consider the joint pdf of two random variable x, y given by f x,y (x,y) = c, where 0 < x < a where a =3.37, and 0 < y < 8.15. find fx (a/2)

It is a rectangle because the opposite sides were parallel and congruent

Given data,

Let the quadrilateral be represented as WXYZ

Now , the line WX is parallel and congruent to the side YZ

So, they have the same slope

And , the line segment WZ is parallel and congruent to the side XY

So , they have the same slope

Therefore , the quadrilateral is a rectangle

Hence , the figure is a rectangle and the opposite sides have same slope

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Answer:  sqrt287

Step-by-step explanation:

sqrt(24^2-17^2)=sqrt287

The 96% confidence interval for the soda cans' diameters manufactured by CO 4 is approximately 51.1 mm to 52.9 mm.

To calculate the 96% confidence interval for the soda cans' diameters, we need to consider the sample mean, standard deviation, and sample size, as well as the appropriate Z-score for the desired level of confidence.

The terms you've provided are:
- Company (CO 4) - A company that manufactures soda cans.
- Diameter of 52 mm - The average diameter of the soda cans.
- Sample size of 18 - The number of soda cans in the sample.
- Standard deviation of 2.3 mm - The measure of dispersion in the sample.

Given the information, we first need to calculate the standard error (SE), which is the standard deviation (2.3 mm) divided by the square root of the sample size (18). This can be calculated as follows:

SE = 2.3 / √18 ≈ 0.54

For a 96% confidence interval, we use a Z-score of 2.05, which means we are 96% confident that the true population means lies within this interval. Now, we can calculate the confidence interval:

Lower limit = Sample mean - (Z-score × SE) = 52 - (2.05 × 0.54) ≈ 51.1 mm
Upper limit = Sample mean + (Z-score × SE) = 52 + (2.05 × 0.54) ≈ 52.9 mm

So, the 96% confidence interval for the soda cans' diameters manufactured by CO 4 is approximately 51.1 mm to 52.9 mm. This means we are 96% confident that the true average diameter of the soda cans produced by the company lies within this range.

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The statement is expressed as:

[tex]y \alpha 1/x[/tex]

To convert to an equation introduce k, the constant of variation.

[tex]y=k * 1/x\\[/tex]

To find k use the condition that [tex]x = 12[/tex] when [tex]y = 5[/tex]

[tex]y=k/x[/tex]

[tex]5=k/12[/tex]

[tex]k=5*12[/tex]

[tex]k=60\\[/tex]

Therefore,  [tex]y =60/x[/tex] is the function.

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The answer is A. True. When working with big data, the sample size is so large that the variability in the data almost completely disappears.

Variability is the degree of difference between values in a set of data. It is a measure of how spread out the values are from the mean or average of the set.

When dealing with smaller datasets, there is typically more variability. This is because a smaller sample size does not represent the entire population of data, and therefore does not provide an accurate representation of the underlying population of data. This variability can make it more difficult to draw valid conclusions from the data.

However, when working with a large sample size, the variability virtually disappears. This is because the data is spread across so many data points that the differences between individual data points become negligible. The result is that the data becomes more consistent and easier to analyze.

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ωc = 1/RC. where R is the resistance of the circuit and C is the capacitance. Once we know ωc, we can substitute it into the equation for vo(t) and evaluate the expression.

When ω=ωc, the output voltage can be found by substituting ωc for ω in the equation for vo(t):

vo(t) = a*cos(ωc*t + ϕ)v

The values of a, ω, and ϕ are given in the problem:

a > 0
-180∘ < ϕ ≤ 180∘
ω = angular frequency of the input signal

So, to find the output voltage when ω=ωc, we need to know the value of ωc. This value depends on the circuit parameters and can be calculated using the formula:

ωc = 1/RC

Without more information about the circuit, we cannot provide a specific answer for the output voltage when ω=ωc.

