can dijkstra's algorithm find the shortest paths when using a directed acyclic graph g = (v,e)

Answer:

50%

Step-by-step explanation:

Above and below the median is always 50%

The shape that has a bigger area is the regular hexagon

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

Both of these shapes are inscribed in a circle

By comparison, the number of sides are

Pentagon = 5 sides

Hexagon = 6 sides

This means that the regular hexagon has a larger area

The large area is as a result of the larger number of sides and longer side length compared to the regular pentagon.

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Diatomic molecules can move through rotation and vibration. Quantum mechanics quantizes energy, and the Schrödinger equation provides solutions for studying molecular energy levels.

When examining molecular energy levels, diatomic molecules exhibit rotational and vibrational motions independent of their center of mass. In quantum mechanics, energy is quantized, meaning it can only be added or subtracted in specific discrete quantities.

The Schrödinger equation provides solutions for various quantum systems. Two solutions, the hydrogen atom and the harmonic oscillator, are particularly useful for studying molecules. The hydrogen atom has a spherically symmetric potential energy, which allows us to determine the allowed values of angular momentum (L) using the equation L^2 = l(l + 1)ℏ^2, where l represents different angular momentum states (l = 0, 1, 2, 3, ...).

Similarly, diatomic molecules have a spherically symmetric potential energy function U(r) = 0 for rotation. As a result, we can utilize the same equation for the allowed angular momentum states as in the case of the hydrogen atom, enabling us to analyze and understand the rotational energy levels of diatomic molecules.

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To compute the partial sums 3, 4, and 5 for a series, we need to add up the first three, four, and five terms of the series, respectively. Let's say the series is denoted by a_n, where n is the index of the term.

For example, if the series is 1, 2, 3, 4, 5, 6, 7, 8, 9, ... (which is an arithmetic series with a common difference of 1), then the partial sums would be:
- The sum of the first three terms (n=1, 2, 3) is 1 + 2 + 3 = 6.
- The sum of the first four terms (n=1, 2, 3, 4) is 1 + 2 + 3 + 4 = 10.
- The sum of the first five terms (n=1, 2, 3, 4, 5) is 1 + 2 + 3 + 4 + 5 = 15.
To find the sum of the series, we need to take the limit of the partial sums as n goes to infinity. In other words, we need to find the value of:
lim n→∞ ∑_(k=1)^n a_k


Without knowing the actual series, it's hard to give a specific answer to this question. However, the process for computing partial sums and finding the sum of a series is the same for any series, so you can apply the same method to whatever series you are given.

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Answer:

The gas with the highest average speed at 350 K is the one with the lowest molar mass. This is because the average speed of a gas molecule is directly proportional to the square root of its temperature and inversely proportional to the square root of its molar mass. So, the gas with the lowest molar mass will have the highest average speed.

Therefore, helium will have the highest average speed at 350 K.

Using z transform, we can find the discrete-time convolution between two sequences, h[n] and x[n]:

1. Take the z-transform of both sequences, h[n] and x[n], separately.

  - Let H(z) be the z-transform of h[n].

  - Let X(z) be the z-transform of x[n].

2. Multiply the z-transforms of the sequences together to obtain the z-transform of the convolution.

  - Y(z) = H(z) * X(z), where * denotes multiplication.

3. Take the inverse z-transform of Y(z) to obtain the discrete-time convolution sequence.

  - y[n] = InverseZTransform(Y(z))

Please note that the z-transform, multiplication, and inverse z-transform operations are specific to the mathematical representation of the sequences in the z-domain. The exact calculations will depend on the specific forms of h[n] and x[n].

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Given that the sum of two unequal numbers is 72 and their difference is 46, we can solve for the two numbers by setting up a system of equations and solving them simultaneously.

Let's assume the two numbers we are trying to find are x and y. Based on the given information, we can establish two equations:

Equation 1: x + y = 72

Equation 2: x - y = 46

To solve this system of equations, we can use the method of substitution or elimination.

