A textbook store sold a combined total of 368 chemistry and history textbooks in a week. The number of history textbooks sold was 52 less than the number of chemistry textbooks sold. How many textbooks of each type were sold?

The Laplace transform is a mathematical tool used to transform a function of time into a function of complex frequency. The inverse Laplace transform does the opposite, transforming a function of complex frequency back into a function of time.

In MATLAB, you can use the "laplace" function to compute the Laplace transform of a given function. The syntax for the "laplace" function is: laplace(f), where f is the function you want to transform.

For example, in Example 1, the function f(t) = 5sin(3t) is defined using MATLAB's symbolic math toolbox by typing ">>symst" to activate symbolic math, followed by ">>f=5* sin(3*t);" to define the function. The Laplace transform of this function is then computed using the "laplace" function as follows: ">>laplace(f)".

Similarly, in Example 2, the function f(t) = (t - 2)^2U(t - 2) is defined using MATLAB's "heaviside" function to represent the unit step function. The Laplace transform of this function is then computed using the "laplace" function as follows: ">>F=laplace(f)".

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To the nearest tenth of a mile, Mr. Miller can see 9.8 miles of the horizon along the arc of his field of vision.

Based on the given information, we can use the formula for the arc length of a circle to find how much of the horizon Mr. Miller can see within his field of vision.

The formula for the arc length of a circle is:

arc length = (angle/360) x 2πr

where angle is the central angle of the arc in degrees, r is the radius of the circle, and 2πr is the circumference of the circle.

In this case, the central angle of the arc is 140 degrees, and the radius of the circle is the distance to the horizon, which is 4 miles. We can substitute these values into the formula:

arc length = (140/360) x 2π x 4

arc length = 0.388 x 8π

arc length = 9.8 miles

Therefore, to the nearest tenth of a mile, Mr. Miller can see 9.8 miles of the horizon along the arc of his field of vision.

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Where c1 and c2 are arbitrary constants determined by initial conditions.

The differential equation given is:

y'' - 6y' + 25y = 0

To find the two linearly independent solutions of the differential equation, we assume a solution of the form:

y = e^(rt)

where r is a constant to be determined. We then substitute this into the differential equation and obtain:

r^2 e^(rt) - 6r e^(rt) + 25 e^(rt) = 0

Dividing both sides by e^(rt), we get:

r^2 - 6r + 25 = 0

This is the characteristic equation of the differential equation, and we can solve for r using the quadratic formula:

r = (6 ± sqrt(6^2 - 4*25)) / 2

r = 3 ± 4i

Therefore, the two linearly independent solutions of the differential equation are:

y1 = e^(3x) cos(4x)

y2 = e^(3x) sin(4x)

To verify that these solutions are linearly independent, we can take the Wronskian of the solutions:

W(y1, y2) = y1y2' - y1'y2

= e^(3x) cos(4x) (3e^(3x) sin(4x) + 4e^(3x) cos(4x)) - e^(3x) sin(4x) (3e^(3x) cos(4x) - 4e^(3x) sin(4x))

= 5e^(6x)

Since the Wronskian is nonzero, the two solutions are linearly independent.

The general solution to the differential equation is then a linear combination of the two linearly independent solutions:

y = c1 e^(3x) cos(4x) + c2 e^(3x) sin(4x)

where c1 and c2 are arbitrary constants determined by initial conditions.

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Answer:

Step-by-step explanation:

The solution of the given equation is x = 2.

Here we have the following equation that we want to solve, it is:

[tex]125^x = \frac{625}{5^{-x}}[/tex]

We want to solve this for x, remember that a negative exponent means that we need to take the inverse, then we can rewrite the right side as:

[tex]125^x = \frac{625}{5^{-x}} = 625*5^x[/tex]

Now we can divide both sides by 5^x to get:

[tex]125^x = \frac{625}{5^{-x}} = 625*5^x\\\\(125/5)^x = 625\\\\25^x = 625\\\\[/tex]

Now we can apply the natural logarithm in both sides, we will get:

[tex]ln(25^x) = ln(625)\\\\x*ln(25) = ln(625)\\x = ln(625)/ln(25)\\\\x = 2[/tex]

That is the solution.

