follows. Assume that a series [a_n]_(n=1)^(∞) is defined such that the terms alternate in sign and are monotonically decreasing in magnitude.

If lim_(n→∞)△a_n = 0, where △a_n denotes the nth term of the series,

then the alternating series [a_n]_(n=1)^(∞) is convergent. We must evaluate if the alternating series is monotonically decreasing and if the absolute value of each term of the series is decreasing as well. If both conditions are met, we may apply the Alternating Series Test (AST). Let's take a look at the given series below:(-1)^n(2^n)/(n!) for n = 1 to infinity The series is given in the general formula for the AST. Because the series is already in the right form, we do not need to test it first.

The terms of the sequence decrease since (n+1)!/(n!) = (n+1), which is a positive number. Furthermore, since (n+1) > n for any natural number n, the sequence decreases monotonically. When we take the absolute value of each term in the series, it is equal to the corresponding term since all terms are positive.

Therefore, the series is convergent according to the Alternating Series Test.

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Answer:15.65

Step-by-step explanation:

Answer:

16c - 48

Step-by-step explanation:

combine like terms : 18c - 2c - 49 + 1

then you get              16c - 48