,. solve kindly

\fEXERCISE 4.36. In Example 4.20, use Eq. 4.16 to verify that the fake spheres with a = 2 and a = , have no umbilical points. (4.16) T. EXAMPLE 4.20 (Fake Spheres). We wish to construct surfaces of revolution with constant Gaussian curvature equal to 1. If K = 1, then Eq. 4.15 says that x" = -x. The following is a solution for every a > 0: (4.17) x(t) = a cos(t). The hypothesis that y is parametrizationgth, so (x')2 + (2')2 = 1, means that z is determined by x as follows: (4.18) = (t) = VI - x'(s)2 ds = 1 - a2 sin? (s) ds. (4.15) K\f3. [10 pts] Consider the simple linear regression model yi = 60+81(xi-x) +ci(i= 1,2,..,n), where x= ni=1 xi. From the least-squares criterion S(80,$1), find the least-squares estimators of 80 and B1 for this model. Hint: Do not expand the term (xi-x). That is, do not expand ni= 1 yi*(xi-x) as nZi=1 yixi- nZi=1 yi x for easier computations. Also remember what nEi=1(xi-x) is. 3. [10 pts] Consider the simple linear regression model yi = Bo + BI(z; - I) + ; (i = 1, 2, ..., n), where I = > I;. From the least-squares criterion S(Bo, 81), find the least-squares estimators of Bo and B, for this model. n Hint: Do not expand the term (r; - I). That is, do not expand ) yi(x; - I) as ) Vidi - 1=1 n it for easier computations. Also remember what E(x - I) is. 1-1 i-1