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The evidence supports the claim that the average life of electric light bulbs is greater than 2000 hours.

a. Null Hypothesis: The average life of electric light bulbs is not greater than 2000 hours. Alternative Hypothesis: The average life of electric light bulbs is greater than 2000 hours.

b. The critical value for a one-tailed test at the 2% level of significance with 63 degrees of freedom is 2.33.

c. The relevant test statistic is:
t = (x - μ) / (s / √n)=[tex]= \frac{(2008 - 2000)}{\frac{12.31}{\sqrt{64}}}= 13.03[/tex]
Since the test statistic is greater than the critical value of 2.33, we can reject the null hypothesis and conclude that there is sufficient evidence to support the claim.

d. The p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true. Using a t-distribution table with 63 degrees of freedom, the p-value is less than 0.01. Since the p-value is less than the significance level of 0.02, we can reject the null hypothesis.

e. Yes, we reject the null hypothesis.

f. No, we do not reject the claim. The evidence supports the claim that the average life of electric light bulbs is greater than 2000 hours.

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The characteristics of the visual representations are:

Dot plots :

Histogram :

Box Plot :

What are these graphs used for ?

Dot plot visually displays individual data points, while simultaneously providing frequency information for each interval. It enables one to easily visualize the median and is ideal for summarizing large data sets; additionally, it allows calculation of the mean.

Histograms present frequency by intervals which make them perfect also for analyzing larger data sets., This graphic allows calculating the mean value- a property that makes histograms an excellent tool in summarization tasks.

Box plots are incredibly useful when processing extensive amounts of data -They have visuals illustrating medians, split four ways. Box plots, similar to Dot plots and Histograms, allow computation of means as well.

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Answer: Dot Plot: individual data points

are seen, mean can be calculated

Histogram: frequency over each

interval is given, best used to summarize

large sets of data

Box Plot: breaks the data

into four equal

parts, best used to summarize

large sets of data, median can be seen

visually

"Got it right on Edmentum"

Explanation:

The median can be seen only on a box plot.

The data is broken into four equal parts on a box plot.

Box plots and histograms are best for large sets of data.

The individual data points are only seen on a dot plot. These points can be used to calculate the mean.

The frequency over each interval is given on a histogram

Answer:

A = 120; C = 95

Step-by-step explanation:

We will need a system of equations to solve for C, the number of child tickets and A, the number of adult tickets.

We know that the sum of the revenue earned from both the child tickets and the adult tickets = the total revenue

Thus, our first equation is 3C + 14A = 1965

We also know that the sum of the total number of child and adult tickets = the the total number of tickets

Thus, our other equation is C + A = 215

We can solve using substitution by first isolating c in the second equation:

[tex]C+A=215\\C=-A+215[/tex]

Now, we can plug in the equation we just made for C in the first equation in our system to solve for A:

[tex]3(-A+215)+14A=1965\\-3A+645+14A=1965\\11A+645=1965\\11A=1320\\A=120[/tex]

Finally, we can solve for C using the second equation in our system by plugging in 120 for A:

[tex]C+120=215\\C=95[/tex]

To calculate the accuracy of the test, we need to create a 2x2 contingency table:

Actual Positive (Meningitis)Actual Negative (No Meningitis)Test PositiveTrue Positive (TP) = 6False Positive (FP) = 3Test NegativeFalse Negative (FN) = 2True Negative (TN) = 69

From the information given, we know that there are 8 actual positive cases (people with meningitis) and 72 actual negative cases (people without meningitis). We also know that there were 3 false positive results (people who tested positive for meningitis but did not have it) and 2 false negative results (people who tested negative for meningitis but actually had it).

