Let X
1
​
,X
2
​
,… denote an iid sequence of random variables, each with expected value 75 and standard deviation 15. a) Use Chebyshev's inequality to find how many samples n we need to guarantee that the sample mean is between 74 and 76 with probability 0.99? b) If each X
i
​
has a Gaussian distribution, how many samples n would we need?

= 2

2 + 3 = 1 0 5 = 1 0

Answer: Here's my real world scenario example:

A cookie shop decided to give away cookies to its customers. Normally, when the shop gives away zero cookies for free, they get 4 customers in a day. When they gave away one cookie for free, they got 8 customers in a day. When they gave away two cookies, they got 16 customers in a day. And when they gave away three cookies, they got 32 customers in a day.

I hope this helps! Feel free to give me Brainliest if you feel this helped. Have a good day, and good luck on your assignment. :)

The gradient of the marginal log-likelihood can be represented as the posterior-expected gradient of the complete-data log-likelihood, and when EM converges, it converges at a local optimum of the MLL.

6.1. To show that the gradient of the marginal log-likelihood can be represented as the posterior-expected gradient of the complete-data log-likelihood, we will apply the chain rule to the logarithm function.

Let's consider the marginal log-likelihood, denoted as L(θ), which is the log probability of the observed data:

L(θ) = log p(x)

Using the chain rule, we can express the gradient of the marginal log-likelihood:

∇_θ L(θ) = ∇_θ log p(x)

Next, let's consider the complete-data log-likelihood, denoted as Q(θ, z), which is the log probability of both the observed data and the unobserved latent variables:

Q(θ, z) = log p(x, z)

The gradient of the complete-data log-likelihood can be expressed as:

∇_θ Q(θ, z)

Now, we want to show that the gradient of the marginal log-likelihood can be represented as the posterior-expected gradient of the complete-data log-likelihood:

∇_θ L(θ) = E_p(z|x) [∇_θ Q(θ, z)]

To prove this, we need to compute the expectation of the gradient of the complete-data log-likelihood with respect to the posterior distribution of the latent variables given the observed data.

Taking the expectation with respect to the posterior distribution, denoted as p(z|x), we have:

E_p(z|x) [∇_θ Q(θ, z)] = ∫ [∇_θ Q(θ, z)] p(z|x) dz

Now, using the property of logarithms, we know that the logarithm of a product is equal to the sum of the logarithms:

log p(x, z) = log p(x|z) + log p(z)

Applying the chain rule to the logarithm function in the complete-data log-likelihood:

∇_θ Q(θ, z) = ∇_θ [log p(x|z) + log p(z)]

= ∇_θ log p(x|z) + ∇_θ log p(z)

Now, substituting this back into the expression for the expected gradient:

E_p(z|x) [∇_θ Q(θ, z)] = ∫ [∇_θ log p(x|z) + ∇_θ log p(z)] p(z|x) dz

= ∫ ∇_θ log p(x|z) p(z|x) dz + ∫ ∇_θ log p(z) p(z|x) dz

= ∇_θ ∫ log p(x|z) p(z|x) dz + ∫ ∇_θ log p(z) p(z|x) dz

= ∇_θ ∫ p(z|x) log p(x|z) dz + ∇_θ ∫ p(z|x) log p(z) dz

= ∇_θ ∫ p(z|x) [log p(x|z) + log p(z)] dz

= ∇_θ ∫ p(z|x) log p(x, z) dz

= ∇_θ ∫ p(z|x) [log p(x, z) - log p(x)] dz

Using the definition of conditional probability, p(z|x) = p(x, z) / p(x), we have:

∇_θ ∫ p(z|x) [log p(x, z) - log p(x)] dz = ∇_θ ∫ p(z|x) log [p(x, z) / p(x)] dz

Since the integral of p(z|x) over all possible values of z equals 1, we can simplify this expression further:

∇_θ ∫ p(z|x) log [p(x, z) / p(x)] dz = ∇_θ E_p(z|x) [log [p(x, z) / p(x)]]

= ∇_θ E_p(z|x) [log p(x, z)] - ∇_θ E_p(z|x) [log p(x)]

Now, we know that the term ∇_θ E_p(z|x) [log p(x)] is zero since it does not depend on θ. Therefore, we are left with:

∇_θ L(θ) = E_p(z|x) [∇_θ Q(θ, z)]

This proves that the gradient of the marginal log-likelihood can be represented as the posterior-expected gradient of the complete-data log-likelihood.

