Answer:
D
Step-by-step explanation:
The factor theorem states if (x - h) is a factor of a polynomial p(x) then
p(h) = 0
However, p(h) ≠ 0 then the value obtained is the remainder.
Thus
p(3) = - 2
When p(x) is divided by (x - 3) then the remainder is - 2
Answer:
0.8
Step-by-step explanation:
Replace the c for 4, since you said that c=4
0.2(4) is what you'll get
Multiply 0.2•4 = 0.8
Answer:
you should take the numeric value _16 to right side.
56392 rounded to the nearest ten thousand is 60,000
Answer:
![l'(\theta) = \frac{1}{\sigma^2} \sum_{i=1}^n (X_i -\theta)](https://tex.z-dn.net/?f=%20l%27%28%5Ctheta%29%20%3D%20%5Cfrac%7B1%7D%7B%5Csigma%5E2%7D%20%5Csum_%7Bi%3D1%7D%5En%20%28X_i%20-%5Ctheta%29)
And then the maximum occurs when
, and that is only satisfied if and only if:
![\hat \theta = \bar X](https://tex.z-dn.net/?f=%20%5Chat%20%5Ctheta%20%3D%20%5Cbar%20X)
Step-by-step explanation:
For this case we have a random sample
where
where
is fixed. And we want to show that the maximum likehood estimator for
.
The first step is obtain the probability distribution function for the random variable X. For this case each
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](https://tex.z-dn.net/?f=%20f%28x_i%20%7C%20%5Ctheta%2C%5Csigma%5E2%29%20%3D%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%7D%5Csigma%7D%20exp%5E%7B-%5Cfrac%7B%28x-%5Ctheta%29%5E2%7D%7B2%5Csigma%5E2%7D%7D%20%2C%20-%5Cinfty%20%5Cleq%20x%20%5Cleq%20%5Cinfty)
The likehood function is given by:
![L(\theta) = \prod_{i=1}^n f(x_i)](https://tex.z-dn.net/?f=%20L%28%5Ctheta%29%20%3D%20%5Cprod_%7Bi%3D1%7D%5En%20f%28x_i%29)
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)](https://tex.z-dn.net/?f=%20L%28%5Ctheta%29%20%3D%20%28%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%20%5Csigma%5E2%7D%7D%29%5En%20exp%20%28-%5Cfrac%7B1%7D%7B2%5Csigma%5E2%7D%20%5Csum_%7Bi%3D1%7D%5En%20%28X_i-%5Ctheta%29%5E2%29)
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](https://tex.z-dn.net/?f=%20l%28%5Ctheta%29%20%3D%20-%5Cfrac%7Bn%7D%7B2%7D%20ln%28%5Csqrt%7B2%5Cpi%5Csigma%5E2%7D%29%20-%20%5Cfrac%7B1%7D%7B2%5Csigma%5E2%7D%20%5Csum_%7Bi%3D1%7D%5En%20%28X_i%20-%5Ctheta%29%5E2)
Now if we take the derivate respect
we will see this:
![l'(\theta) = \frac{1}{\sigma^2} \sum_{i=1}^n (X_i -\theta)](https://tex.z-dn.net/?f=%20l%27%28%5Ctheta%29%20%3D%20%5Cfrac%7B1%7D%7B%5Csigma%5E2%7D%20%5Csum_%7Bi%3D1%7D%5En%20%28X_i%20-%5Ctheta%29)
And then the maximum occurs when
, and that is only satisfied if and only if:
![\hat \theta = \bar X](https://tex.z-dn.net/?f=%20%5Chat%20%5Ctheta%20%3D%20%5Cbar%20X)