Let ${Y}_{1}$ and ${Y}_{2}$ be independent random variables with ${Y}_{1}\sim N(1,3)$ and ${Y}_{2}\sim N(2,5)$ . If ${W}_{1}={Y}_{1}+2{Y}_{2}$ and ${W}_{2}=4{Y}_{1}-{Y}_{2}$ , what is the joint distribution of ${W}_{1}$ and ${W}_{2}$ ?

Elisabeth Esparza
2022-07-18
Answered

Let ${Y}_{1}$ and ${Y}_{2}$ be independent random variables with ${Y}_{1}\sim N(1,3)$ and ${Y}_{2}\sim N(2,5)$ . If ${W}_{1}={Y}_{1}+2{Y}_{2}$ and ${W}_{2}=4{Y}_{1}-{Y}_{2}$ , what is the joint distribution of ${W}_{1}$ and ${W}_{2}$ ?

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polishxcore5z

Answered 2022-07-19
Author has **14** answers

Step 1

First note that $({Y}_{1},{Y}_{2}{)}^{\prime}$ is bivariate normal because ${Y}_{1},{Y}_{2}$ are independent.

Next observe that

$\left(\begin{array}{c}{W}_{1}\\ {W}_{2}\end{array}\right)=A\left(\begin{array}{c}{Y}_{1}\\ {Y}_{2}\end{array}\right)$

$A=\left(\begin{array}{cc}1& 2\\ 4& -1\end{array}\right),$ ,

and use the fact that affine transforms of normal random vectors are normal, i.e.

$X\sim N(\mu ,\mathrm{\Sigma})\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}AX+b\sim N(A\mu +b,A\mathrm{\Sigma}{A}^{\prime}).$ .

First note that $({Y}_{1},{Y}_{2}{)}^{\prime}$ is bivariate normal because ${Y}_{1},{Y}_{2}$ are independent.

Next observe that

$\left(\begin{array}{c}{W}_{1}\\ {W}_{2}\end{array}\right)=A\left(\begin{array}{c}{Y}_{1}\\ {Y}_{2}\end{array}\right)$

$A=\left(\begin{array}{cc}1& 2\\ 4& -1\end{array}\right),$ ,

and use the fact that affine transforms of normal random vectors are normal, i.e.

$X\sim N(\mu ,\mathrm{\Sigma})\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}AX+b\sim N(A\mu +b,A\mathrm{\Sigma}{A}^{\prime}).$ .

asked 2022-06-15

Let's say we have two random variables $Y$ and $X$ used to form regression model

$Y=\alpha +\beta X+\mu $

It also holds that $E(\mu )=0$, $\text{Var}(\mu )={\sigma}_{\mu}^{2}$, $\text{Var}(X)={\sigma}_{X}^{2}$, $\text{Var}(Y)={\sigma}_{Y}^{2}$, $\text{Corr}(X,Y)=r$ and $\text{Corr}(X,\mu )={r}_{X\mu}$. Find $\beta $. I tried to solve this as follows:

For simple linear regression $\beta ={\displaystyle \frac{\text{Cov}(X,Y)}{\text{Var}(X)}}$ and $\text{Corr}(X,Y)={\displaystyle \frac{\text{Cov}(X,Y)}{{\sigma}_{X}{\sigma}_{Y}}}=r$ so that:

$\beta =\frac{\text{Corr}(X,Y)\cdot {\sigma}_{X}{\sigma}_{Y}}{{\sigma}_{X}^{2}}=r\frac{{\sigma}_{Y}}{{\sigma}_{X}}$

Is this as simple as this?

$Y=\alpha +\beta X+\mu $

It also holds that $E(\mu )=0$, $\text{Var}(\mu )={\sigma}_{\mu}^{2}$, $\text{Var}(X)={\sigma}_{X}^{2}$, $\text{Var}(Y)={\sigma}_{Y}^{2}$, $\text{Corr}(X,Y)=r$ and $\text{Corr}(X,\mu )={r}_{X\mu}$. Find $\beta $. I tried to solve this as follows:

For simple linear regression $\beta ={\displaystyle \frac{\text{Cov}(X,Y)}{\text{Var}(X)}}$ and $\text{Corr}(X,Y)={\displaystyle \frac{\text{Cov}(X,Y)}{{\sigma}_{X}{\sigma}_{Y}}}=r$ so that:

$\beta =\frac{\text{Corr}(X,Y)\cdot {\sigma}_{X}{\sigma}_{Y}}{{\sigma}_{X}^{2}}=r\frac{{\sigma}_{Y}}{{\sigma}_{X}}$

Is this as simple as this?

asked 2022-07-07

Now when we take ${Z}_{1}=X+{Y}_{1}$ and ${Z}_{2}=X+{Y}_{2}$ , what can we say about the correlation coefficient between ${Z}_{1}$ and ${Z}_{2}$?

