Question

Let X be a random variable with mean, E(X) = 4 & the standard deviation, SD(X) = 5. Let Z= aX +b. find a so that Z has a standard deviation of 1

Answer 1

\[\
Var(Z)= Var(a x) +0= a^2 Var(X)= a^2 5^2=1\\
a=\frac 1 5
\]

Answer 2

@eliassaab I don't quit understand what this all means, can you please explain?

Answer 3

\[
\text{SD}(X) = \sqrt{\text{Var}(X)}
\]

Answer 4

And \(\text{Var}(X)\) has the property: \[
\text{Var}(aX+b) = \text{Var}(aX)+\text{Var}(b) = a^2\text{Var}(X)+0 = a^2\text{Var}(X)
\]

Answer 5

We want \(\sqrt{\text{Var}(aX+b)} = 1\) which means \(\text{Var}(aX+b) = 1^2\)

Answer 6

This simplifies to \(a^2Var(X) = 1\)

Answer 7

Since \[
5 = \sqrt{\text{Var}(X)} \implies 25 = \text{Var}(X)
\]we say \[
25a^2=1\implies a^2=\frac{1}{25} \implies |a| = \frac{1}{5}
\]