Math / Data & Statistics

Question6 5/5 points Let $X_{1}, X_{2}, \ldots, X_{n}$ be independent and identically distributed random variables from a uniform distribution on the interval $(0, a)$ . The value of a is unknown. Derive the method of moments estimate of the parameter a. $2\left(X_{1}+\ldots+X_{n}\right) / n$ 2* median( $\mathrm{X} 1, \ldots, \mathrm{Xn}$ ) $\operatorname{Max}\left(X_{1}, X_{2}, \ldots, X_{n}\right)$

Studdy Solution

STEP 1

What is this asking? Find a way to estimate the unknown maximum value $a$ from a uniform distribution using the method of moments. Watch out! Don't confuse the method of moments with other estimation methods like maximum likelihood!

STEP 2

1. Understand the uniform distribution
2. Calculate the expected value
3. Set up the method of moments equation
4. Solve for the estimate of $a$

STEP 3

Alright, let's get into it!
We're dealing with a uniform distribution on the interval $(0, a)$ .
This means every value between $0$ and $a$ is equally likely.
It's like picking a random number from a perfectly shuffled deck of cards, where each card represents a number between $0$ and $a$ .

STEP 4

Now, let's talk about the expected value of a uniform distribution on $(0, a)$ .
The expected value, or mean, is the average value you'd expect if you could sample an infinite number of times.
For a uniform distribution, the expected value is given by:
$E(X) = \frac{a}{2}$
This formula tells us that the average value is right in the middle of the interval from $0$ to $a$ .

STEP 5

The method of moments is a way to estimate parameters by equating sample moments to theoretical moments.
Here, we equate the sample mean to the expected value of the distribution.
So, if we have a sample mean $\bar{X}$ , we set:
$\bar{X} = \frac{a}{2}$
This equation is our bridge from the sample data to the unknown parameter $a$ .

STEP 6

Let's solve for $a$ by rearranging the equation from the previous step.
Multiply both sides by 2 to isolate $a$ :
$a = 2 \cdot \bar{X}$
This gives us the method of moments estimate for $a$ .
It's as simple as doubling the sample mean!

STEP 7

The method of moments estimate for the parameter $a$ is given by:
$a = 2 \cdot \bar{X}$

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Studdy Solution

STEP 1

What is this asking? Find a way to estimate the unknown maximum value $a$ from a uniform distribution using the method of moments. Watch out! Don't confuse the method of moments with other estimation methods like maximum likelihood!

STEP 2

1. Understand the uniform distribution
2. Calculate the expected value
3. Set up the method of moments equation
4. Solve for the estimate of $a$

STEP 3

Alright, let's get into it!
We're dealing with a uniform distribution on the interval $(0, a)$ .
This means every value between $0$ and $a$ is equally likely.
It's like picking a random number from a perfectly shuffled deck of cards, where each card represents a number between $0$ and $a$ .

STEP 4

STEP 5

STEP 6

Let's solve for $a$ by rearranging the equation from the previous step.
Multiply both sides by 2 to isolate $a$ :
$a = 2 \cdot \bar{X}$
This gives us the method of moments estimate for $a$ .
It's as simple as doubling the sample mean!

STEP 7

The method of moments estimate for the parameter $a$ is given by:
$a = 2 \cdot \bar{X}$

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Studdy Solution

STEP 1

What is this asking? Find a way to estimate the unknown maximum value aaa from a uniform distribution using the method of moments. Watch out! Don't confuse the method of moments with other estimation methods like maximum likelihood!

STEP 2

1. Understand the uniform distribution2. Calculate the expected value3. Set up the method of moments equation4. Solve for the estimate of aaa

STEP 3

Alright, let's get into it!We're dealing with a **uniform distribution** on the interval (0,a)(0, a)(0,a).This means every value between 000 and aaa is equally likely.It's like picking a random number from a perfectly shuffled deck of cards, where each card represents a number between 000 and aaa.

STEP 4

STEP 5

STEP 6

Let's solve for aaa by rearranging the equation from the previous step.Multiply both sides by 2 to isolate aaa:a=2⋅Xˉa = 2 \cdot \bar{X}a=2⋅XˉThis gives us the **method of moments estimate** for aaa.It's as simple as doubling the sample mean!

STEP 7

The method of moments estimate for the parameter aaa is given by:a=2⋅Xˉa = 2 \cdot \bar{X}a=2⋅Xˉ

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? Find a way to estimate the unknown maximum value $a$ from a uniform distribution using the method of moments. Watch out! Don't confuse the method of moments with other estimation methods like maximum likelihood!

1. Understand the uniform distribution
2. Calculate the expected value
3. Set up the method of moments equation
4. Solve for the estimate of $a$

Alright, let's get into it!
We're dealing with a uniform distribution on the interval $(0, a)$ .
This means every value between $0$ and $a$ is equally likely.
It's like picking a random number from a perfectly shuffled deck of cards, where each card represents a number between $0$ and $a$ .

Let's solve for $a$ by rearranging the equation from the previous step.
Multiply both sides by 2 to isolate $a$ :
$a = 2 \cdot \bar{X}$
This gives us the method of moments estimate for $a$ .
It's as simple as doubling the sample mean!

The method of moments estimate for the parameter $a$ is given by:
$a = 2 \cdot \bar{X}$