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Answer:

yes

Step-by-step explanation:

because it has a dramatic decrease in value

The general solution to the differential equation dv/dt = -32 - kv is:

[tex]v = (Ke^{(-t)} - 32)/k[/tex] if kv > 0

[tex]v = (32 - Ke^{(-t)})/k[/tex] if kv < 0

To solve this differential equation, we need to separate the variables and integrate both sides. We can write:

dv/(32+kv) = -dt

Now, we can integrate both sides. For the left-hand side, we can use the substitution u = 32 + kv, which gives:

dv/u = -dt

Integrating both sides, we get:

ln|u| = -t + C

where C is the constant of integration. Substituting back for u, we get:

ln|32 + kv| = -t + C

To solve for v, we can exponentiate both sides:

[tex]|32 + kv| = e^{(-t+C)} = Ke^{(-t)}[/tex]

where K is another constant of integration.

Taking the absolute value of both sides is necessary because kv can be negative. To solve for v, we need to consider two cases: kv is positive and kv is negative.

If kv is positive, then we have:

[tex]32 + kv = Ke^{(-t)}[/tex]

Solving for v, we get:

[tex]v = (Ke^{(-t)} - 32)/k[/tex]

If kv is negative, then we have:

[tex]-(32 + kv) = Ke^{(-t)}[/tex]

Solving for v, we get:

[tex]v = (32 - Ke^{(-t)})/k[/tex]

Therefore, the general solution to the differential equation dv/dt = -32 - kv is:

[tex]v = (Ke^{(-t)} - 32)/k[/tex] if kv > 0

[tex]v = (32 - Ke^{(-t)})/k[/tex]if kv < 0

where K and k are constants of integration that depend on the initial conditions.

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The function f(x, y, z) that satisfies F = ∇f is f(x, y, z) = xyz + 3xz + 3yz² + 6zk + C, where C is a constant. The integral of ∇f · dr along C is given by:

∫∇f · dr = f(x2, y2, z2) - f(x1, y1, z1) = f(5, 6, 3) - f(3, 0, -1).

To find the function f(x, y, z) such that F = ∇f, we need to determine the gradient of f and equate it to F.

Gradient of f = ∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k

Comparing the components of ∇f with F:

∂f/∂x = yz (1)

∂f/∂y = xz (2)

∂f/∂z = xy + 6z (3)

To solve for f, we integrate each of the partial derivatives with respect to their respective variables.

From equation (1), integrating with respect to x:

f(x, y, z) = xyz + g(y, z), where g(y, z) is a function of y and z.

Taking the partial derivative of f(x, y, z) with respect to y and comparing with equation (2):

∂f/∂y = xz + (∂g/∂y) = xz

∂g/∂y = 0

Integrating g(y, z) with respect to y:

g(y, z) = yz² + h(z), where h(z) is a function of z.

Taking the partial derivative of f(x, y, z) with respect to z and comparing with equation (3):

∂f/∂z = xy + 6z + (∂h/∂z) = xy + 6z

∂h/∂z = 0

Integrating h(z) with respect to z:

h(z) = 6zk + C, where C is a constant.

Substituting the expressions for g(y, z) and h(z) back into f(x, y, z):

f(x, y, z) = xyz + 3xz + 3yz² + 6zk + C

Therefore, the function f(x, y, z) that satisfies F = ∇f is f(x, y, z) = xyz + 3xz + 3yz² + 6zk + C, where C is a constant.

To evaluate the integral of ∇f · dr along the given curve C, we substitute the coordinates of the two endpoints of C into f(x, y, z) and calculate the difference.

Coordinates of the starting point of C: (x1, y1, z1) = (3, 0, -1)

Coordinates of the ending point of C: (x2, y2, z2) = (5, 6, 3)

Substituting the coordinates into f(x, y, z):

f(x1, y1, z1) = f(3, 0, -1)

f(x2, y2, z2) = f(5, 6, 3)

The integral of ∇f · dr along C is given by:

∫∇f · dr = f(x2, y2, z2) - f(x1, y1, z1) = f(5, 6, 3) - f(3, 0, -1)

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The given sequence an = (7n+2)/(8n) converges, and its limit is 7/8

To determine whether the sequence converges or diverges, and to find the limit if it converges, we will analyze the given sequence an = (7n + 2) / (8n).

Step 1: Simplify the sequence by dividing the numerator and the denominator by the highest power of n, which in this case is n^1.

an = (7 + 2/n) / (8)

Step 2: Take the limit of the sequence as n approaches infinity.

lim (n -> ∞) [(7 + 2/n) / 8]

Step 3: As n approaches infinity, the term 2/n approaches 0.

lim (n -> ∞) [(7 + 0) / 8]

Step 4: Simplify the limit.

lim (n -> ∞) [7 / 8] = 7/8

Since the limit exists and is a finite value (7/8), we can conclude that the sequence converges, and its limit is 7/8.

