1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
love history [14]
3 years ago
5

Let X1 and X2 be independent random variables with mean μand variance σ².

Mathematics
1 answer:
My name is Ann [436]3 years ago
7 0

Answer:

a) E(\hat \theta_1) =\frac{1}{2} [E(X_1) +E(X_2)]= \frac{1}{2} [\mu + \mu] = \mu

So then we conclude that \hat \theta_1 is an unbiased estimator of \mu

E(\hat \theta_2) =\frac{1}{4} [E(X_1) +3E(X_2)]= \frac{1}{4} [\mu + 3\mu] = \mu

So then we conclude that \hat \theta_2 is an unbiased estimator of \mu

b) Var(\hat \theta_1) =\frac{1}{4} [\sigma^2 + \sigma^2 ] =\frac{\sigma^2}{2}

Var(\hat \theta_2) =\frac{1}{16} [\sigma^2 + 9\sigma^2 ] =\frac{5\sigma^2}{8}

Step-by-step explanation:

For this case we know that we have two random variables:

X_1 , X_2 both with mean \mu = \mu and variance \sigma^2

And we define the following estimators:

\hat \theta_1 = \frac{X_1 + X_2}{2}

\hat \theta_2 = \frac{X_1 + 3X_2}{4}

Part a

In order to see if both estimators are unbiased we need to proof if the expected value of the estimators are equal to the real value of the parameter:

E(\hat \theta_i) = \mu , i = 1,2

So let's find the expected values for each estimator:

E(\hat \theta_1) = E(\frac{X_1 +X_2}{2})

Using properties of expected value we have this:

E(\hat \theta_1) =\frac{1}{2} [E(X_1) +E(X_2)]= \frac{1}{2} [\mu + \mu] = \mu

So then we conclude that \hat \theta_1 is an unbiased estimator of \mu

For the second estimator we have:

E(\hat \theta_2) = E(\frac{X_1 + 3X_2}{4})

Using properties of expected value we have this:

E(\hat \theta_2) =\frac{1}{4} [E(X_1) +3E(X_2)]= \frac{1}{4} [\mu + 3\mu] = \mu

So then we conclude that \hat \theta_2 is an unbiased estimator of \mu

Part b

For the variance we need to remember this property: If a is a constant and X a random variable then:

Var(aX) = a^2 Var(X)

For the first estimator we have:

Var(\hat \theta_1) = Var(\frac{X_1 +X_2}{2})

Var(\hat \theta_1) =\frac{1}{4} Var(X_1 +X_2)=\frac{1}{4} [Var(X_1) + Var(X_2) + 2 Cov (X_1 , X_2)]

Since both random variables are independent we know that Cov(X_1, X_2 ) = 0 so then we have:

Var(\hat \theta_1) =\frac{1}{4} [\sigma^2 + \sigma^2 ] =\frac{\sigma^2}{2}

For the second estimator we have:

Var(\hat \theta_2) = Var(\frac{X_1 +3X_2}{4})

Var(\hat \theta_2) =\frac{1}{16} Var(X_1 +3X_2)=\frac{1}{4} [Var(X_1) + Var(3X_2) + 2 Cov (X_1 , 3X_2)]

Since both random variables are independent we know that Cov(X_1, X_2 ) = 0 so then we have:

Var(\hat \theta_2) =\frac{1}{16} [\sigma^2 + 9\sigma^2 ] =\frac{5\sigma^2}{8}

You might be interested in
I would really appreciate if I could get some help on this math problem.
beks73 [17]

Answer:

The answer is 16,060.96 round to the nearest dollar is 16,061.

4 0
3 years ago
Read 2 more answers
The sum of two consecutive integers is -37 find the integers
dexar [7]
The answer could possibly be -48+11=-37
6 0
3 years ago
Find the height of the tree
Kipish [7]

Answer:

im guessing 15

Step-by-step explanation:

6 0
2 years ago
Jamie fills an empty horse trough with water she measures and record the water level in centimeters every few minutes in minutes
natali 33 [55]

what does the table look like?

5 0
3 years ago
Jose brought 9 pounds of rice for $4. How many pounds of rice did he get per dollar?​
NikAS [45]

Answer:

1.5 pound ever dollar

Step-by-step explanation:

just calculate

7 0
3 years ago
Read 2 more answers
Other questions:
  • Complete the patterns
    7·1 answer
  • Juan is hiking up a mountain starting at an elevation of 800 800 feet. Every hour, he is 2000 2000 feet higher in elevation.
    15·2 answers
  • Suppose Jamal swam the 100-meter butterfly race in 57.79 seconds. The next time he swam the same race, he swam it in 54.83 secon
    10·1 answer
  • 3. Evaluate c^3 − 2(c + 3d) for c = −2 and d = 1.<br> A. −10<br> B. 6<br> C. −18<br> D. 10
    7·1 answer
  • Write as an expression: Twice the square of the product of x and y
    7·1 answer
  • The perimeter of a square is 28ft. can you conclude that area of the square is 49 square feet? Explain.
    9·1 answer
  • Integrate. Choose the best approach and the answer. LaTeX: \int\sin^3x\:dx ∫ sin 3 ⁡ x d x a. use LaTeX: \sin^2x=\frac{1}{2}\lef
    15·1 answer
  • The length of a rectangular prism is three times its width. The height is two times the length. If
    7·1 answer
  • What the answer for 3/5 + 1/4
    8·2 answers
  • Graph the function using the techniques of shifting, compressing, stretching, and/or reflecting f(x)=(x+3)^3-2
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!