Probabilities for a shifted normal

Topic: Functions of random variables and limit theorems · more questions on this topic
‹ Two standard normal probabilities An affine image of an exponential density ›

Let Y\sim\operatorname{N}(1,4). Using \Phi(0.5)\approx 0.6915 and \Phi(1)\approx 0.8413, evaluate \mathbb{P}(Y\le 3) and \mathbb{P}(-1\le Y\le 2).

Worked answer (click to show)

Write Y=1+2X with X standard normal. Then \mathbb{P}(Y\le 3)=\mathbb{P}(X\le 1)=\Phi(1)\approx 0.8413, and

\mathbb{P}(-1\le Y\le 2)=\mathbb{P}(-1\le X\le 1/2)=\Phi(1/2)-(1-\Phi(1))\approx 0.6915-0.1587=0.5328.