Normal Distribution Problem Involving Events

Question

Suppose that X~(2,9) and calculate the probability of the following events:

a) x^2+x-2 > 0;
b) abs(x-2) < 1

Answer 1

For both cases, consider $Y=X-2$, a normal distribution with mean $0$.

a) $x^2+x-2=(y+2)^2+y+2-2=(y+2)^2>0$, so $P(X^2+X-2>0)=1$

b) Use the cumulative to assess $P(-{1\over3}<Y<{1\over3})$ or $P(-{1\over9}<Y<{1\over9})$ depending on what your notation means for the variance of the normal.