Consider the probability mass function for the number of rejected quality control items (X) in one random day in a manufacturing factory. Х X f(x)=P(X= x) 3A/20 F(x)=P(X< x) 0 0 1 1 2 0.05 0.05 7 B/20 2 3 3 3 4 4 0.1 4 5 ол PMF CDF a) Complete the above probability mass table (PMF) and the corresponding cumulative distribution table (CDF) (15 points) b) Find P(X = 5). (5 points) c) Find the probability of two or fewer rejected items in a random day. (10 points) d) Calculate expected value of the number of rejected items per day. (10 points) e) Calculate the variance and the standard deviation of rejected items per day. (10 points)

Shannon found a stack of 100 collectible fantasy horse cards in her desk that she had forgotten about. She randomly looked at 20 cards and got 8 centaur, 7 pegasus, and 5 unicorn cards.

Based on the data, estimate how many unicorn cards are in the stack.

a) Let A denote the event that 4 candies taken have different tastes, B denote the event that 2 candies have the same taste, and C denote the event that 6 candies include 2 candies with marzipan, 2 candies with nuts and 2 candies with caramel.

b) The probability of event A is 1/210

The probability of event B is 5/126

The probability of event C is 3/70

To find the probability of event A, we need to count the number of ways to choose 4 candies out of 10, where each candy has a different taste. Thus, the probability of event A is given by:

To find the probability of event B, we need to count the number of ways to choose 2 candies of the same taste and 2 candies of different tastes out of 10. There are 4 choices for the taste of the 2 candies that are the same, and 6 choices for the taste of the other 2 candies. Thus, the probability of event B is given by:

To find the probability of event C, we need to count the number of ways to choose 2 candies with marzipan, 2 candies with nuts, and 2 candies with caramel out of 10. There are (3 choose 2) = 3 ways to choose 2 candies with marzipan, (2 choose 2) = 1 way to choose 2 candies with nuts, and (4 choose 2) = 6 ways to choose 2 candies with caramel. Thus, the probability of event C is given by:

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The Venn diagram can be used to visualize the different probabilities and relationships between events, and Bayes' theorem can be used to calculate conditional probabilities.

Let's start by drawing a Venn diagram to represent the probabilities given :

               _________________

              /                 \

             /                   \

            /        A∩B          \

           /                       \

          /_________________________\

         /                          \

        /                            \

       /          A\B               \

      /                              \

     /______________     ____________\

                    \   /

                     \ /

                      |

                      B

We know that:

P(A fails) = 0.35, which means P(A works) = 0.65

P(A and B fail) = 0.22

P(B fails alone) = 0.3, which means P(B works) = 0.7

To find (i) P(A fails alone), we need to subtract the probability of A and B failing together from the probability of A failing:

P(A fails alone) = P(A fails) - P(A and B fail) = 0.35 - 0.22 = 0.13

Therefore, the probability of A failing alone is 0.13.

To find (ii) P(A fails given that B has failed), we need to use Bayes' theorem:

P(A fails | B fails) = P(A∩B) / P(B fails)

We already know that P(A∩B) = 0.22 and P(B fails) = 0.3. So, we can substitute these values to get:

P(A fails | B fails) = 0.22 / 0.3 = 0.7333...

Therefore, the probability of A failing given that B has failed is approximately 0.7333.

In summary, the Venn diagram can be used to visualize the different probabilities and relationships between events, and Bayes' theorem can be used to calculate conditional probabilities.

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The price of the trailer was $74,041.54.

Let's start by finding the present value of the perpetual annuity payments of $4,100 per year, using the formula:

PV = PMT / r

where:

PV is the present value

PMT is the payment per period

r is the interest rate per period

Since the payments are made annually and the interest rate is 14.3% per year, the interest rate per period is also 14.3%. Thus:

PV = $4,100 / 0.143 = $28,671.33

At a current interest rate of 14.3%, Amazon Rafting would need to invest this much in order to receive permanent $4,100 yearly payments.

Now, using the following calculation, we can determine the price of the caravan using the present value of the perpetuity and the future value of the special payment:

[tex]FV = PV * (1 + r)^n + SP[/tex]

where:

FV is the future value

PV is the present value of the perpetuity

r is the interest rate per period (14.3% per year)

n is the number of periods (5 years)

SP is the special payment of $8,700 over 5 years

Substituting the values we have:

[tex]FV = $28,671.33 * (1 + 0.143)^5 + $8,700[/tex]

FV = $28,671.33 * 1.8333 + $8,700

FV = $74,041.54

Therefore, the price of the trailer was $74,041.54.

