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

Question

anonymous · Answer

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

summersnow8 · Answer

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

anonymous · Answer

$$
\text{SD}(X) = \sqrt{\text{Var}(X)}
$$

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

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