Consider the simple linear regression model yi=β0+β1xi+ϵi, where ϵi's are independent n(0,σ2) random variables. therefore, yi is a normal random variable with mean β0+β1xi and variance σ2. moreover, yi's are independent. as usual, we have the observed data pairs (x1,y1), (x2,y2), ⋯⋯, (xn, yn) from which we would like to estimate β0 and β1. in this chapter, we found the following estimators β1^=sxysxx,β0^=y−β1^x¯¯¯. where sxx=∑i=1n(xi−x¯¯¯)2,sxy=∑i=1n(xi−x¯ ¯¯)(yi−y). show that β1^ is a normal random variable. show that β1^ is an unbiased estimator of β1, i. e., e[β1^]=β1. show that var(β1^)=σ2sxx.

See proof below.

Step-by-step explanation:

If we assume the following linear model:

And if we have n sets of paired observations the model can be written like this:

And using the least squares procedure gives to us the following least squares estimates for and for  :

Where:

Then is a random variable and the estimated value is . We can express this estimator like this:

Where and if we see careful we notice that and

So then when we find the expected value we got:

And as we can see is an unbiased estimator for

In order to find the variance for the estimator we have this:

And we can assume that since the observations are assumed independent, then we have this:

And if we simplify we got:

And with this we complete the proof required.

answer:

(-8,7)

explanation:

we know that  

f

(

−

3

)

=

7

therefore, if we wish to get the output as 7, in the function, we need to have our input to be -3.

looking at the function  

f

(

x

+

5

)

, what should the value of x be, such that the input in the function becomes 7?

we simply equate the terms,

x

+

5

=

−

3

x

=

−

8

thus, the point on the new function would be  

(

−

8

,

7

)