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If Sheila’s test had a z-score of 2.2 then her test score was 89.6%.

We can use the formula for calculating the z-score of a value,

z = (x - μ) / σ, value we want to convert to a z-score is x, mean of the distribution is μ,  standard deviation of the distribution is σ and z score is z. In this case, we know that the average mark on the test was 72%, which means μ = 72. We also know that the standard deviation was 8%, which means σ = 8. We know that Sheila's z-score was 2.2,

We can rearrange the formula to solve for x,

x = μ + zσ

Substituting in the values we know,

x = 72 + 2.2 * 8

x = 72 + 17.6

x = 89.6

Therefore, Sheila's test score was 89.6%.

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Given: The solution to [tex]x^3[/tex] = [tex]-2-i[/tex] In polar form Is:

[tex]2 < 75^o, 2 < 195^o, 2 < 315^o[/tex]

Answer:

[tex]\large \boxed{\mathrm{ion \ even \ no \ fr}}[/tex]

Step-by-step explanation:

DO IT YOUR SELF [tex]\large \boxed{\mathrm{BOZO}}[/tex]

Answer:

(1,-1)

Step-by-step explanation:

Substituting the given point into the equation

[tex]-5/6x=y+1/6\\-5/6(1) = -1 + 1/6\\-5/6 = -5/6[/tex]

To correctly use the wolframalpha command to match the function z = x-y/1+x^2+y^2 given by Problem 30 on Page 392 with a graph and a contour map on Page 393, you can follow the steps below:

1. Go to the wolframalpha website.
2. In the search bar, type "plot z = x-y/(1+x^2+y^2)" and hit enter.
3. The website will generate a 3D graph of the function.
4. To match the graph C and contour map II, click on the "More" button below the graph and select "Contour plot."
5. In the new window, select the second option from the left, which is the contour map.
6. Adjust the settings as necessary to match the colors and levels of the contour map on Page 393.
7. To match the graph C and contour map I, follow the same steps as above, but select the first option for the contour map.
8. To match the graph D and contour map I, click on the "More" button below the graph and select "Contour plot."
9. In the new window, select the first option from the left, which is the contour map.
10. Adjust the settings as necessary to match the colors and levels of the contour map on Page 393.
11. To match the graph D and contour map II, follow the same steps as above, but select the second option for the contour map.

It's important to correctly use parentheses in the wolframalpha command to ensure that the website understands the order of operations. In this case, we want to divide y by the sum of 1, x^2, and y^2 before subtracting it from x. Therefore, we need to enclose the denominator in parentheses, like this:

plot z = x-(y/(1+x^2+y^2))

By following these steps and using the correct wolframalpha command, you can match the function with the appropriate graph and contour map.

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The value of correct option for maps ΔABC to a triangle that is similar, but not congruent, to ΔABC are,

⇒  dilation with scale factor 2 about the origin

We have to given that;

To find correct option for maps ΔABC to a triangle that is similar, but not congruent, to ΔABC

Since, We know that;

For any translation the condition of congruency is not change.

But for any type of dilation condition of congruency for triangles are change.

Thus, The value of correct option for maps ΔABC to a triangle that is similar, but not congruent, to ΔABC are,

⇒  dilation with scale factor 2 about the origin

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To determine the quadrant(s) in which the solutions lie for an angle in the range of 0 < θ < 2π (or 0 to 360 degrees), there are several steps to follow.

Firstly, we need to identify the reference angle. This is the angle formed between the terminal arm of the angle and the x-axis in the standard position.

Next, we need to determine the sign of the angle, which is based on whether the terminal arm is located in the positive or negative x-axis, and the positive or negative y-axis.

Then, we need to use the inverse trigonometric functions (such as sin^-1, cos^-1, or tan^-1) to determine the exact angle measure. This step is important because it ensures that we obtain the angle measure within the desired range of 0 < θ < 2π.

Once we have the exact angle measure, we can determine the quadrant(s) in which the solution lies. This is based on the signs of the trigonometric functions in each quadrant. For example, if the sine and cosine are positive, the angle lies in the first quadrant. If the sine is positive and the cosine is negative, the angle lies in the second quadrant. If the sine and cosine are negative, the angle lies in the third quadrant. And if the sine is negative and the cosine is positive, the angle lies in the fourth quadrant.

