[5 points] suppose we run the perceptron algorithm with w initialized to be an arbitrary unit vector. suppose that the algorithm is then given the same vector x (whose label is 1) over and over again. how many mistakes can it make in the worst case?
express your answer in terms of ||x||2. (hint: derive the result from first principles, do not attempt to use the general result we proved in class).

Evaluate this using ! 0.25^2 x 2.4 + 0.25 x 2.4^2 − 0.25 x 2.4 x 0.65

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.

Maddie's rectangular kitchen has an area of 10 square yards and a length of 3 3/4 yards.how many yards wide is maddie's kitchen?