Using the elimination method, we can add Equation 1 and Equation 2 to eliminate the y term:

(x + y) + (x - y) = 72 + 46

2x = 118

x = 118/2

x = 59

Substituting the value of x into Equation 1:

59 + y = 72

y = 72 - 59

y = 13

Therefore, the two numbers are 59 and 13.

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Answer: 1/1

Step-by-step explanation:to multiply fractions, multiply straight across. (4*3)/(15*8)=12/120, this reduces to 1/10.

you could also reduce from top to bottom before multiplying. 4/15 *3/8. 4/8*3/15=1/2*1/5=1/10

Answer:

Step-by-step explanation:

Given μA = $2.78, σA = $0.25, μB = $2.45, σB = $0.20, you want ...

The probabilities of interest are found using the CDF function of a suitable calculator or spreadsheet.

(a) P(A < $2.50) ≈ 0.1314

(b) P(B < $2.50) ≈ 59.87%

(c) P(B > $2.78) ≈ 0.0495

__

Additional comment

We note that you have provided your own answers to these questions. The answer you give for question B is not given as the percentage requested.

<95141404393>

Step-by-step explanation:

Exterior angles of all polygons sum to 180 degrees

180 - 58 - 29-73-71-62 = n    degrees

Green's theorem can be used to evaluate the line integral along the triangular path from (0, 0) to (2, 0) to (2, 1) to (0, 0) of the function xy dx + y^3 dy.

Green's theorem relates a line integral around a closed curve to a double integral over the region enclosed by the curve. The theorem states that the line integral of a vector field F along a simple closed curve C is equal to the double integral of the curl of F over the region D enclosed by C. In this case, we are given the line integral of the function xy dx + y^3 dy along the triangular path.

To evaluate the line integral using Green's theorem, we first need to find the curl of the vector field associated with the function. The curl of F = (P, Q) is given by ∂Q/∂x - ∂P/∂y, where P and Q are the components of the vector field.

In this case, P = xy and Q = y^3. Taking the partial derivatives, we get ∂Q/∂x = 0 and ∂P/∂y = x. Therefore, the curl of F is 0 - x = -x.

Now, we can evaluate the double integral of the curl of F over the region D enclosed by the triangular path. The region D is a triangle with vertices (0, 0), (2, 0), and (2, 1). By integrating -x over this region, we can find the value of the line integral.

Performing the double integral and simplifying the result will give us the final answer for the line integral along the given path using Green's theorem.

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the dimensions of the larger rectangle are (50/3) x's for the width and (75/3) x's for the length.

To determine the original rectangle, we need to find the dimensions of the larger rectangle. The given rectangle has a width of 10 x's and a length of 15 x's. Since it is stated that the given rectangle is 3/5 of the larger rectangle, we can set up the following equations:

Width of the larger rectangle: (10 x's) = (3/5) × (width of the larger rectangle)

Length of the larger rectangle: (15 x's) = (3/5) × (length of the larger rectangle)

Solving these equations, we can find the dimensions of the larger rectangle. Let's denote the width of the larger rectangle as W and the length as L. We have:

W = (10 x's) × (5/3) = (50/3) x's

L = (15 x's) × (5/3) = (75/3) x's

By scaling the given rectangle with the fraction 3/5, we can determine the dimensions of the original rectangle as (50/3) x's for the width and (75/3) x's for the length.

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The probability of the three independent events all occurring in three consecutive dice rolls is 1/216.

Assuming a fair six-sided die, the probability of any single roll resulting in a specific number is $1/6$. Since the events are independent, the probability of all three events occurring in three consecutive rolls is the product of the probabilities of each individual event.

Therefore, the probability of getting a specific number on three consecutive rolls is:

$P = (1/6) * (1/6) * (1/6) = 1/216$

So the probability of the three independent events all occurring in three consecutive dice rolls is 1/216.

