All Questions
17
questions
1
vote
0
answers
27
views
Mean term in simple linear regression
I am trying to derive the expression for the $E(y_i \epsilon_i)$ in simple linear regression. I substitute using $Cov(X,Y) = E(XY) - E(X)E(Y)$, so $E(y_i \epsilon_i) = Cov(y_i , \epsilon_i)- (E(y_i)...
- 111
5
votes
1
answer
500
views
Zero Covariance vs Independence of Slope and Intercept Estimators in Linear Models with Least Squares
$\newcommand{\Cov}{\operatorname{Cov}}$Problem Statement: Under the assumptions of Exercise 11.16, find
$\Cov\big(\hat\beta_0,\hat\beta_1\big).$ Use this answer to show that
$\hat\beta_0$ and $\hat\...
- 5,932
0
votes
0
answers
76
views
Zero conditional expectation implying zero covariance?
Proof: E[X|Y]=0 implies COV[X,Y]=0
I was thinking maybe the law of total covariance or tower rule but couldn't come up with the proof
- 11
0
votes
0
answers
59
views
Finding Cov(X,Y) given pdf(X,Y)
Could you please guide me in the right direction for the problem below?
I don't know if I am right, but here is a headstart
Cov(X,Y) = E(XY) - E(X)E(Y)
= $\int_{0}^{\infty}\int_{0}^{\infty} ...
- 101
1
vote
1
answer
48
views
Proving covariance
is this the right method of proving:
\begin{align} Cov(a_0+a_1R_1+a_2R_2, Q_1)=a_1Cov(R_1,Q_1)+a_2(R_2,Q_1) \end{align}
By
\begin{align} =Cov(a_0+a_1R_1+a_2R_2, Q_1) \end{align}
\begin{align} =Cov(...
- 235
0
votes
2
answers
97
views
Proving covariance using (Pearson) correlation coefficient
Could anyone guide or show me how to prove the covariance
$$2\cdot Cov\left(\frac{X}{\sigma_X},\frac{Y}{\sigma_Y}\right)=\frac{2}{\sigma_X\sigma_Y}Cov(X,Y)$$
with
$$Cov(X,Y) = E(X − E(X))(Y − E(Y ))...
- 235
2
votes
0
answers
190
views
Understanding Autocovariance under Gaussian Random Process
I'm recently been trying to understand time series better,and would really appreciate if someone can show me this:
I found this online under a lecture slide by J. McJames of Portland Univ., and I ...
9
votes
2
answers
11k
views
Covariance of two sample means
I am trying to derive the covariance of two sample means and get confused at one point. Given is a sample of size $n$ with paired dependent observations $x_i$ and $y_i$ as realizations of RVs $X$ and $...
- 6,624
2
votes
1
answer
193
views
Finding $Cov(2X+7, X^2 +3X - 12)$
So I have this pdf, $f(x)=3x^2$ for $x\in (0,1)$ and I need to find $Cov(2X+7, X^2+3X-12)$. My main concern about how I answer this is, what is the joint pdf for these two distributions? I guess it'...
- 489
5
votes
1
answer
114
views
Question regarding covariance
I'm trying to prove a theorem, where it is given that each $X_i$ is independent and identically distributed with mean $\mu$ and variance $\sigma^2$. Within this theorem, I have multiple sub-results to ...
- 381
4
votes
1
answer
162
views
Sample covariance mean-corrected vector proof
Prove that $$(n-1)S = X^TX -{1\over{n}}(X^T\vec1)(\vec1^TX) = X^TX-n\vec{\bar x}\vec{\bar x}^T$$
My attempt so far goes like this
$$S = {1\over{n-1}}X_m^TX_m$$
Edit:
Where $X_m$ is the mean ...
- 699
2
votes
1
answer
56
views
Covariance help
Consider an experiment in which a fair coin is tossed 10 times in a row (the tosses are independent of each other). Let X denote the number of heads observed and let Y=X^2. Find the covariance between ...
- 41
8
votes
2
answers
11k
views
Covariance term in simple linear regression
I am trying to derive the expression for the variance of $\hat{\beta_0}$ in simple linear regression. I substitute $\bar{y} - \hat{\beta_1} \bar{x}$ for $\hat \beta_0$, but in the intermediate steps ...
- 119
2
votes
1
answer
3k
views
Covariance with conditional expectation
Suppose $X$ and $Y$ are random variables, $E(Y^2) < \infty$ and
$\varepsilon = Y - E(Y|X)$ so that $Y = E(Y|X) + \varepsilon$.
Given that $E(\varepsilon | X) = E(\varepsilon) = 0$, show that
$Cov(\...
- 165
6
votes
1
answer
338
views
Why is Pearson's correlation coefficient defined the way it is?
$$
r = \frac{{\rm Cov}(X,Y)}{ \sigma_{X} \sigma_{Y}}
$$
I do not understand this equation at all. Where does it come from?
From my personal understanding ${\rm Cov}(X,Y)$ comes from that fact that $...
- 406