Probability distritubion of linear function

Question

Given a variable X belongs to gaussian distribution $N(\mu, \sigma)$. How to find the distribution of linear function $y=ax+b$? My answer is that the linear distribtion will belong the $N(a\mu,\sigma)$, Is it correct. How to prove it. Please prove help me. Thank you so much

Answer 1

Since normal distribution belongs to location-scale family, $Y=aX+b$ is normally distributed if $X$ is.

I'm afraid your answer is wrong:

$$\mu_Y=E[Y]=E[aX+b]=aE[X]+b=a\mu+b$$ And $$\sigma_Y^2=V(Y)=E[(Y-\mu_Y)^2] \\=E[(aX+b-(a\mu+b))^2] \\= E[(aX-a\mu)^2] \\ =a^2E[(X-\mu)^2] \\ =a^2V(X) \\=a^2\sigma^2 \\ \implies \sigma_Y=a\sigma$$ Thus $$\therefore X\sim\mathcal{N}(\mu,\sigma^2), Y=aX+b\implies Y\sim \mathcal{N}(a\mu+b, a^2\sigma^2)$$