Using this information, we can calculate the accuracy of the test as:

Accuracy = (TP + TN) / (TP + TN + FP + FN)

Accuracy = (6 + 69) / (6 + 69 + 3 + 2)

Accuracy = 75 / 80

Accuracy = 0.9375 or 93.75%

Therefore, the accuracy of the meningitis test is 93.75%.

0 is a single digit whole number for y that supports Doni's claim.

2 is a single digit whole number for y that does not supports Doni's claim.

Doni claims that [tex]\frac{3^y}{2^y} \leq 1[/tex]

We have to find a single digit whole number for y that supports Doni's claim.

Let 0 be the single digit whole number for y that supports Doni's claim.

1/1≤1

Now let us find  single digit whole number for y that does not support Doni's claim.

2 be the whole number

9/4≤1

2.25≤1

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The exact values of the cosecant, secant, and cotangent ratios of 5π/6 are 2, -2/√3, and -√3, respectively.

To find the exact values of the cosecant, secant, and cotangent ratios of an angle of 5π/6, we need to use the definitions of these trigonometric functions and the values of the sine, cosine, and tangent of this angle.

First, we can find the sine and cosine of 5π/6 using the unit circle or reference angles:

sin(5π/6) = sin(π/6) = 1/2

cos(5π/6) = -cos(π/6) = -√3/2

Then, we can use the definitions of the cosecant, secant, and cotangent ratios:

cosec(5π/6) = 1/sin(5π/6) = 1/(1/2) = 2

sec(5π/6) = 1/cos(5π/6) = -2/√3

cot(5π/6) = cos(5π/6)/sin(5π/6) = (-√3/2)/(1/2) = -√3

Therefore, the exact values of the cosecant, secant, and cotangent ratios of 5π/6 are 2, -2/√3, and -√3, respectively.

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The given relationship in the task content is not a function as more than one output value is paired with the same input value.

Recall that a relationship is said to be a function only if one output value is attached to each input value of the relationship.

On this note, by observation; the pair of coordinates (6, 3) and (6, 9) implies that two output values are assigned to the same input value. Consequently, the given relationship is not a function.

The complete and correct sentence is therefore; Since one input value is paired with two output values; the relationship is not a function.

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The simplified difference is:

3y/8 - 5/6y = (-y)/24

Note that there are no non-permissible values in this case.

a. 4x/(2x+5) + 10/(2x+5)

To add these two expressions, we need to find a common denominator. In this case, the common denominator is (2x+5):

4x/(2x+5) + 10/(2x+5) = (4x+10)/(2x+5)

Now we can simplify the numerator by factoring out a 2:

(4x+10)/(2x+5) = 2(2x+5)/(2x+5)

And we can cancel out the common factor of (2x+5):

2(2x+5)/(2x+5) = 2

Therefore, the simplified sum is:

4x/(2x+5) + 10/(2x+5) = 2

Note that the non-permissible value is x = -2.5, since this value would make the denominator equal to zero.

b. 3y/8 - 5/6y

To subtract these two expressions, we also need a common denominator. In this case, the common denominator is 24y:

3y/8 - 5/6y = (9y^2)/(24y) - (20y)/(24y)

We can simplify the first term in the numerator by canceling out a common factor of 3:

([tex]9y^2[/tex])/(24y) = (3y)/8

So the subtraction becomes:

3y/8 - 5/6y = (3y)/8 - (10y)/12

Now we can find a common denominator of 24:

(3y)/8 - (10y)/12 = (9y)/24 - (10y)/24

Simplifying the numerator gives:

(9y)/24 - (10y)/24 = (-y)/24

Therefore, the simplified difference is:

3y/8 - 5/6y = (-y)/24

Note that there are no non-permissible values in this case.

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The value of variable x and y in the right triangle are 4 and 4√3 units respectively.

A right angle triangle is a triangle that has one of its angles as 90 degrees.

