exercise:0a82eb3e0d: Difference between revisions
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(Created page with "<div class="d-none"><math> \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}</math></div> Let <math>X</math> and <math>Y</math> be random variables with positive variance. The ''correlation'' of <math>X</math> and <math>Y</math> is defined as <math display="block"> \rho(X,Y) = \frac{{\rm cov}(X,Y)}{\sqrt{V(X)V(Y)}}\ . </math> <ul><li...") |
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Let <math>X</math> and <math>Y</math> be random variables with positive variance. The ''correlation'' of <math>X</math> and <math>Y</math> is defined as | |||
The ''correlation'' of <math>X</math> and <math>Y</math> is defined as | |||
<math display="block"> | <math display="block"> | ||
\rho(X,Y) = \frac{{\rm cov}(X,Y)}{\sqrt{V(X)V(Y)}}\ . | \rho(X,Y) = \frac{{\rm cov}(X,Y)}{\sqrt{V(X)V(Y)}}\ . | ||
</math> | </math> | ||
<ul><li> Using | <ul style="list-style-type:lower-alpha"><li> Using [[exercise:0876775b21 |Exercise]](c), show that | ||
<math display="block"> | <math display="block"> | ||
Latest revision as of 22:44, 14 June 2024
Let [math]X[/math] and [math]Y[/math] be random variables with positive variance. The correlation of [math]X[/math] and [math]Y[/math] is defined as
[[math]]
\rho(X,Y) = \frac{{\rm cov}(X,Y)}{\sqrt{V(X)V(Y)}}\ .
[[/math]]
- Using Exercise(c), show that
[[math]] 0 \leq V\left( \frac X{\sigma(X)} + \frac Y{\sigma(Y)} \right) = 2(1 + \rho(X,Y))\ . [[/math]]
- Now show that
[[math]] 0 \leq V\left( \frac X{\sigma(X)} - \frac Y{\sigma(Y)} \right) = 2(1 - \rho(X,Y))\ . [[/math]]
- Using (a) and (b), show that
[[math]] -1 \leq \rho(X,Y) \leq 1\ . [[/math]]