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To demonstrate that the limit lim(x,y)→(0,0) [tex](x^2)/(x^2y^2 + (x-y)^2)[/tex] does not exist, we can use two different paths:

Path 1: Let y = x In this path, we substitute y with x in the expression: lim(x,x)→ [tex](0,0) (x^2)/(x^2x^2 + (x-x)^2)[/tex] = lim(x,x)→ [tex](0,0) (x^2)/(x^4)[/tex]As x approaches 0, the expression simplifies to: lim(x→0) [tex](x^2)/(x^4)[/tex] = lim(x→0) [tex]1/x^2[/tex] When x approaches 0, [tex]1/x^2[/tex] goes to infinity.

Therefore, the limit along this path does not exist.

Path 2: Let y = 0 In this path, we substitute y with 0 in the expression: lim(x,0)→(0,0)[tex](x^2)/(x^2(0)^2 + (x-0)^2)[/tex] = lim(x,0)→ [tex](0,0) (x^2)/(x^2)[/tex] As x approaches 0, the expression simplifies to: lim(x→0) (x^2)/(x^2) = lim(x→0) 1

When x approaches 0, the expression equals 1, which is a finite value.

Since the limits along Path 1 and Path 2 are not equal, we can conclude that the limit lim(x,y)→(0,0) [tex](x^2)/(x^2y^2 + (x-y)^2)[/tex] does not exist.

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If the first equation is 3x + 2y = 5, a second equation that would make this system have no solution is: A. 6x + 4y = 10.

If the first equation is y = 2x - 5, a second equation that would make this system have no solution is: A. 4x - 2y = 10.

The statement that correctly describes the solution to this system of equations is: A. there is no solution.

The statement that correctly describes the solution to this system of equations is: A. there is no solution.

The statement that correctly interprets Xaysha's solution is: C. there is no solution since 5 = 7 is a false statement.

The statement that correctly interprets Sophina's solution is: A. the solution is x = 0.

In Mathematics, no solution is sometimes referred to as zero solution, and an equation is said to have no solution when the left hand side and right hand side of the equation are not the same or equal.

This ultimately implies that, a system of equations would have no solution when the line representing each of the equations are parallel lines and have the same slope or coincide i.e both sides of the equal sign are the same and the variables cancel out.

For Sophina's solution, we have;

7x = 6x

7x - 6x = 0

x(7 - 6) = 0

x = 0

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Answer:

7/24

Step-by-step explanation:

A = 7/8 × 2/3 × 1/2 = 14/48 = 7/24

The required answer is = (n/(n+1))theta

theta = (n/(n+1))theta the sample maximum is an unbiased estimator of theta.

One unbiased estimator of theta is the sample maximum, which is defined as max(x1,...,xn). To show that this estimator is unbiased, we need to calculate its expected value and show that it equals theta.

Since x1,...,xn are i.i.d. random variables from the uniform distribution on the interval [0,theta], their probability density function is f(x) = 1/theta for 0 <= x <= theta and 0 otherwise.

The probability that the sample maximum is less than or equal to a given value x is the probability that all n samples are less than or equal to x. Since the samples are i.i.d., this probability is (x/theta)^n.

Therefore, the cumulative distribution function of the sample maximum is F(x) = (x/theta)^n for 0 <= x <= theta and 0 otherwise.

The probability density function of the sample maximum is the derivative of its cumulative distribution function, which is f(x) = (n/theta)(x/theta)^(n-1) for 0 <= x <= theta and 0 otherwise.

The expected value of the sample maximum is the integral of x times its probability density function from 0 to theta, which is

E[max(x1,...,xn)] = integral from 0 to theta of x*(n/theta)(x/theta)^(n-1) dx

= (n/theta) integral from 0 to theta of x^n/theta^n dx

= (n/theta) * [x^(n+1)/(n+1)theta^n)]_0^theta

= (n/(n+1))theta
Therefore, the sample maximum is an unbiased estimator of theta.

