How can we derivate with a constant in trying to prove that correlation coefficient is between -1 and 1?

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Douglas Woodard · Accepted Answer

Step 1We suppose we have random variables X and Y with known distributions.Then   Var  ⁡  [  2  X  +  Y  ] is the variance of a variable   2  X  +  Y  . . The variance is a real number.Also   Var  ⁡  [  3  X  +  Y  ] is the variance of another variable; that variance also is a real number.Similarly   Var  ⁡  [  −  17  X  +  Y  ]  .In fact, you can take any real number a, put it in the expression  Var  ⁡  [  a  X  +  Y  ]  ,and out comes a real number, which happens to be the variance of the random variable you&#039;ve just put together.Or in other words we have a function f such that for any real number a,  f  (  a  )  =  Var  ⁡  [  a  X  +  Y  ]  .In fact it is a differentiable function, so you can take its derivative with respect to a.We know that  f  (  a  )  =      a    2    Var  ⁡  [  X  ]  +  2  a  Covar  ⁡  [  X  ,  Y  ]  +  Var  ⁡  [  Y  ]  .And since X and Y are already known distributions that don&#039;t depend on a, the expressions   Var  ⁡  [  X  ] ,   Covar  ⁡  [  X  ,  Y  ]  , , and   Var  ⁡  [  Y  ] , which are determined just by those known distributions, are constants.

How can we derivate with a constant in trying to prove that correlation coefficient between -1 and 1

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