If two variates X and Y are connected by the relation aX+b/c, where a, b, c are constants such that ac < o, then

Question

If two variates X and Y are connected by the relation \(\frac{aX+b}{c}\), where a, b, c are constants such that ac < o, then

A. σY = \(\frac{a}{c}\) σX 

B.  σY = \(-\frac{a}{c}\) σX

C.  σY = \(\frac{a}{c}\)​​​​​​​ σX + b

D. none of these

Answer 1

Given, Y = \(\frac{aX+b}{c}\) 

To Find: Write the expression for the standard deviation of Y.

Explanation: We have Y = \(\frac{aX+b}{c}\) 

Mean (y) = \(\frac{\Sigma y_i}{n}\) 

We can write as: Mean (y) 

= 

Mean (y) = \(\frac{a\Sigma \bar X}{nc}+\frac{nb}{nc}\) 

Var(X) = \(\Sigma\frac{(x_i-\bar X)^2}{n}\) 

Then, Var (Y) = \(\Sigma\frac{(y_i-\bar Y)^2}{n}\) 

Now, Substitute the value of y_i and Y, then we get

 

Var(Y) = \(\Big(\frac{a}{c}\Big)^2\sigma ^2\) 

SD(σ) = \(\sqrt{\Big(\frac{a}{c}\Big)^2\sigma ^2}\) 

(Yσ) = \(|\frac{a}{c}|\sigma\)