Chapter 5.7, Problem 13E

Let $X_1,X_2,\mathrm{...},X_{36}$ be a random sample of size 36 from the geometric distribution with pmf $f\left(x\right)={\left(\frac{1}{4}\right)}^{x-1}\left(\frac{3}{4}\right),x=1,2,3,\mathrm{....}$.

Approximate

(a) $P\left(46\le {\displaystyle \underset{i=1}{\overset{36}{\sum }}{X}_i}\le 49\right)$.

(b) $P\left(1.25\le \bar{X}\le 1.50\right)$.

HINT: Observe that the distribution of the sum is of the discrete type.