If $X \sim N(μ,σ^2)$ how to find the PDF of $Y = 2X$

Question

Given a random variable $X$ that follows the Gaussian Distribution (i.e $X \sim N(μ,σ^2)$), I need to find the Probability Density Function (PDF) of $Y = 2X$.

It's been some years since I've done something similar so I am kinda stuck. How do I approach this problem?

Of course, $f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

Obviously $Y \sim \frac{N(μ,σ^2)}{2} $

Therefore $f_Y(y) =f_{2X}(y) = ?$

Note: Please do not give me a specific answer, just a general guideline, as I would like to solve this by myself.

Answer 1

Here are some guidelines:

• The PDF is the derivative of the CDF.
• $F_Y(z)=P(Y \leq z) = P(X \leq z/2)=F_X(z/2)$.