In summary, to determine the quadrant(s) in which the solutions lie for an angle in the range of 0 < θ < 2π, we need to identify the reference angle, determine the sign of the angle, use the inverse trigonometric functions to find the exact angle measure, and then use the signs of the trigonometric functions in each quadrant to determine the quadrant(s) in which the solution lies.
A general description of the steps used to determine the quadrant(s) in which the solutions lie for an angle in the range of 0 < θ < 2π (or 0 to 360 degrees) involves understanding the angle, reference angle, and quadrant relationships. Here are the steps:

1. Convert the angle (θ) into standard position, which means placing the vertex at the origin and the initial side along the positive x-axis. If the angle is given in degrees, convert it to radians (if needed) using the conversion factor: 1 radian = 180/π degrees.

2. Identify the reference angle (α). The reference angle is the acute angle formed between the terminal side of the angle (θ) and the x-axis. To find the reference angle, use the following rules:
- If θ is in the first quadrant, α = θ
- If θ is in the second quadrant, α = π - θ
- If θ is in the third quadrant, α = θ - π
- If θ is in the fourth quadrant, α = 2π - θ

3. Determine the quadrant(s) in which the angle (θ) lies using the reference angle (α) and the inverse trigonometric functions.

The inverse trigonometric functions (e.g., sin⁻¹, cos⁻¹, and tan⁻¹) can help in finding the corresponding angle(s) for a given trigonometric function value. Depending on the function and value, one or two quadrants may be determined as solutions.

4. Once the quadrant(s) are identified, the solutions for the angle (θ) can be written using the reference angle (α) and the relevant inverse trigonometric function.

By following these steps, you can effectively determine the quadrant(s) in which the solutions lie for an angle within the specified range.

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True. The orthogonal projection of vector x onto the line determined by non-zero vector y is indeed a vector. The projection creates a new vector that lies on the line determined by y and is orthogonal (perpendicular) to y.

2. If the orthogonal projection of x onto the line determined by non-zero y is the vector, then x - is orthogonal to y: True

If the orthogonal projection of x onto the line determined by non-zero y is the vector, then the difference between x and the projection (x - projection) will be orthogonal to y. This is because the projection is the closest point on the line determined by y to x, and thus the difference vector will be perpendicular to the line.

3. For unit vector y ∈ R, the projection matrix for orthogonal projection onto the line determined by y is yy^T: True

For a unit vector y, the projection matrix P for the orthogonal projection onto the line determined by y is given by P = yy^T, where y^T is the transpose of the vector y. This matrix is used to calculate the orthogonal projection of a vector onto the line determined by y.

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The equation represents the total cost of turkey and amount is y= 3.25x.

We have,

The turkey costs $3. 25 per pound.

let x be the amount in pounds and y be the total cost in dollar.

Then, the relation between x and y

Total cost = 3.25 (amount in  poind)

y = 3.25 x

Thus, the required equation is y= 3.25x.

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The critical region would be in the tail of the null distribution corresponding to the alpha level (0.05), which is the upper 5%.

We have,

The critical region would be in the upper 5% of the null distribution.

This is because alpha is the probability of making a type I error (rejecting the null hypothesis when it is actually true),

In this case,

We are looking for evidence that the population mean is greater than 1.2.

Therefore,

The critical region would be in the tail of the null distribution corresponding to the alpha level (0.05), which is the upper 5%.

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To obtain the prediction [tex]w_*^Tx[/tex] for a test sample x, we need to substitute [tex]w = X^[/tex]T\alpha into

a) To find the minimizer of the objective function f(w), we need to differentiate it with respect to w, set it equal to zero, and solve for w.

First, we expand the norm term:

[tex]||Xw - y||^2 = (Xw - y)^T(Xw - y) = w^TX^TXw - 2y^TXw + y^Ty[/tex]

Taking the derivative of f(w) with respect to w and setting it equal to zero, we get:

[tex]2(X^TXw - X^Ty)[/tex] + 2\lambda w = 0

Rearranging the terms, we have:

[tex]X^TXw[/tex] + \lambda Iw [tex]= X^Ty[/tex]

which implies that:

[tex](X^TX[/tex] + \lambda I)w [tex]= X^Ty[/tex]

To obtain the minimizer, we need to solve for w, which gives us:

[tex]w = (X^TX + \lambda I)^{-1}X^Ty[/tex]

To justify that [tex]X^TX[/tex] + \lambda I is invertible for lambda > 0, we can use the SVD decomposition of X:

