Calculation:

The negative binomial distribution for a random variable Y is,

p(y)=(y−1r−1)prqy−r,      y=r,r+1,r+2,...,0≤p≤1 where q=1−p

The negative binomial distribution for a random variable Y−1 is,

p(y−1)=((y−1)−1       r−1)prq(y−1)−r=(y−2r−1)prqy−r−1 where q=1−p

Take the ratio of p(y) and p(y−1).

p(y)p(y−1)=(y−1r−1)prqy−r(y−2r−1)prqy−r−1=(y−1)!(r−1)!(y−1−(r−1))!prqy−r(y−2)!(r−1)!(y−2−(r−1))!prqy−rq−1=((y−1)(y−2)!(y−r)!)((y−2)!(y−r−1)!)q−1=((y−1)(y−r)(y−r−1)!)(1(y−r−1)!)(1q)=(y−1y−r)q

Hence, it sis showed that if y≥r+1,  p(y)p(y−1)=(y−1y−r)q.

Calculation:

Consider the inequality, p(y)p(y−1)>1

(y−1y−r)q>1(y−1)q>(y−r)yq−q>y−ryq−y>q−r

y(q−1)>q−r        [multiply inequality with−1]−y(q−1)<−(q−r)y(1−q)<r−qy<r−q1−q

Hence, it is showed that p(y)p(y−1)>1 if y<r−q1−q.

Consider the inequality, p(y)p(y−1)<1

(y−1y−r)q<1(y−1)q<(y−r)yq−q<y−ryq−y<q−r

y(q−1)<q−r        [multiply inequality with−1]−y(q−1)>−(q−r)y(1−q)>r−qy>r−q1−q

Hence, it is showed that p(y)p(y−1)<1 if y>r−q1−q.

Calculation:

From part (b), p(y)>p(y−1) if y<r−q1−q.

The value of q is,

q=1−p=1−0.5=0.5

Substitute, r=7 and q=0.5 in the inequality

y<7−0.51−0.5y<6.50.5y<13

Hence, y takes values less than 13.