The variable x and y can be as follows:

cos 60° = adjacent / hypotenuse

cos 60° = x / 8

cross multiply

x = 8 cos 60°

x = 8 × 0.5

x = 4 units

Let's find the value of y as follows:

sin 60° = opposite / hypotenuse

Therefore,

sin 60° =  y / 8

cross multiply

y = 8 sin 60

y = 8 × √3 / 2

y = 4√3 units  

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Why don't you and Julián eat? Why not have something to eat?

How old am I? You have come nine years.

In the first sentence, Mario's answer is using the auxiliary verb "have" to suggest that they should eat something. The phrase "have something to eat" is a common way to express this idea.

In the second sentence, Mario's answer is using the verb "come" in the sense of "reach" or "attain". So, he is saying that his little sister has reached the age of nine years old. This is a colloquial way of answering the question about age, rather than saying "you are nine years old".

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Full Question: Mario's little sister is very curious and always wants to know why people do certain things. Complete Mario's answers with have and come. 1. Why don't you and Julián eat? Why not ______ hungry. 2. How old am I? You ____ nine years

Answer:5

explanation:

you are doing great finding the slopes:

Answer: The box is large enough to ship the coat.

15000cm to the power of 3>10000cm to the power of 3

Step-by-step explanation:

V box=25x30x20

         =750+20

         =15000cm to the power of 3

So the box is large enough to ship the coat

Answer:

Calcualate 40% (0.40) times 20 to get that there are 8 boys in the class. The rest must be girls, so 20 - 8 gives you 12 girls. 25% (0.25) times the 8 boys gives you that 2 of the boys wear glasses. 50% (0.5) times the 12 girls tives you that 6 girls wear glasses. Add together the 2 boys and the 6 girls that wear glasses to get that a total of 8 students wear glasses.

Step-by-step explanation:

The coordinates of the vertices after a reflection over the x-axis would be:

Reflecting a figure over the x-axis causes the y-coordinates of its vertices to change signs, while their respective x-coordinates remain unchanged.

To apply this principle to the given triangle, we flip the y-coordinate of point A from -3 to 3, that of point B from -1 to 1 and also for point C changing from -2 to 2 respectively. The outcome is as follows: A (0, 3), B (0, 1) and C (6, 2). It's noted that the x-coordinate remains constant across all points of the newly transformed shape.

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The probability that you are not one of the first four students chosen is 0.75 or 75%.

For the first selection, the probability that you are not chosen is:

(16 - 1) / 16 = 15 / 16

For the second selection, the probability that you are not chosen is:

(16 - 1 - 1) / (16 - 1) = 14 / 15

For the third selection, the probability that you are not chosen is:

(16 - 1 - 1 - 1) / (16 - 1 - 1) = 13 / 14

For the fourth selection, the probability that you are not chosen is:

(16 - 1 - 1 - 1 - 1) / (16 - 1 - 1 - 1) = 12 / 13

To find the probability that you are not one of the first four students chosen, we multiply these probabilities together:

(15/16) x (14/15) x (13/14) x (12/13) = 0.75

Therefore, the probability that you are not one of the first four students chosen is 0.75 or 75%.

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a. At (0,2] f is uniformly continuous.

b. At (2,∞) f is uniformly continuous.

c. At (0,∞) f is uniformly continuous.

A function connects an input with an output. It is analogous to a machine with an input and an output. And the output is somehow related to the input. The standard manner of writing a function is f(x) "f(x) =... "

(a) Let f(x) = x³ on (0,2]. Let ε > 0 be given. We need to find a δ > 0 such that |x - y| < δ implies |f(x) - f(y)| < ε for all x,y in (0,2]. Note that |f(x) - f(y)| = |x³ - y³| = |x - y||x² + xy + y²|. Since x,y ∈ (0,2], we have x² + xy + y² ≤ 12. Thus, if we choose δ = ε/12, then for any x,y ∈ (0,2] such that |x - y| < δ, we have |f(x) - f(y)| < ε. Hence, f is uniformly continuous on (0,2].

