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
Nina [5.8K]
4 years ago
10

Let X1, X2, ... , Xn be a random sample from N(μ, σ2), where the mean θ = μ is such that −[infinity] < θ < [infinity] and

σ2 is a known positive number. Show that the maximum likelihood estimator for θ is θ^ = X.
Mathematics
1 answer:
Sliva [168]4 years ago
6 0

Answer:

l'(\theta) = \frac{1}{\sigma^2} \sum_{i=1}^n (X_i -\theta)

And then the maximum occurs when l'(\theta) = 0, and that is only satisfied if and only if:

\hat \theta = \bar X

Step-by-step explanation:

For this case we have a random sample X_1 ,X_2,...,X_n where X_i \sim N(\mu=\theta, \sigma) where \sigma is fixed. And we want to show that the maximum likehood estimator for \theta = \bar X.

The first step is obtain the probability distribution function for the random variable X. For this case each X_i , i=1,...n have the following density function:

f(x_i | \theta,\sigma^2) = \frac{1}{\sqrt{2\pi}\sigma} exp^{-\frac{(x-\theta)^2}{2\sigma^2}} , -\infty \leq x \leq \infty

The likehood function is given by:

L(\theta) = \prod_{i=1}^n f(x_i)

Assuming independence between the random sample, and replacing the density function we have this:

L(\theta) = (\frac{1}{\sqrt{2\pi \sigma^2}})^n exp (-\frac{1}{2\sigma^2} \sum_{i=1}^n (X_i-\theta)^2)

Taking the natural log on btoh sides we got:

l(\theta) = -\frac{n}{2} ln(\sqrt{2\pi\sigma^2}) - \frac{1}{2\sigma^2} \sum_{i=1}^n (X_i -\theta)^2

Now if we take the derivate respect \theta we will see this:

l'(\theta) = \frac{1}{\sigma^2} \sum_{i=1}^n (X_i -\theta)

And then the maximum occurs when l'(\theta) = 0, and that is only satisfied if and only if:

\hat \theta = \bar X

You might be interested in
Evaluate z^2−3 z+4 , when z=−4
Maru [420]

Answer:

8

Step-by-step explanation:

=z²-3z+4 when z is 4

=4²-3(4)+4

=16-12+4

=8

6 0
3 years ago
Read 2 more answers
a wooden block measures 2 inches by 2 inches by 3 inches. A wedge is cut off from each corner of the block by slicing at points
Alla [95]

Answer:

Total number of edges for resulting solid = 12

Step-by-step explanation:

A wooden block measures 2 inches by 2 inches by 3 inches. A wedge is cut off from each corner of the block by slicing at points that are 1 inch from each corner.

We have to find the number of edges in the resulting solid after we cut the edges as told above.

There are two faces along the height which will become sharpened after we take out the cuts.

So the resulting shape would be like that of a cylinder with its both ends pointed.

So total number of edges = 4 + 4 + 4 = 12

5 0
3 years ago
Thomas invested $1498.00 at 6% simple interest per annum calculate How long it will be before his investment earns $449.40
valentinak56 [21]

Answer:

t = 6 years

Step-by-step explanation:

Use the simple interest formula:  i = prt, where p is the principal, r is the interest rate as a decimal fraction, and is the elapsed time, in years.

Here we want to know how long it will take for the interest alone to reach $449.40.  We first solve i = prt for t, obtaining t = i/(pr).

Here, the length of time is t = ($449.40) / (0.06*$1498.00).  This works out to

t = 5.9947, or approximately 6 years.

t = 6 years

8 0
3 years ago
Emelina wrote the equation of a line in point-slope form as y-4 = 3(x-2) What is Emelina's equation in slope-intercept form? *
Sergio [31]

Answer:

y = 3x - 2

Step-by-step explanation:

5 0
3 years ago
Please Help!!! I give a medal to anyone to tries!!!
poizon [28]

Well, the way I see it is that both Mr. Romano, and Ms. Guerra are correct but Professor McCoy is incorrect because he said (x+2) when it should be (x-2).


The factor theorem states that:
If f(a)=0, then (x-a) is a factor


The remainder theorem states that:
If (x-a) is a factor of f(x), then f(x) / (x-a)  = 0


So if 2 is indeed a zero of f(x), then a factor must be (x-2) according to the fist which supports Ms. Guerra and also if (x-2) is indeed a factor as Ms. Guerra says then we know that f(x) / (x-a) = 0 which supports Mr. Romano

Professor McCoy is wrong because he used (x + 2) when it should be (x-2). I know this because according to the factor theorem if f(a)=0, then (x-a) is a factor. And the remainder theorem says if (x-a) is a factor of f(x), then f(x)/x-a =0.

5 0
3 years ago
Other questions:
  • Anyone know whats 9 + 10 ??
    7·1 answer
  • I REALLY NEED HELP !!! HOW DO YOU DO THIS?
    5·1 answer
  • Absolute value of 5=|3×-2|
    11·2 answers
  • Lori recently quit her job and decided to start her own company, which will sell makeup to small beauty salons. What should she
    9·1 answer
  • A pot of boiling water at 212 degrees Fahrenheit on the stove has less thermal energy than a hot tub full of water at 100..4 deg
    8·1 answer
  • Find the percentage change from 2004 to 2006. Round your answer to whole persentages
    8·1 answer
  • -2 • f(-6) -7 • g(-7)
    13·1 answer
  • A metal rod will be cut into pieces that are each 1/56 meters long. The rod is 7/8
    9·1 answer
  • What is the absolute maximum or minimum of f(x) = 3x^2-5x-9
    13·2 answers
  • Urgent Please Help Me!!!!!!!!!!!!
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!