6.2. The fact that EM converges to a local optimum of the MLL can be shown using the result from 6.1.

In the EM algorithm, the E-step involves computing the posterior distribution of the latent variables given the observed data, and the M-step involves maximizing the expected complete-data log-likelihood with respect to the model parameters.

By maximizing the expected complete-data log-likelihood, we are effectively maximizing the posterior-expected complete-data log-likelihood. From 6.1, we know that the gradient of the marginal log-likelihood is equal to the posterior-expected gradient of the complete-data log-likelihood.

Since EM iteratively updates the parameters by maximizing the expected complete-data log-likelihood, it follows that the updates are driven by the gradients of the marginal log-likelihood. As a result, EM converges to a local optimum of the marginal log-likelihood.

Therefore, when EM converges, it converges at a local optimum of the MLL.

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Answer:  Choice C)  \(\frac{8+2\sqrt{6x}}{8-3x}\)

======================================

Work Shown:

\(y = \frac{4}{4-\sqrt{6x}}\\\\y = \frac{4(4+\sqrt{6x})}{(4-\sqrt{6x})(4+\sqrt{6x})}\\\\y = \frac{4(4+\sqrt{6x})}{4^2 - (\sqrt{6x})^2}\\\\y = \frac{4(4+\sqrt{6x})}{16-6x}\\\\y = \frac{2*2(4+\sqrt{6x})}{2(8-3x)}\\\\y = \frac{2(4+\sqrt{6x})}{8-3x}\\\\y = \frac{8+2\sqrt{6x}}{8-3x}\\\\\)

This shows why choice C is the answer.

----------------

Notes:

A function is considered continuous at a point if its limit exists at that point and is equal to the function's value at that point.

a function is continuous at a point if it has no gaps, jumps, or holes in its graph at that point. Since the integral of f(x) from 2 to 10 is zero, and f(x) is continuous and non-negative on this interval, it follows that f(x) must be identically zero on [2, 10].

Therefore, f(6) = 0

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Answer:

41 °

Step-by-step explanation:

A = x     b = y     c = z  = 41 °

Answer:

Hope the picture will help you

Answer:

here is my answer. check it out

In geometry, the segment addition postulate states that given 2 points A and C, a third point B lies on the line segment AC if and only if the distances between the points satisfy the equation AB + BC = AC.

Answer: 201.06

Step-by-step explanation: A=πr2=π·82≈201.06193in²

━━━━━━━☆☆━━━━━━━

▹ Answer

201.06 in

▹ Step-by-Step Explanation

A = πr²

A = π(8)²

A ≈ 201.06 in

Hope this helps!

CloutAnswers ❁

━━━━━━━☆☆━━━━━━━

Answer:

x=82.7

Step-by-step explanation:

3x+6-6=248

3x=248

x=82.7

Answer:  

Step-by-step explanation:

3x   +  6  −  6  =  248

3x  +  6  +  −  6  =  248

(  3x  )   +  (   6   +  −  6  )  =  248      (ℂ  )

3x  =  248

3x  /  3  = 248  /  3

x   =   248  /  3

x  =  82.6

Option:-

\( \: \)

Given:-

\( \: \)

Solution:-

\( \: \)

\( \: \)

\( \: \)

━━━━━━━━━━━━━━━━━━━━━━━

hope it helps ⸙

Answer:

a) 3x + 2

Step-by-step explanation:

Now we have to,

→ Find the quotient of the following.

Given problem,

→ (9x + 6) ÷ 3

Let's find the required quotient,

→ (9x + 6) ÷ 3

→ (9x + 6)/3

→ (9x/3) + (6/3)

→ (3x) + (2)

→ 3x + 2

Hence, the option (a) is correct.