(b) Let f(x) = 1/2 on (2,∞). Let ε > 0 be given. We can choose any δ > 0 since for any x,y ∈ (2,∞), we have |f(x) - f(y)| = 0 < ε. Thus, f is uniformly continuous on (2,∞).

(c) Let f(x) = (x-1)/(x+1) on (0,∞). Let ε > 0 be given. We need to find a δ > 0 such that |x - y| < δ implies |f(x) - f(y)| < ε for all x,y in (0,∞). Note that |f(x) - f(y)| = |(x-1)/(x+1) - (y-1)/(y+1)| = |(x-y)(2/(x+1)(y+1))|. Thus, if we choose δ = ε/2, then for any x,y in (0,∞) such that |x - y| < δ, we have |f(x) - f(y)| = |(x-y)(2/(x+1)(y+1))| < ε. Hence, f is uniformly continuous on (0,∞).

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Answer:

option B: x= 18

Step-by-step explanation:

To solve 1/3x - 1 = 5, we can start by adding 1 to both sides of the equation:

1/3x - 1 + 1 = 5 + 1

Simplifying:

1/3x = 6

Multiplying both sides by 3:

3(1/3x) = 3(6)

Simplifying:

x = 18

Therefore, the solution is x = 18, which is option B.

The simplified form of the given expression written into one power is 3⁰

From the question, we are to simplify the given expression into one power

The given expression is

[tex]\frac{(3^{2})^{2}}{3 \ \cdot \ 3^{3}}[/tex]

To simplify the expression, we will implore the laws of indices

Simplifying the expression

[tex]\frac{(3^{2})^{2}}{3 \ \cdot \ 3^{3}}[/tex]

[tex]\frac{(3^{2\times 2})}{3^{1+3}}[/tex]

[tex]\frac{3^{4}}{3^{4}}[/tex]

Applying the division law of indices

[tex]3^{4-4}[/tex]

[tex]3^{0}[/tex]

Hence, the simplified expression is 3⁰

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1. What is the probability a randomly selected bag will have a positive test? Give your answer to four decimal places is 0.1032

2. Given a randomly selected bag has a positive test, what is the probability it actually contains a large amount of liquid is 0.3780

3. Given a randomly selected bag has a positive test, what is the probability it does not contain a large amount of liquid is  0.6219

Let's characterize the taking after occasions:

A: The pack contains huge sums of fluid.

B: The test is positive.

We are given the taking after probabilities:

P(A) = 0.04

P(B | A) = 0.98

P(B | not A) = 0.07

1. To discover the likelihood of a positive test, we are able to utilize the law of adding up to likelihood:

P(B) = P(B | A) P(A) + P(B | not A) P(not A)

= 0.98 * 0.04 + 0.07 * 0.96

= 0.1032

So the likelihood of a haphazardly chosen pack having a positive test is 0.1032 (adjusted to four decimal places).

2. To discover the likelihood that a sack really contains large amounts of fluid given a positive test, we are able to utilize Bayes' hypothesis:

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

= 0.98 * 0.04 / 0.1032

= 0.3780

So the likelihood that a pack really contains expansive sums of fluid given a positive test is 0.3780 (adjusted to four decimal places).

3. To discover the likelihood that a sack does not contain expansive sums of fluid given a positive test, ready to utilize Bayes' hypothesis again:

P(not A | B) = P(B | not A) P(not A) / P(B)

= 0.07 * 0.96 / 0.1032

= 0.6219

So the likelihood that a pack does not contain expansive sums of fluid given a positive test is 0.6219 (adjusted to four decimal places). 

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When you are given an equation for a function, it is important to know whether the function is increasing or decreasing. A function is said to be increasing if the value of the function increases as the input increases. Conversely, a function is said to be decreasing if the value of the function decreases as the input increases.