X = U\Sigma [tex]V^T[/tex]

where U and V are orthogonal matrices and \Sigma is a diagonal matrix with the singular values of X. Then, we have:

[tex]X^TX = V\Sigma^TU^TU\Sigma V^T = V\Sigma^T\Sigma V^T[/tex]

Since \Sigma[tex]^T[/tex]\Sigma is also a diagonal matrix with non-negative entries, adding lambda I to [tex]X^TX[/tex] ensures that all eigenvalues are strictly positive, and hence the matrix is invertible.

b) Substituting [tex]X^Tu[/tex] for w in [tex]X^TXw[/tex] + \lambda Iw [tex]= X^Ty[/tex], we get:

[tex]X^TX(X^Tu)[/tex] + \lambda [tex]IX^Tu = X^Ty[/tex]

Simplifying, we get:

[tex]X^T[/tex]([tex]X^TX[/tex] + \lambda I)u =[tex]X^Ty[/tex]

Thus, we have:

[tex]u = (X^TX + \lambda I)^{-1}X^Ty[/tex]

Substituting this into [tex]X^Tu[/tex], we get:

[tex]w = X^T(X^TX + \lambda I)^{-1}X^Ty[/tex]

c) Since [tex]w = X^T[/tex] \alpha and [tex]X^T[/tex] represents a linear combination of the columns of X, we can say that w is a linear combination of the columns of X, and hence is "in the span of the data."

d) Substituting [tex]w = X^T[/tex]\alpha into the expression for a, we get:

[tex]a = \frac{1}{\lambda}(X^Ty - X^TX(X^T\alpha))[/tex]

Multiplying both sides by \lambda and rearranging, we get:

[tex]X^TX[/tex]\alpha + \lambda\alpha [tex]= X^Ty[/tex]

This can be rewritten as:

(\lambda I [tex]+ X^TX)[/tex]\alpha [tex]= X^Ty[/tex]

To obtain the expression for \alpha, we can simply solve for \alpha:

[tex]\alpha = (\lambda I + X^TX)^{-1}X^Ty[/tex]

Note that X^TX is the Gram (kernel) matrix for the standard vector dot product.

e) Substituting [tex]w = X^[/tex]T\alpha into the expression for Xw, we get:

[tex]Xw = XX^T\alpha[/tex]

Using the kernelized form of \alpha, we have:

[tex]Xw = XX^T(\lambda I + XX^T)^{-1}y[/tex]

f) To obtain the prediction [tex]w_*^Tx[/tex] for a test sample x, we need to substitute [tex]w = X^[/tex]T\alpha into

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Answer:

Step-by-step explanation:

Why did many Americans consider César Chávez a hero in the '60s and '70s?

Answer:

The surface area is 70 yards

Step-by-step explanation:

The formula for surface area is (SA)=2lw+2lh+2hw. Meaning it would be 2 times (8 times 3 + 8 times 1 + 3 times 1) which equals 70 yards.

Answer:

The area of the triangle is one-half the area of the rectangle. So the correct answer is C.

Answer:

z = 10

Step-by-step explanation:

Lets explain this to you,

Step one: Solve with the parenthesis first (2 times z is 2z and 2 times -8.5 is

-17)

Step two: Subtract 2z from both sides (0.3z - 2z = - 1.7) (2z - 2z = 0)

Step three: Divide negative 1.7 on both sides (-1.7 = -17)

Step four: Simplify - 1.7 / 17 which will finally be 10

There is an easier way but im also in class so I'll do it later!!!!

Answer :x^15

Step-by-step explanation:

You would combine components so it would be x^10x^5 you would add 5+10 and then you would get your answer

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We need to solve the equation 1 - Φ((L - Cm)/0.025) = 0.9265 for L. This can be done using a standard normal table or calculator.

(a) Let X be the length of the cylindrical bar. Then X ~ N(Cm, 0.25^2). We need to find P(X < 11.75).

Z = (X - Cm)/0.25 follows standard normal distribution.

P(X < 11.75) = P((X-Cm)/0.25 < (11.75-Cm)/0.25) = P(Z < (11.75-Cm)/0.25)

Using a standard normal table or calculator, we get P(Z < (11.75-Cm)/0.25) = Φ((11.75-Cm)/0.25)

where Φ is the cumulative distribution function of the standard normal distribution.

(b) The sample mean length, X, follows normal distribution with mean Cm and standard deviation σ/√n, where n = 100 is the sample size. So, X ~ N(Cm, 0.25/√100) = N(Cm, 0.025). Therefore, the mean of the sample mean length is Cm and the standard deviation of the sample mean length is 0.025.