A data set's attributes are enumerated through descriptive statistics. You can use inferential statistics to test a hypothesis or determine whether your data can be applied to a larger population.

Descriptive statistics concentrate on describing the features of a dataset that are readily evident (a population or sample). In contrast, inferential statistics concentrate on drawing conclusions or generalisations from a sample of data in a larger dataset.

The information from a research sample is described and condensed using descriptive statistics. We can draw conclusions about the larger population from which we drew our sample using inferential statistics.

The area of statistics known as descriptive statistics is focused on providing a description of the population being studied. A type of statistics known as inferential statistics concentrates on inferring information about the population from sample analysis and observation.

Hence we get the required answer.

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Answer:

The given differential equation is:

12y" - 12y' + 30y = 0

To find the general solution, we first need to find the auxiliary equation. We assume a solution of the form:

y(t) = e^(rt)

where r is a constant. Differentiating this equation twice with respect to t, we get:

y'(t) = re^(rt)

y''(t) = r^2 e^(rt)

Substituting these into the differential equation, we get:

12r^2 e^(rt) - 12re^(rt) + 30e^(rt) = 0

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

12r^2 - 12r + 30 = 0

Simplifying this equation by dividing both sides by 6, we get:

2r^2 - 2r + 5 = 0

This is a quadratic equation with complex roots, which are given by the formula:

r = (2 ± sqrt(-4))/4 = (1 ± i√6)/2

Therefore, the general solution to the differential equation is:

y(t) = c1 e^((1+i√6)/2)t + c2 e^((1-i√6)/2)t

where c1 and c2 are constants determined by the initial conditions or boundary conditions of the problem.

Therefore , the solution of the given problem of equation comes out to be c1*e((1/2)*t)*cos((3/2)t) + c2e((1/2)*t)*sin((3/2)*t) = y(t).

The equation y=mx+b serves as the basis for a linear regression model. B is the slope, and m is the y-intercept. The above line is commonly referred to as the "mathematical problems with two variables," even though y and y are distinct components. Bivariate linear equations only contain two variables. The application areas of linear functions do not have clear-cut solutions.

Here,

For the specified differential equation, the auxiliary equation is:

=> 12r² - 12r + 30 = 0

When we multiply both parts by 6, we get:

=> 2r² - 2r + 5 = 0

Given that the discriminant is negative, the solutions of this quadratic equation are complicated:

=> b² - 4ac = (-2) ² - 4(2)(5) = -36

The following are the roots:

=> r = (2 ± √(-36))/(4) = (1 ± 3i)/2

The differential equation has the following general solution:

=> Y(t) = C1E + C2E (R1T) (r2t)

where c1 and c2 are arbitrary values and r1 and r2 are the auxiliary equation's roots.

When we replace the stems, we obtain:

=>  c1*e((1/2)*t)*cos((3/2)t) + c2e((1/2)*t)*sin((3/2)*t) = y(t).

As a result, the differential equation has the following general solution:

=> c1*e((1/2)*t)*cos((3/2)t) + c2e((1/2)*t)*sin((3/2)*t) = y(t).

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A regression model can only predict values that are lower or higher than the actual value. As a result, the only way to determine the model's accuracy is through residuals.

What is regression ?

Regression is a statistical technique used in the fields of finance, investing, and other disciplines that aims to establish the nature and strength of the relationship between a single dependent variable (often represented by Y) and a number of independent variables (known as independent variables).

The most popular variation of this method is linear regression, which is also known as simple regression or ordinary least squares (OLS).

Based on a line of best fit, linear regression determines the linear relationship between two variables.

The slope of a straight line used to represent linear regression thus indicates how changing one variable affects changing another..

In a linear regression connection, the value of one variable when the value of the other is zero is represented by the y-intercept. Regression without linearity.

Hence, A regression model can only predict values that are lower or higher than the actual value.