To determine whether a function is increasing or decreasing, you need to look at the sign of its first derivative. The first derivative of a function is the rate of change of the function with respect to its input. If the first derivative is positive, the function is increasing, and if it is negative, the function is decreasing. If the first derivative is zero, the function may have a local maximum or a local minimum.

To explain this in more detail, let's take the example of the function f(x) = x^2. To find the first derivative of this function, we need to differentiate it with respect to x. This gives us f'(x) = 2x. We can see that f'(x) is positive for x > 0, which means that the function f(x) = x^2 is increasing for x > 0. Similarly, f'(x) is negative for x < 0, which means that the function f(x) = x^2 is decreasing for x < 0.

In summary, to determine where a function is increasing or decreasing, you need to look at the sign of its first derivative. If the first derivative is positive, the function is increasing, and if it is negative, the function is decreasing. If the first derivative is zero, the function may have a local maximum or a local minimum.

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To evaluate the given integral by changing to polar coordinates, we first need to determine the limits of integration in polar form. The region R is in the first quadrant and is bounded by the circles with the center of the origin and radii 2 and 3. In polar coordinates, the equation of a circle centered at the origin is given by r = a, where a is the radius.

So, the equations of the two circles are:

r = 2  and  r = 3

Since the region R is between these two circles, the limits of integration for r are:

2 ≤ r ≤ 3

To determine the limits of integration for θ, we need to consider the quadrant in which the region R lies. Since R is in the first quadrant, we have:

0 ≤ θ ≤ π/2

Now, we can express the integrand sin(x^2 + y^2) in terms of polar coordinates:

sin(x^2 + y^2) = sin(r^2)

Therefore, the integral in polar coordinates is:

∫∫R sin(x^2 + y^2) dA = ∫ from 0 to π/2 ∫ from 2 to 3 sin(r^2) r dr dθ

This integral can be evaluated using standard techniques of integration.
To evaluate the integral using polar coordinates, we first need to express the given region R and the integrand in terms of polar coordinates. In polar coordinates, x = r*cos(θ) and y = r*sin(θ), so x^2 + y^2 = r^2.

The region R is in the first quadrant and is bounded by the circles with radii 2 and 3. In polar coordinates, this translates to 0 ≤ θ ≤ π/2, 2 ≤ r ≤ 3.

Now we can rewrite the integral as:

integral_integral_R sin(x^2 + y^2) dA
= integral (θ=0 to π/2) integral (r=2 to 3) sin(r^2) * r dr dθ

Now we can evaluate the integral step by step:

1. Integrate with respect to r:
integral (θ=0 to π/2) [(-1/2)cos(r^2)] (from r=2 to r=3) dθ
= integral (θ=0 to π/2) [(-1/2)(cos(9) - cos(4))] dθ

2. Integrate with respect to θ:
[(-1/2)(cos(9) - cos(4))]*(θ evaluated from 0 to π/2)
= [(-1/2)(cos(9) - cos(4))] * (π/2)

So the final answer is:

(π/2)(-1/2)(cos(9) - cos(4))

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(i) The transfer function of the system is:

H(z) = (0.4)^z / (0.2)^z - 0.3(0.4)^(z-1) / (0.2)^{z-1} - 2

(ii) The difference equation characterizing the system is:
y[n] = (0.4)^n x[n] - 0.3(0.4)^(n-1) x[n-1]

(i) To determine the transfer function of the system, we can take the Z-transform of both the input and output:

X(z) = (0.2)^z / (z - 0.4)
Y(z) = (0.4)^z / (z - 0.4) - 0.3(0.4)^(z-1) / (z - 0.4)

Then we can solve for the transfer function H(z) by dividing Y(z) by X(z):

H(z) = Y(z) / X(z)
    = (0.4)^z / (z - 0.4) - 0.3(0.4)^(z-1) / (z - 0.4) * (z - 0.4) / (0.2)^z
    = (0.4)^z / (0.2)^z - 0.3(0.4)^(z-1) / (0.2)^{z-1} - 2

So the transfer function of the system is H(z) = (0.4)^z / (0.2)^z - 0.3(0.4)^(z-1) / (0.2)^{z-1} - 2.

