Covariance for a bivariate normal distribution

statistics normal-distribution covariance bivariate-distributions statistical-inference

Once you proved that $\mathbb{Cov}[Y-\rho X,X]=0$ you are done because (using the fact given in the text) this is equivalent to prove that $(Y-\rho X)\perp\!\!\!\perp X $ thus

$$\mathbb{E}[(Y-\rho X)^{10}X^3]=\mathbb{E}[(Y-\rho X)^{10}]\cdot\mathbb{E}[X^3]=0$$

Being $\mathbb{E}[X^{2n+1}]=0$, $ \forall n$

Related

Show that the dimention of the intersection of projective linear sub-spaces of dimentions $d_1$ and $d_2$ of $\mathbb{P}^n$ is bigger than $d_1+d_2-n$
if $k > 1$ and $m\ge 1$ are such that $a^k = (2^m-1)(2^m+1)$, then does $a | 2^m-1$ or $a| 2^m+1$
Halmos set definition of projection onto the first coordinate, notation confusion.
If $F=\{(x,y,z)\in\mathbb R^3:f(x,y,z)=0\}$ is non-empty and $\frac{\partial f}{\partial x}\neq0$ on $F$, can $F$ be finite?
Why the answer for double integral is coming as zero?
How to perform integration by parts when the upper integral limit is infinity?
Trouble with the Jacobian for a camera projection matrix.
Derivative of L2 Norm of Matrix
Representing first order sentences as conceptual graphs
Symmetric random walk on the integers
Dividing Rs.69 among 115 students, each girl gets 50 paise less than a boy; each boy gets twice the paise as each girl. How many girls in the class?
Assumption for separate continuity implies joint continuity