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To calculate the discharge through the circular channel, we can use Manning's equation, which relates the flow rate (Q) to the channel properties and flow conditions. Manning's equation is given by:

Q = (1/n) * A * R^(2/3) * S^(1/2)

where:

Q is the discharge (flow rate)

n is Manning's coefficient (0.014 in this case)

A is the cross-sectional area of the channel

R is the hydraulic radius of the channel

S is the slope of the channel bed

First, let's calculate the cross-sectional area (A) of the circular channel. The diameter of the channel is given as 6m, so the radius (r) is half of that, which is 3m. Therefore, the area can be calculated as:

A = π * r^2 = π * (3m)^2 = 9π m^2

Next, let's calculate the hydraulic radius (R) of the channel. For a circular channel, the hydraulic radius is equal to half of the diameter, which is:

R = r = 3m

Now, we can calculate the slope (S) of the channel bed. The given slope is 1 in 600, which means for every 600 units of horizontal distance, there is a 1-unit change in vertical distance. Therefore, the slope can be expressed as:

S = 1/600

Finally, we can substitute these values into Manning's equation to calculate the discharge (Q):

Q = (1/0.014) * (9π m^2) * (3m)^(2/3) * (1/600)^(1/2)

Using a calculator, the discharge can be evaluated to get the final result.

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.m The correct equation to determine the amount of material used, m, to construct the fully enclosed tent is:

                         c. =2(8∙9)+2(7.5∙8)+(12∙9∙6)

To determine the amount of material used to construct the fully enclosed tent, we need to consider the surface area of the tent. The tent is fully enclosed, so we need to calculate the area of all the sides.

Option a. =(8∙9)+2(7.5∙8) calculates the area of the top and two sides of the tent. This does not include the front and back of the tent, so it is not the correct equation.

Option b. =(8∙9)+2(7.5∙8)+2(12∙9∙6) calculates the area of the top, two sides, front and back of the tent, but it also includes an extra term of 2(12∙9∙6) which is not necessary for a fully enclosed tent. This option overestimates the amount of material used.

Option c. =2(8∙9)+2(7.5∙8)+(12∙9∙6) calculates the area of the top, bottom, and all four sides of the tent. This is the correct equation to determine the amount of material used in a fully enclosed tent.

Option d. =3(8∙9)+2(12∙9∙6) overestimates the amount of material used because it includes an extra term of 3(8∙9) which is not necessary for a fully enclosed tent.

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Always check the quantity in each package to determine the appropriate number of hotdog bun packages needed for the given number of hotdog packages.
The dependent variable in this situation is the number of packages of hotdog buns needed, as it relies on the number of packages of hotdogs (the independent variable).

We want to know the dependent variable when considering the number of packages of hotdog buns needed for a certain number of packages of hotdogs.

Let me explain this relationship:
First, let's define the terms involved.

The dependent variable is the variable that depends on another variable (called the independent variable).

In this case, we are examining the relationship between the number of packages of hotdogs and the number of packages of hotdog buns needed.
The independent variable is the number of packages of hotdogs.

This is because the number of hotdog buns needed depends on the number of hotdogs you have.
The dependent variable, in this case, is the number of packages of hotdog buns needed.

This variable depends on the number of packages of hotdogs, as you need a certain amount of buns to accommodate the hotdogs.

To determine the relationship between these two variables, we would likely use a proportion or a ratio.

For example, if one package of hotdogs contains 10 hotdogs and one package of hotdog buns contains 10 buns, then the ratio is 1:1, meaning you need one package of hotdog buns for every package of hotdogs.
This relationship may vary depending on the number of hotdogs and buns in each package.

Remember to examine the proportion between the two variables to ensure you have the correct amount of buns for the hotdogs.

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Answer:

518

Step-by-step explanation:

Since you are rounding to the nearest whole number.

