The distribution of independent random variables X and Y are $X\sim \mathcal{N}(24,2^2)$ and $Y \sim \mathcal{N}(25,3^2)$. Find the distribution of $2X-Y$ and $\mathbb{P}[X-Y<40]$.
I am doing self studying and I just reached the topic Working with normal distribution on the chapter Linear combinations of random variables and I came to this topic and am stuck I don't really know what to do. Any help would be much appreciated. Thank You!
Best Answer
There is an important fact that a linear span of INDEPENDENT normally distributed random variables is still normally distributed.
For your case, let $Z=2X-Y$, then $Z$ is normally distributed with mean $E[Z]=2E[X]-E[Y]$ and variance $\mbox{Var}(Z) = \mbox{Var}(2X)+\mbox{Var}(-Y)$.
However, it is important to point out that if the given normally distributed random variables are NOT independent, their linear combination needs not be normally distributed.