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The indicated probabilities from the probability tree are as follows:

(A) P(M∩S) = 0.9 * 0.5 = 0.45

(B) P(R) = 0.9

In the probability tree, we have two branches originating from the initial event, denoted by R and M. The probability of event R occurring is given as 0.9, which means P(R) = 0.9.

Moving down the R branch, we encounter another event denoted by M, with a probability of 0.5. Now, to calculate the probability of the intersection of events M and S, denoted by M∩S, we multiply the probabilities of M and S,

which gives us 0.9 * 0.5 = 0.45.

Therefore, the probability of event M and S both occurring, P(M∩S), is 0.45.

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Complete question here:

Compute the indicated probabilities by referring to the probability tree. 0.9 R 0.5M (A) P(MnS) (B) P(R) 0.6R 0.5 N (A) P(MnS)(Type an integer or a decimal.) (B) P(R) = (Type an integer or a decimal.)

Answer:

Step-by-step explanation:

You want to know angles x, y, and z in the given figure where parallel lines 'a' and 'b' are crossed by a transversal. The sum of these angles is 290°.

Angles y and z are called consecutive interior angles. As such, they are supplementary, so their sum is 180°.

  x + y + z = 290°

  x + 180° = 290°

  x = 110°

Angles x and y are vertical angles, so are congruent.

  y = x = 110°

Then z is found from ...

  y + z = 180°

  110° + z = 180°

  z = 70°

The measures of x, y, and z are 110°, 110°, and 70°, respectively.

<95141404393>

The probability of drawing a D, then drawing an H (with replacement) from the given deck is 2/49.

To calculate the probability of drawing a D, then drawing an H, replacing the first card, we need to know the total number of cards in the deck and the number of D and H cards in the deck.

Since you mentioned the deck consists of the letters OHDBOHG, we'll assume there are 7 cards in total.

The probability of drawing a D on the first draw, assuming all cards are equally likely to be drawn, is 1 out of 7 since there is only one D card in the deck.

Since we are replacing the first card, the deck remains the same for the second draw. So, the probability of drawing an H on the second draw, assuming all cards are equally likely to be drawn, is also 1 out of 7 since there is only one H card in the deck.

To find the overall probability, we multiply the probabilities of the individual events:

Probability = (1/7) * (1/7) = 2/49

Therefore, the probability of drawing a D, then drawing an H (with replacement) from the given deck is 2/49.

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The statement supported by the bar chart is option d) The percentage of all animals at fair Y that are goats is equal to the percentage of all animals at fair X that are goats.

Explanation:

To determine which statement is supported by the bar chart, analyze the data shown. The bar chart provides the number of different types of animals at two county fairs: fair X and fair Y. It also gives the total number of animals at each fair.

Statement a) cannot be determined from the bar chart

as it does not provide specific numbers for each type of animal.

Statement b) cannot be determined

as the number of chickens at each fair is not given.

Statement c) cannot be determined

as the ratio between sheep and horses is not provided.

Statement d) can be supported by the bar chart

by comparing the percentage of goats at each fair. If the percentage of all animals that are goats is the same at both fairs, then statement d) is true.

Statement e) cannot be determined from the bar chart

as it does not provide the percentage of goats at each fair.

Therefore, based on the information provided by the bar chart, the statement supported is option d).

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a)  , the slope in the x-direction at the point (-2, 1, 3) is -4

b) The slope in the y-direction at the point (-2, 1, 3) is -2.

To find the slopes of the surface in the x-direction and y-direction at the point (-2, 1, 3) for the function h(x, y) = x^2 - y^2, we need to calculate the partial derivatives with respect to x and y.

(a) Slope in the x-direction:

The partial derivative of h(x, y) with respect to x, denoted as ∂h/∂x or h_x, gives the slope of the surface in the x-direction.

∂h/∂x = ∂/∂x (x^2 - y^2)

= 2x

Substituting the point (-2, 1, 3) into the partial derivative:

∂h/∂x = 2(-2)

= -4

Therefore, the slope in the x-direction at the point (-2, 1, 3) is -4.