(ii) To determine the difference equation characterizing the system, we can use the formula for the output y[n] of a discrete-time LTI system with input x[n]:

y[n] = sum{k=0}{N-1} h[k] x[n-k]

where h[k] is the impulse response of the system. In this case, the impulse response can be found by setting x[n] = delta[n], the unit impulse function, and solving for y[n]:

h[n] = y[n] / delta[n]
    = (0.4)^n - 0.3(0.4)^(n-1)

So the difference equation characterizing the system is:

y[n] = (0.4)^n x[n] - 0.3(0.4)^(n-1) x[n-1]

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From multiplcation operation, Alex has enough paint to cover 1,000 ft² of his apartment. The true statement is 2 gallons of paint will cover 1068 ft², Alex have enough paint of quantity 2 gallons.

We have Mr. Alex painted his apartment. Area of his apartment'walls = 178 ft²

Quantity of paint used by him to paint his apartment'walls with area 178 ft² =[tex] \frac{1}{3} \: \: gallons[/tex]

Total quantity of paint used in all

= 2 gallons

We have to check the provide paint is enough or not to cover 1,000 ft² of his apartment. Let the required paint for 1000 ft² be x gallons. Using multiplcation, 1/3 gallons quantity of paint will cover the area of apartment = 178 ft², so, 1 gallons quantity of paint will cover the area of apartment = 178 ×3 ft²= 534 ft²

Now, 2 gallons quantity of paint will cover the area of apartment = 2× 534 ft² = 1068 ft²> 1000 ft²

But he wants to paint 1000 ft² of his apartment in 2 gallons quantity (x=1.9 gal ). So, he has enough paint to paint his apartment.

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Complete question:

The above figure complete the question.

Alex painted 178 ft2 of his apartment’s walls with 1/3 gallon of paint. He has 2 gallons of paint in all. If he wants to cover 1,000 ft2 of his apartment, does he have enough paint? Complete a true statement

The expected value of Y is: 1.8

The student made a mistake by using the formula for the variance of XY instead of the expectation of XY^2.

(a) E(Y) = (0.2)(2) + (0.5)(1) + (0.3)(4) = 1.8

(b) Since X and Y are independent, the distribution of X + Y is the convolution of their respective distributions. That is, if Z = X + Y, then for any real number z,

F(z) = P(Z ≤ z) = P(X + Y ≤ z) = ∑P(X = i, Y ≤ z − i) for i = 0, 1, 2, ...

Now, since X has a Poisson distribution with parameter 2 and Y takes values 2, 3, and 4 with probabilities 0.2, 0.5, and 0.3 respectively, we have

P(X = i, Y = j) = P(X = i)P(Y = j) = e^(-2) (2^i/i!)p_j for i = 0, 1, 2, ... and j = 2, 3, 4

where p_2 = 0.2, p_3 = 0.5, and p_4 = 0.3. Then, for z ≥ 2,

F(z) = P(Z ≤ z) = ∑_{i=0}^{z-2} P(X=i, Y≤z-i) = ∑_{i=0}^{z-2} e^(-2) (2^i/i!) ∑_{j=2}^{min(z-i, 4)} p_j

(c) P(X > Y) = ∑_{i=0}^∞ ∑_{j=0}^{i-1} P(X=i, Y=j) = ∑_{i=0}^∞ ∑_{j=0}^{i-1} e^(-2) (2^i/i!) p_j

(d) The calculation is incorrect. It should be E[XY^2] = E[X]E[Y^2] = 2(0.2)(2^2) + 2(0.5)(3^2) + 2(0.3)(4^2) = 11.6. The student made a mistake by using the formula for the variance of XY instead of the expectation of XY^2.

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