518.1 would need to be 518.5 or higher to be 519 so anything below that would just round up the 518

If a categorical variable that can take the values from the set {Red, Blue, Green, Yellow} is included as an independent variable in a linear regression, the number of dummy variables that are created is two

When a categorical variable that has n categories is to be included as an independent variable in a linear regression analysis, it must be converted to n - 1 dummy variables. The reason for this is that including all n categories as dummy variables would cause perfect multicollinearity in the regression analysis, making it impossible to estimate the effect of each variable.In this case, the set of categories {Red, Blue, Green, Yellow} has four categories. As a result, n - 1 = 3 dummy variables are required to represent this variable in a linear regression. This is true since each category is exclusive of the others, and we cannot assume that there is an inherent order to the categories.The dummy variable for the first category is included in the regression model by default, and the remaining n - 1 categories are represented by n - 1 dummy variables. As a result, the number of dummy variables that are required to represent the categorical variable in the regression model is n - 1.

Thus, if a categorical variable that can take the values from the set {Red, Blue, Green, Yellow} is included as an independent variable in a linear regression, the number of dummy variables that are created is two .

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Answer:

The second option

Step-by-step explanation:

The rate of change needs to stay constant.

The rate of change can be explained by looking at the "rise" (y) over the "run"(x)

The rate of change for the second option is 2.  We know this because...

Rise(2)

----------

Run(1)

Try looking at one of the points, now go up 2 and over 1, you should now be at the second point, do it again but twice, you should now be at the third point. A proportional relationship must also go through the origin (0,0)

Answer: He has earned $200 interest and the balance of his account is $1,200.

Step-by-step explanation:

Given: Principal : P = $1,000

Simple interest rate : r = 5% = 0.05

Time: t = 4 years

Simple interest = \(Prt\)

\(=1000\times0.05\times4=200\)

i.e. Simple interest = $200

Balance of account = Principal + Simple interest

=  $1,000 + $200

= $1,200

Hence, he has earned $200 interest and the balance of his account is $1,200.

To find out how long it will take for the egg to hit the ground, we need to determine the value of t when the height (h) of the egg is zero. In other words, we need to solve the equation:

-16t^2 + 30 = 0

To solve this quadratic equation, we can use the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = -16, b = 0, and c = 30. Substituting these values into the quadratic formula, we get:

t = (± √(0^2 - 4*(-16)30)) / (2(-16))

Simplifying further:

t = (± √(0 - (-1920))) / (-32)

t = (± √1920) / (-32)

t = (± √(64 * 30)) / (-32)

t = (± 8√30) / (-32)

Since time cannot be negative in this context, we can disregard the negative solution. Therefore, the time it will take for the egg to hit the ground is:

t = 8√30 / (-32)

Simplifying this further, we get:

t ≈ -0.791 seconds

The negative value doesn't make sense in this context since time cannot be negative. Therefore, we discard it. So, the egg will hit the ground approximately 0.791 seconds after being dropped.

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When we want to estimate a population parameter then we use hypothesis test.

What is hypothesis test?

In hypothesis testing, an analyst examines a statistical sample with the aim of demonstrating the plausibility of the null hypothesis.

By measuring and reviewing a representative sample of the population under study, statistical analysts test a hypothesis. All analysts use a random population sample to test the null hypothesis and the alternative hypothesis.

Main body:

As the researchers want to understand the difference between to states they need to do a hypothesis test.

Hence a hypothesis test is done.

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Answer:

0.35

Step-by-step explanation:

Step 1:

7/20 = 35/100

Step 2:

35 ÷ 100

Answer:

0.35

Hope This Helps :)

Answer:

The car is 13 years old

Step-by-step explanation:

Lets say Car is x and the Truck is y, so the trucks age = x + 3 = y

    Lets find the value of the Car or x

x + 3 = 1/2(x + 7)------- >

first we will divide x by 1/2 and 7 by 1/2, So...

x + 3 = (x / 1/2 + 7 / 1/2)------- >

x + 3 = 1/2x + 3.5------->

next Isolate variable x,

x + 1/2x = 3 + 3.5

1/2x = 6.5, So...

x = 13 and in turn, y = 13 + 3

:)

If resources are unlimited, population will increase exponentially. A logistic growth happens as those resources are less readily available.

Exponential Growth:

Logistic Growth:

Thus, there are several key distinctions between populations growing logistically and those growing exponentially.

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Answer:

9(g+3h) = 36

Step-by-step explanation: