How to compute probabilities of normally distributed variables?

Question

Let $X_1, X_2, \ldots, X_{16}$ be independent with $N(3,4)$ distributions and $\bar{X}$ denote the sample mean. Find:

1. $P(-8 < 2 \bar{X} < -4)$.
2. A number $K$ such that $P(-K < 2 \bar{X} < K) = 0.99$.
3. $P(-10 < 3 X_1+X_7 < -8)$.
4. A number $K$ such that $P(3 X_2+X_5 < K) = 0.3$.

If someone could give me some help or hints on where to start with (a) or any of the problems that would be really helpful and I'd really appreciate it.

Answer 1

For $k$ in some set of indices, let each $\alpha_k$ be a real constant, $Y_1,\ldots,Y_n$ independent normal random variables. A very useful property of such variables is that their linear combinations are normal. Therefore, using properties of the mean and variance operators, we get $$\sum_k\alpha_kY_k\sim N\Bigg(\sum_k\alpha_kE[Y_k], \sum_k\alpha_k^2V[Y_k]\Bigg)\quad.$$

By setting the indices and constants appropriately — and looking up quantiles on the even numbered items —, you will obtain every answer.