First, I'll mention one problem that I am working on over this break. Consider the standard plane in two-space. Then in, say, the first quadrant, consider the "wedge" shaped region

*W *bounded by a line through

*(0,0)* of slope

*m*, where

*m > 0, *and the

*x*-axis. Let

*X* denote some infinite set of points in

*W* which has

*(0,0)* as a limit point. Clearly it is possible to find a single variable function

*f* whose graph runs though an infinite subset of

*W* such that

*f* is continuous on the entire real line*f* is differentiable except for possibly at *x = 0*.

Here is how one can obtain such an *f* : Select from *W *some infinite subset *{xi}* with the following property: *r(i+1)where {r(j)}* is some strictly decreasing sequence which converges to 0. Next, construct an polygonal infinite arc by connecting the points *xi* to each other via line segments, and then setting *f(0)=0*, and then choosing a real number sequence *ti* where *ti* converges to 0 and then setting *xi = (ti, f(ti))*. Clearly such an *f* is continuous. To make *f* differentiable: let *yi* denote the midpoint of the segment connecting *xi* to *x(i+1)*. Then do a three point spline smoothing process to obtain a smooth segment through *y(i-1), xi, y(i)* such that the smooth arc has derivative equal to the slope of the segment containing *yi* for each *i*. This modification makes *f* smooth, except for possibly at *t = 0 *

So the question is: how does one make *f* differentiable from the right at *t = 0 *; that is, I want {limit as dt goes to zero) of *(f(0+dt)-0)/dt* to exist. The type of thing I am worried about would be things like *f(t) = tsin(1/t)* which has a limit as *t* goes to zero, but cannot be made to be differentiable there. I have some rather cumbersome ideas to try out over the next 4-5 days.

Now, about football predictions: I made some bowl game predictions here:

http://blueollie.blogspot.com/2006/01/bowls-and-my-prediction-nds-season.html

and my record was 17-9 straight up (on correctly picking the winner) and 15-10-1 (on correctly picking the winner vs. the published point spread). Example: at the Insight Bowl, Arizona State was favored to be Rutgers by 11.5 points. I picked Arizona State to win (which they did) but I picked Rutgers to cover ( that is, either win or lose by no more than 11 points). The final score was Arizona State 45, Rutgers 40, so I won "straight up" and also won "against the spread". On the other hand, Michigan was an 11 point favorite against Nebraska in the Alamo Bowl. I picked Michigan to win, but Nebraska to cover. Nebraska won 32-28, so I missed the "straight up" pick, but won the "spread" pick.

So, the question obtains: am I a good at making football predictions? I define "good" as being "better than chance". In statistics, one can safely say that they are better than random chance if the probability of, say, a coin flip doing as well is less than or equal to 5%.

So I decided to do a hypothesis test:

Ho: my predictions are no better than a random coin flip

Ha: my predictions are better than a random coin flip.

I used the binomial distribution (26 bowl games, a coin flip has .5 probability of being correct each time, so how many times will a coin be right?). In the days of old, it was usual to use the normal approximation to the binomial or to use tables; but now-a-days there are handy calculators such as this one: http://www.stat.sc.edu/~west/applets/binomialdemo.html

So if *X* is the random variable that denotes the number of times a coinflip would be correct out of 26 trials, we see that *P(X>= 17) = .0843* and *P(X>=18) = .0378.* Now we see that *x = 15* for the spread betting and *x = 17* for the straight up betting (note: my point spread picks were correct 15 times and *P(X>=15) = .2786. ) *So that tells us that in neither case (straight up or by the spread) gives convincing statistical evidence that I can do better than a coin flip, though I was "close" to being better than a coin flip with my "straight up" predictions.