The independent random variables $X$ and $Y$ satisfy $X\sim N(500,100)$ and \(Y\sim N(250,\sig

Question

The independent random variables $X$ and $Y$ satisfy $X\sim N(500,100)$ and $Y\sim N(250,\sigma^2)$, where $\sigma>0$. Let $T=X+2Y$. (a) State the distribution of $T$, giving its parameters in terms of $\sigma$. (b) Find the greatest possible value of $\sigma$ such that $P(940<T<1060)\geq0.98$. (c) It is also required that $P(|T-1000|<30)\leq0.80$. Find the range of values of $\sigma$ for which both requirements are satisfied.

Accepted Answer

(a) $T\sim N(1000,100+4\sigma^2)$; (b) $11.9$; (c) $10.58285\ldots\leq\sigma\leq11.88698\ldots$ (endpoints $10.6$ and $11.9$, 3 s.f.).

Bounds on variation in a weighted normal sum

Bounds on variation in a weighted normal sum

Solution:

Solution locked