(c) We need to find P(10.99 < X < 11.01), where X is the sample mean length.

Z = (X - Cm)/(0.025) follows standard normal distribution.

P(10.99 < X < 11.01) = P((10.99 - Cm)/(0.025) < Z < (11.01 - Cm)/(0.025))

Using a standard normal table or calculator, we get P((10.99 - Cm)/(0.025) < Z < (11.01 - Cm)/(0.025)) = Φ((11.01 - Cm)/(0.025)) - Φ((10.99 - Cm)/(0.025))

(d) Let L be the length such that 92.65% of the sample means are more than L. This means we need to find the value of L such that P(X > L) = 0.9265.

Z = (X - Cm)/(0.025) follows standard normal distribution.

P(X > L) = P(Z > (L - Cm)/0.025) = 1 - Φ((L - Cm)/0.025)

Therefore, we need to solve the equation 1 - Φ((L - Cm)/0.025) = 0.9265 for L. This can be done using a standard normal table or calculator.

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The CPMP mathematics methodology seems to be more effective on average at helping students solve word problems compared to the traditional method.

There is a greater variation in performance among students in the CPMP group. We can compare the CPMP and traditional math program groups' performance on solving word problems by looking at the mean and standard deviation.

1) Observe the mean scores:

- CPMP: 57.4

- Traditional: 53.9

The CPMP group has a higher mean score than the traditional group, indicating that students in the CPMP group performed better on average.

2) Observe the standard deviations:

- CPMP: 32.1

- Traditional: 28.5

The CPMP group has a higher standard deviation than the traditional group, meaning that the scores in the CPMP group are more spread out. This suggests there's a greater range of performance levels among students in the CPMP group.

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Therefore, the area of the surface S is approximately 1.14 square units.

Here We can parametrize the  hyperbolic-paraboloid surface S as follows:

r(u,v) = (u, v, [tex]u^2[/tex] - v)

Here u is restricted to the interval [−1, 1] and v is restricted to the interval [−5, 5].

The area of the surface, we need to compute the magnitude of the cross product of the partial derivatives of r with respect to u and v:

|ru x rv| = |(1, 0, 2u) x (0, 1, -1)| = |(2u, 1, 0)| = [tex]\sqrt{(4u^2 + 1)}[/tex]

Therefore, the area of the surface is given by the double integral:

A = ∬S dS = ∫[tex]-5^5 * -1^1 \sqrt{ (4u^2 + 1)}[/tex] du/dv

We can evaluate this integral by making the substitution w = [tex]2u^2 + 1,[/tex] ,which gives:

A = ∫[tex]1^2 *1/4 \sqrt{w} dw[/tex]

= [tex](2/3) * w^{({3/2)}} |1^2 *1/4[/tex]

= [tex](2/3)(2 * \sqrt{5} - 1)[/tex]

So the area of the surface S is approximately 1.14 square units.

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PART A: The component form of vector is: u = <4, 8> and v = <4, 7>

PART B: u + v = <8, 15>

PART C: 5u - 2v = <12, 26>

The component form of a vector is <x, y>.

PART A:

Looking at the graph, vector u is the displacement from (2, -6) to (6, 2). Thus,

u = (6, 2) - (2, -6)

u = (6-2, 2-(-6))

u = (4, 8)

In component form, u = <4, 8>

Vector v is the displacement from (7, 8) to (11, 1). Thus,

v = (11, 1) - (7, 8)

v = (11-7, 8-1)

v = (4, 7)

In component form, v = <4, 7>

PART B:

u + v = <4, 8> + <4, 7>

u + v = <4+4, 8+7>

u + v = <8, 15>

PART C:

5u - 2v = 5*<4, 8> - 2*<4, 7>

5u - 2v = <20, 40> - <8, 14>

5u - 2v = <20-8, 40-14>

5u - 2v = <12, 26>

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The vertex that is not part of the image is:

A:(-4,6)

The translations to an image are represented as follows:

The vector notation of a translation is given as follows:

{x ± a, y ± a}

We have the next polygon is a translation of the first along the vector (3,-3).

The rule applied to each vertex of the image is:

(x, y) → (x + 3, y - 3).

Now, We have the vertices are:

(-7,3),(-4,6),(-1,3) and (-4,0).

Applying the translation rule, the vertices of the image are as follows:

(-4, 0), (-1, 3), (2, 0), (-1, -3).

