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Let $X,Y,Z$ be 3 random variables. If the correlation between $X$ and $Y $ is $c_1\ge 0$ and the correlation between $Y$ and $Z$ is $c_2\ge 0$, what is the maximum and minimum possible correlation between $X$ and $Z$ in terms of $c_1$ and $c_2$? Show your work.

So what I've figured out is that from the geometric interpretation of this, the min is -1 and max is 1. Use the dot product: x goes 'right', y goes 'up' z goes 'left'. This implies that the product between X and Y is 0 and Y and Z is 0, but X and Z is -1 (they are opp directions). So the minimum must be -1.

The max must be 1 since we know that is the max value of correlation and there is no upper bound here.

So we know the absolute max and min, but I'm having trouble sorting it in terms of $c_1$ and $c_2$ here.

thanks

Gregory
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vootoo
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  • In general, if $c_j=\cos\theta_j,,\theta_j\in[0,,\pi]$, the extrema on the third correlation are $\cos(\theta_1\mp\theta_2)=c_1c_2\pm\sqrt{(1-c_1^2)(1-c_2^2)}$. This should give you an idea as to why. – J.G. Mar 04 '21 at 21:30
  • our bounds are not [0,pi] though? @J.G. – vootoo Mar 04 '21 at 21:34
  • Any cosine has the right value to be taken as the cosine of an angle from $0$ to $\pi$. – J.G. Mar 04 '21 at 22:05

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