(b) Slope in the y-direction:

The partial derivative of h(x, y) with respect to y, denoted as ∂h/∂y or h_y, gives the slope of the surface in the y-direction.

∂h/∂y = ∂/∂y (x^2 - y^2)

= -2y

Substituting the point (-2, 1, 3) into the partial derivative:

∂h/∂y = -2(1)

= -2

Therefore, the slope in the y-direction at the point (-2, 1, 3) is -2.Learn

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The series ∑((-1)^n * n)/(n^3 * 2^n) is absolutely convergent.

To determine the convergence of the series ∑((-1)^n * n)/(n^3 * 2^n), we can use the ratio test.

Using the ratio test, we evaluate the limit of the absolute value of the ratio of consecutive terms:

lim(n→∞) |((-1)^(n+1) * (n+1))/((n+1)^3 * 2^(n+1))| / |((-1)^n * n)/(n^3 * 2^n)|

Simplifying, we get:

lim(n→∞) |(-1)^(n+1) * (n+1) * n^3 * 2^n| / |((-1)^n * (n+1)^3 * 2^(n+1))|

Since the absolute values of the terms simplify and cancel out, we have:

lim(n→∞) |(-1)^(n+1) * (n+1) * n^3 * 2^n| / |((-1)^n * (n+1)^3 * 2^(n+1))|

= lim(n→∞) (n^3 * 2^n) / ((n+1)^3 * 2^(n+1))

We can simplify further by dividing both the numerator and the denominator by n^3 * 2^n:

lim(n→∞) (n^3 * 2^n) / ((n+1)^3 * 2^(n+1))

= lim(n→∞) (n / (n+1))^3 * (1/2)

As n approaches infinity, the term (n / (n+1))^3 approaches 1, and the term (1/2) is a constant.

Therefore, the limit simplifies to:

lim(n→∞) (n / (n+1))^3 * (1/2)

= (1/2)

Since the limit of the absolute value of the ratio is less than 1 (specifically, 1/2), according to the ratio test, the series is absolutely convergent.

In conclusion, the series ∑((-1)^n * n)/(n^3 * 2^n) is absolutely convergent.

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The point (-2, 5) translated down 4 units would result in the new coordinates (-2, 1).

When a translation is performed, the entire shape or point is shifted in a specified direction.

In this case, since we are translating down, we need to decrease the y-coordinate of the point by 4 units.

Starting with the original point (-2, 5), we move 4 units downward along the y-axis. Since we are subtracting 4 units from the y-coordinate, the new y-coordinate becomes 5 - 4 = 1.

Therefore, the translated point would be (-2, 1).

This means that the point originally located at (-2, 5) has been shifted downward by 4 units and is now located at (-2, 1).

The x-coordinate remains the same since the translation was only performed along the y-axis.  

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The solution to the given PDE with the provided BCs and ICs involves finding the eigenfunctions and eigenvalues through separation of variables and then using the Fourier series expansion to determine the coefficients that satisfy the initial conditions.

The given partial differential equation (PDE) is a wave equation in one dimension, represented as utt = 25ux x, where u is a function of two variables x and t. This equation describes the behavior of waves propagating in the x-direction.

The boundary conditions (BC) state that u(0,t) = u(1,t) = 0, which means that the function u is zero at both ends of the interval x = 0 and x = 1. These boundary conditions enforce the idea that there are no reflections or transmissions at the boundaries.

The initial conditions (IC) specify the initial behavior of the wave. Here, u(x,0) = 9sin(2πx) represents the initial displacement of the wave, and ut(x,0) = 4sin(3πx) represents the initial velocity of the wave.

To solve this problem, we can use the method of separation of variables. We assume a solution of the form u(x,t) = X(x)T(t), where X(x) represents the spatial component and T(t) represents the temporal component.

By substituting this solution into the wave equation, we obtain two ordinary differential equations: X''(x)/X(x) = T''(t)/(25T(t)) = -λ².