For this case, is it possible to find the correlation coefficient as function of ${\sigma}_{x}$ and ${\sigma}_{y}$?

For this case, is it possible to find the correlation coefficient as function of ${\sigma}_{x}$ and ${\sigma}_{y}$?

asked 2022-06-04

Short version, I need to find a regression to this: $a\equiv t\phantom{\rule{0.444em}{0ex}}(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}\mathrm{\Delta})$, $a$ and $\mathrm{\Delta}$ are the unknowns constants.

Any idea where I should start looking?

Some context, because I may be wording it in a confusing way: I am trying to find the tempo of time-stamped events ${t}_{i}$ for some real time musical analysis. They have a typical interval of $\mathrm{\Delta}$, but there isn't an event at every "tick", so no linear regression, and there may be more than one event for a given "tick". In other words, ${t}_{n+1}-{t}_{n}$ may be $0$ or any $m\mathrm{\Delta}$.

Any idea where I should start looking?

Some context, because I may be wording it in a confusing way: I am trying to find the tempo of time-stamped events ${t}_{i}$ for some real time musical analysis. They have a typical interval of $\mathrm{\Delta}$, but there isn't an event at every "tick", so no linear regression, and there may be more than one event for a given "tick". In other words, ${t}_{n+1}-{t}_{n}$ may be $0$ or any $m\mathrm{\Delta}$.

asked 2022-05-01

Suppose you have two sequences of complex numbers ${a}_{i}$ and ${b}_{i}$ indexed over the integer numbers such that they are convergent in ${l}^{2}$ norm and a has norm greater than b in the sense

$\mathrm{\infty}>\sum _{i}|{a}_{i}{|}^{2}\ge \sum _{i}|{b}_{i}{|}^{2}.$

Suppose moreover they are uncorrelated over any time delay, meaning

$\sum _{i}{a}_{i}\overline{{b}_{i-n}}=0\phantom{\rule{1em}{0ex}}\mathrm{\forall}n\in \mathbb{Z}.$

Is it true that the polinomial $a(z)=\sum _{i}{a}_{i}{z}^{-i}$ is greater in absolute value than $b(z)=\sum _{i}{b}_{i}{z}^{-i}$ for any unit norm complex number z?

$\mathrm{\infty}>\sum _{i}|{a}_{i}{|}^{2}\ge \sum _{i}|{b}_{i}{|}^{2}.$

Suppose moreover they are uncorrelated over any time delay, meaning

$\sum _{i}{a}_{i}\overline{{b}_{i-n}}=0\phantom{\rule{1em}{0ex}}\mathrm{\forall}n\in \mathbb{Z}.$

Is it true that the polinomial $a(z)=\sum _{i}{a}_{i}{z}^{-i}$ is greater in absolute value than $b(z)=\sum _{i}{b}_{i}{z}^{-i}$ for any unit norm complex number z?

asked 2022-06-30

Consider a signal that is a sum of sinusoids, e.g.

$x(t)=Asin(at)+Bcos(bt)$

Is there an easy and general way to get an analytical solution for the autocorrelation of x(t)?

Is the best way to simply plug x(t) into the autocorrelation formula?

$x(t)=Asin(at)+Bcos(bt)$

Is there an easy and general way to get an analytical solution for the autocorrelation of x(t)?

Is the best way to simply plug x(t) into the autocorrelation formula?

asked 2022-05-08

If I have built two linear regression models over sets $A$ and $B$, and now want a linear regression over set $A\cup B$.Is there a way to reuse what I already have?

asked 2022-05-23

A random sample of size $n$ from a bivariate distribution is denoted by $({x}_{r},{y}_{r}),r=1,2,3,...,n$. Show that if the regression line of $y$ on $x$ passes through the origin of its scatter diagram then

$$\overline{y}\sum _{r=1}^{n}{x}_{r}^{2}=\overline{x}\sum _{r=1}^{n}{x}_{r}{y}_{r}$$

where $(\overline{x},\overline{y})$ is the mean point of the sample.

$$\overline{y}\sum _{r=1}^{n}{x}_{r}^{2}=\overline{x}\sum _{r=1}^{n}{x}_{r}{y}_{r}$$

where $(\overline{x},\overline{y})$ is the mean point of the sample.