1. ## Statistics help needed

Im stuck with understanding course material on probability distributions.
I need some help working out and understanding :-

if there is an exact negative linear relationship between Y and X -

Y = a - bX,

a and b are positive-valued parameters, and X and Y are R.V.s

The value of the correlation coefficient corresponding to X and Y is equal to –1

2. Originally Posted by bill2010
Im stuck with understanding course material on probability distributions.
I need some help working out and understanding :-

if there is an exact negative linear relationship between Y and X -

Y = a - bX,

a and b are positive-valued parameters, and X and Y are R.V.s

The value of the correlation coefficient corresponding to X and Y is equal to –1

What is the actual problem you have here ...?

3. Its not a very clear question.

I think its saying that if there is a negative linear relationship between X and Y, that it must have a correllation coefficient of -1.
Basically it has to just be shown.

4. Originally Posted by bill2010
Its not a very clear question.

I think its saying that if there is a negative linear relationship between X and Y, that it must have a correllation coefficient of -1.
Basically it has to just be shown.

Have you tried starting from the definition: $\displaystyle \rho = \frac{Cov(X, Y)}{\sigma_X \sigma_Y}$.

5. Originally Posted by mr fantastic
Have you tried starting from the definition: $\displaystyle \rho = \frac{Cov(X, Y)}{\sigma_X \sigma_Y}$.
$\displaystyle Y = b - aX$

$\displaystyle \Rightarrow E(Y) = b - a E(X)$ .... (1)

and

$\displaystyle \sigma_Y = a \sigma_X$ .... (2) since $\displaystyle a > 0$ is given.

$\displaystyle Cov(X, Y) = E(XY) - E(X)E(Y) = E(X (b - aX)) - E(X) \cdot [b - a E(X)]$

using (1)

$\displaystyle = E(bX - aX^2) - b E(X) + a [E(X)]^2 = b E(X) - a E(X^2) - b E(X) + a [E(X)]^2$

$\displaystyle = a ([E(X)]^2 - E(X^2)) = - a Var(X) = -a \sigma^2_X$ .... (3)

Substitute (2) and (3) into the formula for $\displaystyle \rho$: $\displaystyle \rho = \frac{-a \sigma_X^2}{a \sigma_X^2} = -1$.

6. Thanks again Mr F