Solving the spatial equation X''(x)/X(x) = -λ², subject to the boundary conditions X(0) = X(1) = 0, we find that the eigenfunctions are Xn(x) = sin(nπx), and the corresponding eigenvalues are λn = nπ.

Solving the temporal equation T''(t)/(25T(t)) = -λ², we obtain Tn(t) = A_nsin(λnt) + B_ncos(λnt), where A_n and B_n are constants determined by the initial conditions.

Finally, we can express the general solution as the superposition of all the eigenfunctions: u(x,t) = Σ[A_nsin(λnt) + B_ncos(λnt)]sin(nπx), where the sum is taken over all possible values of n.

To find the specific solution that satisfies the given initial conditions, we can use the Fourier series expansion of the initial conditions and match the coefficients with the general solution.

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The series converges. To determine the convergence or divergence of the series:

∑[infinity] (−1)^n ln(n) / n^2

We can use the alternating series test. The alternating series test states that if a series is of the form:

∑[infinity] (-1)^n b_n

where b_n > 0 for all n and b_n is a decreasing sequence, then the series converges if the limit of b_n as n approaches infinity is 0.

In the given series, we have b_n = ln(n) / n^2.

First, let's check if b_n is positive for all n. Since ln(n) is positive for n > 1 and n^2 is also positive, the ratio ln(n) / n^2 is positive for n > 1.

Next, we need to show that b_n is a decreasing sequence. To do this, we can consider the ratio of consecutive terms:

b_{n+1} / b_n = [ln(n+1) / (n+1)^2] / [ln(n) / n^2]

= (ln(n+1) / n^2) * (n^2 / (n+1)^2)

= (ln(n+1) / n^2) * (1 / (1+1/n)^2)

Since ln(n+1) is a logarithmic function, it grows at a slower rate than any positive power of n. Therefore, the first term ln(n+1) / n^2 decreases as n increases. The second term (1 / (1+1/n)^2) is always less than or equal to 1.

Thus, the ratio b_{n+1} / b_n is less than or equal to 1 for all n > 1. This shows that the sequence b_n is decreasing.

Now, we need to evaluate the limit of b_n as n approaches infinity:

lim(n->∞) ln(n) / n^2

= lim(n->∞) [ln(n) / n] / n

= (0 / ∞) / ∞ (using L'Hôpital's rule)

= 0

Since the limit of b_n as n approaches infinity is 0, the alternating series test tells us that the series:

∑[infinity] (−1)^n ln(n) / n^2

converges.

Therefore, To determine the convergence or divergence of the series:

∑[infinity] (−1)^n ln(n) / n^2

we can use the alternating series test. The alternating series test states that if a series is of the form:

∑[infinity] (-1)^n b_n

where b_n > 0 for all n and b_n is a decreasing sequence, then the series converges if the limit of b_n as n approaches infinity is 0.

In the given series, we have b_n = ln(n) / n^2.

First, let's check if b_n is positive for all n. Since ln(n) is positive for n > 1 and n^2 is also positive, the ratio ln(n) / n^2 is positive for n > 1.

Next, we need to show that b_n is a decreasing sequence. To do this, we can consider the ratio of consecutive terms:

b_{n+1} / b_n = [ln(n+1) / (n+1)^2] / [ln(n) / n^2]

= (ln(n+1) / n^2) * (n^2 / (n+1)^2)

= (ln(n+1) / n^2) * (1 / (1+1/n)^2)

Since ln(n+1) is a logarithmic function, it grows at a slower rate than any positive power of n. Therefore, the first term ln(n+1) / n^2 decreases as n increases. The second term (1 / (1+1/n)^2) is always less than or equal to 1.

Thus, the ratio b_{n+1} / b_n is less than or equal to 1 for all n > 1. This shows that the sequence b_n is decreasing.

