Problem 1: Write a Python computer program which uses gradient information to maximize the following function:
f(x) = 2sin(x)+3
What solution is obtained? Does this solution vary as a function of the initial guess and if so why?
Problem 2: Mary walks to work 70% of all days and gets the train to work on all other days. The weather is only good on 65% of all days while all remaining days are rainy. Mary only walks to work on 20% of all rainy days. Mary walked to work today. Use Bayes rule to determine the probability it was rainy today. In your answer state all steps taken to obtain your solution.
Problem 3: Consider the data set in the table below:
|
ID
|
X
|
Y
|
Z
|
Class
|
|
1
|
1
|
1
|
1
|
T
|
|
2
|
1
|
0
|
0
|
T
|
|
3
|
0
|
0
|
1
|
F
|
|
4
|
1
|
1
|
0
|
F
|
|
5
|
0
|
0
|
1
|
T
|
|
6
|
0
|
1
|
0
|
T
|
|
7
|
0
|
1
|
0
|
F
|
|
8
|
1
|
0
|
0
|
F
|
|
9
|
0
|
1
|
1
|
T
|
|
10
|
0
|
0
|
1
|
F
|
(a) Estimate the conditional probabilities P(X=1|Class=T) and P(Z=1|Class=F). In your answer state all steps taken to obtain your solution.
(b) Compare P(X=1), P(Y=1) and P(X=1, Y=1). State the relationship between X and Y. In your answer state all steps taken to obtain your solution.
(c) Compare P(X=1, Y=1|Class=T) with P(X=1|Class=T) and P(Y=1|Class=T). Are the random variables X and Y conditionally independent given the class? In your answer state all steps taken to obtain your solution.
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