Postby **sachem** » Thu Nov 06, 2008 7:06 pm

Question 58:

f(z)=f(x+iy)=u(x,y)+i*v(x,y) where u and v are real valued. so, v(x,y)=0 for all x and y, since f(z) maps the entire complex plane to the real line.

Now use the fact that f(z) is analytic - this means that the cauchy equalities must hold : du/dx=dv/dy, and du/dy=-dv/dy

so du/dx = 0, and then u=h(y)+c for some function h of y.

and du/dy=0 so then u=g(x)+c for some function g of x.

Then u(x,y)=c for some constant c.

So, f(z)=c...everything goes to a single point. I am pretty sure this is correct but please correct me if I'm wrong. Note that given the real part or the imaginary part ( u or v ) of an *analytic* function, you can figure out the other part ( up to a constant ).

60)

The expected value of the outcome {6} is 1/6*360 = 60. (for a binomial distribution)

The standard deviation is sigma^2= npq=360*1/6*5/6=50. So the standard deviation is root(50) which is 5root2...so we can approximate this with a N(60, 5root2) distribution and instead find P(z>10/5root2) or P(z>root2)=1-P(z<root2)..

I think at this point you would have simply had to have memorized the fact that for the standard normal distribution, p(z<one standard deviation away) = .84 so p(z>one standard devation away) is .16. Funny enough, that is the number that they gave.

If you're curious, the numbers are P(z less than mean+0 standard deviations) is .5, 1 standard deviation is .84, 2 standard deviations is .98, and 3 is .999

Cheers, hope that helped.