The lone coordinate without a vertex of the image is (-4,6)

The correct option is (A)

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a. E[Y] = (0)(0.2) + (2)(0.5) + (4)(0.3) = 1.8 is the expected value for Y.

b. The cumulative distribution function of X+Y is P(X+Y = k) = Σ P(X=i)P(Y=k-i).

c. P(X=Y) is 0.3654.

d. Calculation is not correct. 3.6 is the correct value of E[XY].

In mathematics, a variable is defined as an alphabetic character that expresses a numerical value or number. A variable is used to represent an unknown quantity in algebraic equations.

(a) The expected value of Y can be calculated as E[Y] = (0)(0.2) + (2)(0.5) + (4)(0.3) = 1.8.

(b) To find the cumulative distribution function of X+Y, we first note that the sum of two independent random variables has a probability mass function given by the convolution of their respective probability mass functions. That is,

P(X+Y = k) = Σ P(X=i)P(Y=k-i)

where the sum is taken over all possible values of i such that both P(X=i) and P(Y=k-i) are nonzero. Using this formula, we can compute the cumulative distribution function of X+Y as:

F(x) = P(X+Y ≤ x) = Σ P(X+Y = k) for k ≤ x

     = Σ Σ P(X=i)P(Y=k-i) for k ≤ x

     = Σ P(X=i) Σ P(Y=k-i) for k ≤ x

     = Σ P(X=i) [tex]F_Y[/tex](x-i)

where [tex]F_Y[/tex](x) is the cumulative distribution function of Y. Since X has a Poisson(2) distribution, we can compute the cumulative distribution function of X+Y as:

F(x) = Σ P(X=i) F_Y(x-i)

     = Σ [tex]e^{(-2)} (2^i / i!) (0.2P(Y=x-i=0) + 0.5P(Y=x-i=2) + 0.3P(Y=x-i=4))[/tex]

where P(Y=x-i=k) is the probability mass function of Y.

(c) P(X=Y) can be calculated as:

P(X=Y) = Σ P(X=i, Y=i)

         = Σ P(X=i)P(Y=i) (since X and Y are independent)

         = Σ [tex]e^{(-2)} (2^i / i!) (0.2)(0) + (0.5)(e^(-2))(2^i / i!) + (0.3)(e^{(-2)})(2^i / i!)^2[/tex]

         = [tex]e^{(-4)} (0 + 0.5(2e^2/2) + 0.3(4e^2/4))[/tex]

         = 0.3654

(d) The student's calculation is not correct. To see why, let's first note that E[XY] can be computed as:

E[XY] = E[E[XY|X]] = E[XE[Y|X]]

where E[Y|X] is the conditional expected value of Y given X. Since X and Y are independent, we have E[Y|X] = E[Y] = 1.8. Therefore,

E[XY] = E[XE[Y|X]] = E[X(1.8)] = 2(1.8) = 3.6

So the correct value of E[XY] is 3.6, which is twice the value calculated by the student. The mistake the student made was in assuming that E[XY] is equal to the product of E[X] and E[Y]. This is only true if X and Y are uncorrelated, which is not the case here since X and Y are independent but not identically distributed.

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Experiment: "Favorite Cake Flavor"

Hypothesis: Different people have different favorite cakes flavors.

Now, Data collection enables us to identify the cakes flavors both popular and unpopular amongst users.

Furthermore, data analysis based on factors such as age, gender, and location allows for determining if disparities exist between consumer flavor preferences.

The findings of this study would be beneficial to cakes manufacturers and retailers, as they could enhance comprehension of preferred buyer choices and eventuate their products accordingly.  

Conclusively, improved knowledge of preferred flavor profiles would facilitate refining marketing methodologies to attract a broader target audience.

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The mechanic worked for 5.5 hours to install the new water pump.

Let's assume the mechanic worked for "x" hours to install the water pump.

The cost of the repair work will include the cost of the water pump as well as the labor charges.

The cost of the water pump is given as $249.98.

The labor charges can be calculated by multiplying the hourly rate by the number of hours worked.

So, the labor charges will be $109 multiplied by "x" hours, which is $109x.

Adding the cost of the water pump to the labor charges will give us the total cost of the repair work.

Therefore, we can write the equation as:

$849.48 = $109x + $249.98

Simplifying the equation, we get:

$599.50 = $109x

Dividing both sides by $109, we get:

x = 5.5

Therefore, the mechanic worked for 5.5 hours to install the new water pump.

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