Variance algebra

Question

This might seem very simple but I'm having some trouble getting to the answer. If I have a random variable that's normally distributed $$X\sim N(30, 3^2)$$ and another random var. $$Y \sim N(20, (2.5)^2) $$ and I want to find the $P(2X+5Y<175)$ how would I go about doing this?

*They are both independent

Answer 1

1. Compute the expectation of $2X + 5Y$.
2. Compute the variance and standard deviation of $2X + 5Y$.
3. Express the distribution of $2X + 5Y$ as a normal random variable $J$.
4. Express the probability $ P(J<175)$ in terms of the standardized random variable $Z$.
5. Look up the answer in a $Z$-table.