Now, we need to evaluate the limit of b_n as n approaches infinity:

lim(n->∞) ln(n) / n^2

= lim(n->∞) [ln(n) / n] / n

= (0 / ∞) / ∞ (using L'Hôpital's rule)

= 0

Since the limit of b_n as n approaches infinity is 0, the alternating series test tells us that the series:

∑[infinity] (−1)^n ln(n) / n^2

converges.

Therefore, the series converges.

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 The radius of convergence,  R, is 1

To find the radius of convergence, R, of the series, we can use the formula:

R = 1 / lim(n→∞) |(aₙ₊₁ / aₙ)|

In this case, we have the series [∑ from n = 0 to ∞] (x - 8)^n(n^8 + 1).

To apply the ratio test, let's compute the limit of |(aₙ₊₁ / aₙ)| as n approaches infinity:

lim(n→∞) |[(x - 8)^(n + 1)(n^8 + 1)] / [(x - 8)^n(n^8 + 1)]|

Simplifying, we can cancel out (n^8 + 1) terms:

lim(n→∞) |x - 8|

For the series to converge, the limit above must be less than 1. Therefore, we have:

|x - 8| < 1

This inequality implies that x must be within a distance of 1 from 8. Hence, we have:

7 < x < 9

Therefore, the interval of convergence, I, is (7, 9), and the radius of convergence, R, is half the length of the interval:

R = (9 - 7) / 2 = 2 / 2 = 1

Thus, the radius of convergence, R, is 1.

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Answer:

70 ounces

Step-by-step explanation:

26÷13=2

so 1 candy bar weighs 2 ounces

35 candy bars will then equal 35×2

35×2=70 ounces

Answer: 70 ounces

Step-by-step explanation:

       First, we will find the weight per bar. We will do this by dividing 26 ounces by 13 bards.

               26 ounces / 13 candy bars = 2 ounces per bar

       Next, we will multiply this value of ounces per bar by 35 candy bards.

               35 candy bards * 2 ounces per bar = 70 ounces

If the total spending of the family is $54,500, then the expected spending on others is $5400.00, The correct option is B.

To calculate the amount spent on "Other," we must determine the fraction of the total expenses corresponding to "Other." According to the graph, "Other" accounts for 1/10 of the total expenses.

To find the amount spent on "Other," we multiply the fraction by the total expenditure:

Amount spent on "Other" = (1/10) * $54,000

Now let's calculate it:

Amount spent on "Other" = (1/10) * $54,000 = $5,400.00

Therefore, the correct answer is B) $5,400.00.

The provided question is incomplete, I think the question is,

Suppose your family spent $54,000 on the items in the graph above the graphs shows( Clothing = 1/ 20, Housing=3/10, Education= 1/10, Other= 1/10, Food= 1/5, Transportation 1/4). How much might we expect was spent on other?

A) $2700.00

C) $4725.00

B) $5400.00

D) $4050.00

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The correct answer is: b. Quantitative, because numerical values, found by either measuring or counting, are used to describe the data.

The variable "Happiness after graduation (on a scale of 1 to 10)" represents a quantitative variable. The scale of 1 to 10 assigns numerical values to measure the level of happiness reported by individuals. The use of numerical values indicates a quantitative variable, as the responses are quantified on a numerical scale.

The data collected from individuals are numerical measurements that can be analyzed and compared using mathematical operations such as averaging, calculating the range, and performing statistical analyses. Additionally, the scale from 1 to 10 implies an ordinal level of measurement, where the values have an inherent order or ranking. This allows for comparisons between different levels of happiness, identifying higher or lower ratings on the scale.

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Step-by-step explanation:

4 ft 2 in =  50 inches

50 inches  * 2.54 cm / in = 127 cm    

It is found that Julie's height is 127 centimeters.

If an event can occur in m different ways and if following it, a second event can occur in n different ways, then the two events in succession can occur in m × n different ways.

Given that Julie is 4 feet 2 inches tall and we need to change to centimeters.

