Let X be a random variable with distribution $N\left(0,1\right)$
$$\therefore f_X\left(x\right) = \dfrac{1}{\sqrt{2\pi}}e^{-\left(x^2/2\right)}$$
Y be another random variable such that,
$$f_Y\left(x\right) = \dfrac{1}{\sqrt{2\pi}}e^{-\left(x - \mu\right)^2/2\sigma^2}$$
and Z is such that,
$$f_Z\left(x\right) = \dfrac{1}{\sigma \sqrt{2\pi}}e^{-\left(x - \mu\right)^2/2\sigma^2}$$
By location-scale transformation I can write Y as $\sigma X + \mu$.
Simple calculations yield,
$$EX = 0$$ thus, $$EY = E\left(\sigma X + \mu\right) = \mu$$
and,
$$EZ = \dfrac{EY}{\sigma} = \dfrac{\mu}{\sigma} $$
But $EZ$ should be equal to $\mu$. Where am I going wrong?
There is no such random variable $Y$ since $f_Y$ is not a density function. What is true is $Z$ has same distribution as $\mu +\sigma X$ and $EZ=\mu+\sigma EX=\mu+0=\mu$.