To find an unbiased estimator of theta, given that x1, ..., xn are i.i.d. random variables from the uniform distribution on the interval [0, theta], follow these steps:

1. Determine the sample maximum: Since the data is from a uniform distribution, the sample maximum (M) can be used as a starting point. M = max(x1, ..., xn).

2. Calculate the expected value of M: The expected value of the sample maximum, E(M), is given by the formula E(M) = n * theta / (n + 1).

3. Find an unbiased estimator: To find an unbiased estimator of theta, we need to adjust the expected value of M so that it equals theta. We can do this by solving for theta in the equation E(M) = theta:

theta = (n + 1) * E(M) / n

4. Replace E(M) with the sample maximum M: Since we are using the sample maximum as our estimator, we can replace E(M) with M in the equation:

theta_hat = (n + 1) * M / n

The unbiased estimator of theta is theta h at = (n + 1) * M / n, where M is the sample maximum and n is the number of i.d. random variables.

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The simulation that could be used to fairly represent the situation is A. Use a computer to randomly generate 4 numbers from 1 to 3. Each time 1 appears, it represents a strawberry taffy.

The probability of each taffy being strawberry is 0.25, so the probability of all 3 taffies being strawberry is:

0.25 * 0.25 * 0.25 = 0.015625 or approximately 1.56%

Therefore, the likelihood of all taffies being strawberry is very low.

The simulation that could be used to fairly represent the situation is to use a computer to randomly generate 4 numbers from 1 to 3. Each time 1 appears, it represents a strawberry taffy. This simulates the probability of each taffy being strawberry being 0.25 or 25%.

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The percentage of middle schoolers is given as 45.1%

A percentage is a fractional expression, written as parts of every one-hundred, represented by the sign "%".

For instance, assume there are 20 red balls in a bag that comprises of 100 balls – providing us an opportunity to ascertain that the portion of red balls available in the aforementioned sack is twenty percent.

If you wish to calculate a percentage of numbers, then you must first divide the part by the entirety and multiply the result with one hundred. As an example, if we want to calculate what amount of hundred is constituted by twenty, then we simply divide twenty by hundred which gives us 0.2, and then further imply multiplication of it by 100 which equals twenty percent.

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The determinant of the linear transformation t(f)=2f 3f' from p2 to p2 is 36.

To find the determinant of the linear transformation t(f)=2f 3f' from p2 to p2, we first need to represent the transformation as a matrix.

Let's start by choosing a basis for p2, say {1,x,x²}. Then, the linear transformation t can be represented by the matrix

[2 0 0]
[0 3 0]
[0 0 6]

To find the determinant of this matrix (and hence the determinant of the linear transformation), we can use the formula for the determinant of a 3x3 matrix:

det(A) = a11(a22a33 - a23a32) - a12(a21a33 - a23a31) + a13(a21a32 - a22a31)

Plugging in the entries of our matrix, we get:

det(t) = 2(3×6 - 0×0) - 0(2×6 - 0×0) + 0(2×0 - 3×0)
      = 36

Therefore, the determinant of the linear transformation t(f)=2f 3f' from p2 to p2 is 36.

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Sum of the series [infinity] ln n² / (8 + 7n²)² is π² / 6.

Series [infinity] 1/n² convergent to π² / 6.

We can use the limit comparison test.

First, note that both ln n² and (8 + 7n²)² are positive for all n ≥ 1.

Let a_n = ln n² / (8 + 7n²)².

Then, consider the series b_n = 1/n².

We know that b_n is a convergent p-series with p = 2.

Next, we take the limit of the ratio of a_n and b_n as n approaches infinity:

lim (n→∞) a_n / b_n = lim (n→∞) (ln n² / (8 + 7n²)²) / (1/n²)

Using L'Hôpital's rule twice, we can simplify this limit as follows:

= lim (n→∞) [(2/n) / (-28n / (8 + 7n²))]

= lim (n→∞) -14n / (8 + 7n²)

Since the numerator and denominator both approach infinity as n approaches infinity, we can apply L'Hôpital's rule again:

= lim (n→∞) -14 / (14n)

= 0

Since the limit is finite and positive, we can conclude that the series [infinity] ln n² / (8 + 7n²)² converges by the limit comparison test.