First we change her height to inches:

[tex]\sf 1 \ Foot = 12 \ Inches[/tex]

Therefore, 4 feet [tex]\sf = 4\times 12 = 48[/tex] inches

The total height in inches = 48 inches + 2 Inches = 50 inches

Now, we have gotten her height in inches, change to centimeters.

We have that:

[tex]\sf 1 \ inch = 2.54 \ \bold{centimeters}[/tex]

[tex]\sf 50 \ inches = 50 \times 2.54 \ cm[/tex]

[tex]\sf= 127\ centimeters[/tex].

Hence, Julie's height is 127 centimeters.

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It's 16.

The difference between two consecutive numbers is one more than between the previous two numbers.

1+1=2+2=4+3=7+4=11+5=16+6=22

Locust swarms are known for their devastating impact on agricultural crops and vegetation. The population of a swarm of locusts can grow at a rate that is proportional to the fourth power of the cubic root of its current population.

This means that the rate of growth is highly dependent on the current population size. As the population increases, the rate of growth also increases exponentially.
For example, if the current population is 1000 locusts, the rate of growth will be proportional to the fourth power of the cubic root of 1000, which is approximately 31.62. This means that the population will increase at a rapid rate, and if measures are not taken to control it, it can lead to significant damage to crops and vegetation.
It is essential to monitor the population of locust swarms regularly and take appropriate measures to control their growth. This can include the use of insecticides, implementing early warning systems, and carrying out surveillance activities to detect and monitor any potential outbreaks. By doing so, we can help to mitigate the impact of locust swarms and ensure food security for communities affected by these pests.

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The joint probability mass function is determined to be p(x,y) = (2/3)xy, the conditional probability p(x > 1 | y = 1) is 1/2, E(X) is 10/3, and E(Y) is 2.

To find the value of k that makes the given function a probability mass function, we need to ensure that the sum of all probabilities over the entire sample space is equal to 1.

Let's calculate the sum of probabilities:

∑∑ p(x, y) = ∑∑ k(xy)

Since the function is defined as zero when x ≠ 1 and 2y ≠ 1, we only need to consider the cases where x = 1 and 2y = 1:

∑∑ p(x, y) = k(1 * y) + k(1 * (1/2))

To make this sum equal to 1, we need:

k(y + 1/2) = 1

Since this equation holds for all values of y, we can choose a value of y that satisfies the equation. Let's choose y = 1:

k(1 + 1/2) = 1

k(3/2) = 1

k = 2/3

So, the value of k that makes the function a probability mass function is 2/3.

Now let's find p(x > 1 | y = 1):

p(x > 1 | y = 1) = p(x = 2 | y = 1) / p(y = 1)

To calculate p(x = 2 | y = 1), we use the joint probability mass function:

p(x = 2 | y = 1) = k(2 * 1) = 2/3

To calculate p(y = 1), we sum the probabilities over all x values:

p(y = 1) = ∑ p(x, 1) = p(1, 1) + p(2, 1) = k(1 * 1) + k(2 * 1) = 2/3 + 2/3 = 4/3

Therefore, p(x > 1 | y = 1) = (2/3) / (4/3) = 1/2.

To find E(X), we need to calculate the expected value of X using the joint probability mass function:

E(X) = ∑∑ x * p(x, y)

= 1 * p(1, 1) + 2 * p(2, 1)

= 1 * (k * 1 * 1) + 2 * (k * 2 * 1)

= 1 * (2/3 * 1 * 1) + 2 * (2/3 * 2 * 1)

= 2/3 + 8/3

= 10/3

Therefore, E(X) = 10/3.

To find E(Y), we need to calculate the expected value of Y using the joint probability mass function:

E(Y) = ∑∑ y * p(x, y)

= 1 * p(1, 1) + 1 * p(1, 2)

= 1 * (k * 1 * 1) + 1 * (k * 1 * 2)

= 1 * (2/3 * 1 * 1) + 1 * (2/3 * 1 * 2)

= 2/3 + 4/3

= 6/3

Therefore, E(Y) = 2.

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