To find its sum, we can use a known result that the series [infinity] 1/n² converges to π² / 6.

Therefore, the sum of the series [infinity] ln n² / (8 + 7n²)² is π² / 6.

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(1) To show that X has a subsequence X' such that either X' is in A or X' is in B, we can use the fact that A and B are subsets of the metric space (M,d) to construct two subsequences, one consisting of terms from A and the other consisting of terms from B.

Let X_A be the subsequence of X that consists of all terms in A, and let X_B be the subsequence of X that consists of all terms in B. If either of these subsequences is infinite, then we are done. Otherwise, both A and B are finite sets, and we can construct a subsequence X' by interleaving the terms from X_A and X_B in any way we choose.

For example, suppose A = {a1, a2, a3} and B = {b1, b2}. Then X_A = (a1, a2, a3) and X_B = (b1, b2), and we can construct the subsequence X' = (a1, b1, a2, b2, a3). This subsequence has terms from both A and B, but we can easily extract a sub-subsequence consisting only of terms from A or only of terms from B if we wish.

(2) To show that the union of two sequentially compact sets in a metric space (M,d) is sequentially compact, we need to show that every sequence in the union has a convergent subsequence. Let A and B be two sequentially compact subsets of M, and let X be a sequence in A ∪ B. By (1), X has a subsequence X' that is either in A or in B.

If X' is in A, then it has a convergent subsequence by the sequential compactness of A. This subsequence is also a subsequence of X and therefore converges in A ∪ B. If X' is in B, then it has a convergent subsequence by the sequential compactness of B, and we can again argue that this subsequence converges in A ∪ B.

Therefore, every sequence in A ∪ B has a convergent subsequence, and so A ∪ B is sequentially compact.
(1) Since X = (xk) is a sequence in A ∪ B, each term xk is either in A or in B. Divide the terms of X into two subsequences: X_A consisting of terms in A, and X_B consisting of terms in B. At least one of these subsequences must be infinite (since a finite subsequence cannot exhaust the entire sequence X).

Without loss of generality, assume X_A is infinite. Then X_A is a subsequence of X consisting only of terms in A. Let X' = X_A. Then X' is a subsequence of X such that X' is in A. Similarly, if X_B were infinite, we could construct a subsequence X' in B.

(2) To show that the union of two sequentially compact sets in a metric space (M, d) is sequentially compact, we need to show that any sequence in the union has a convergent subsequence.

Let A and B be two sequentially compact sets in (M, d), and let X = (xk) be a sequence in A ∪ B. By part (1), we know that X has a subsequence X' such that either X' is in A or X' is in B. Without loss of generality, assume X' is in A.

Since A is sequentially compact, X' has a convergent subsequence X'' in A. Thus, X'' is a convergent subsequence of X in A ∪ B. Similarly, if X' were in B, we would have a convergent subsequence in B. Therefore, the union A ∪ B is sequentially compact.

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The CLI option on the MODEL statement of an MLR analysis in PROC GLM produces confidence intervals for the mean response at all predictor combinations in the dataset. b

The CLI option stands for "Confidence Level of Intervals," and it specifies the level of confidence for the confidence intervals produced.

By default, the CLI option is set to 0.95, which means that the confidence intervals produced will have a 95% level of confidence.

These confidence intervals provide a range of values within which the true mean response at a particular combination of predictor values is expected to fall with a specified level of confidence.

They can be useful for assessing the uncertainty associated with the estimated mean response at different combinations of predictor values and for making inferences about the relationships between predictors and the response variable.

The CLI option, which stands for "Confidence Level of Intervals," defines the degree of confidence in the confidence intervals that are generated.

The CLI option's default value of 0.95 designates a 95% degree of confidence for the confidence intervals that are generated.

With a given degree of confidence, these confidence intervals show the range of values within which the real mean response for a specific set of predictor values is anticipated to fall.

They can be helpful for determining the degree of uncertainty surrounding the predicted mean response for various combinations of predictor values and for drawing conclusions on the connections between predictors and the response variable.

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The closer the value of r to O the greater the variation around the line of best fit. Different... Are there guidelines to interpreting Pearson's correlation coefficient? Yes, the following guidelines have been proposed: ...

Answer:

$0.70

Step-by-step explanation:

Let the cost of an orange be represented by o and the cost of a mango be represented by m.

According to the problem, we can set up the following system of equations:

2o + 5m = 4.5

4o + 3m = 4.1

We can solve for o and m using elimination or substitution. Here's one way to solve it using elimination:

Multiply the first equation by 4 and the second equation by -2 to eliminate o:

8o + 20m = 18

-8o - 6m = -8.2

14m = 9.8

m = 0.7

Substitute m = 0.7 into one of the equations to solve for o:

2o + 5(0.7) = 4.5

2o + 3.5 = 4.5

2o = 1

o = 0.5

Therefore, the cost of an orange is $0.50 and the cost of a mango is $0.70.

Hope this helps!

An expression to find the distance between the two jellyfish is B - A.

The distance is 2.0 meters.

In Mathematics and Geometry, a number line simply refers to a type of graph with a graduated straight line which comprises both positive and negative numbers that are placed at equal intervals along its length.

This ultimately implies that, a number line primarily increases in numerical value towards the right from zero (0) and decreases in numerical value towards the left from zero (0).

Therefore, the required expression for the distance between the two jellyfish is given by;

Distance = B - A

Distance = 0.5 - (-1.5)

Distance = 0.5 + 1.5

Distance = 2.0 m.

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The series [infinity] ∑k=0 k 10 k converges since the terms of the series approach zero as k increases, and the series satisfies the ratio test.

The ratio of successive terms is 10, which is less than 1, indicating that the series converges.

To show this using the digits of π, we can express the series as [infinity] ∑k=0 c k 10 k , where c k represents the kth digit of π. Since the digits of π are bounded and do not increase indefinitely, the terms of the series also approach zero.

Additionally, the ratio of successive terms can be expressed as c k+1 / c k 10, which is less than 1 for all k, indicating convergence. Therefore, the series [infinity] ∑k=0 k 10 k converges, both in general and when expressed using the digits of π.

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Answer:

BC/CD = DE/EF

The slope of this line is 2/3. From B, go up two units to C, then right three units to D.

The probability that Lin gets another turn is 1/60

Here, Lin will get a turn if both the cube and card have the same number.

A cube is numbered as 1, 2, 3, 4, 5, and 6

And the deck of 10 cards numbered 1 through 10

We can observe that there are only 6 (1 to 6) numbers which can follow above condition.

So, the chances of getting 6 in dice equals to 1/6

and the chances of getting 6 in card equals 1/10

Thus, the chance of getting both at once would be,

1/6 × 1/10

= 1/60

Therefore, the required probability is 1/60

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The maximum likelihood estimate of θ is 0.8.

The likelihood function for the sample is:

L(θ) = fY(2.3;θ) x fY(1.9;θ)  x fY(4.6;θ)

      = (2.3 e^(-2.3/theta) / (6 θ theta)) (1.9e^(-1.9/theta) / (6 theta)) (4.6 e^(-4.6/theta) / (6 theta))

Taking the natural logarithm of both sides, we get:

ln(L(θ)) = ln(2.3) - (2.3/θ) - 3ln(θ) - ln(6) + ln(1.9) - (1.9/θ) - 3ln(θ) - ln(6) + ln(4.6) - (4.6/θ) - 3ln(θ) - ln(6) = 3ln(2.3) - (2.3/θ) - 3ln(θ) - ln(6) + 3ln(1.9) - (1.9/θ) - 3ln(θ) - ln(6) + 3ln(4.6) - (4.6/θ) - 3ln(θ) - ln(6)

To find the maximum likelihood estimate of theta, we need to maximize this expression with respect to theta. We can do this by taking the derivative of the expression with respect to theta, setting it to zero, and solving for theta:

d/dθ ln(L(θ)) = (2.3/θ) - (3/θ) + (1.9/θ) - (3/θ) + (4.6/θ) - (3/θ) = 0 Simplifying this expression, we get:

= (2.3/θ) + (1.9/θ) + (4.6/θ)

= (9/theta)

Multiplying both sides by θ, we get:

2.3 + 1.9 + 4.6 = 9θ

Solving for θ , we get:

θ= (2.3 + 1.9 + 4.6) / 9 = 0.8

Therefore, the maximum likelihood estimate of θ is 0.8.

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