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Tutor profile: Amadej Kristjan K.

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Amadej Kristjan K.
Mathematics Tutor
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Questions

Subject: Machine Learning

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Question:

Suppose we are trying to use a machine learning algorithm to perform a classification task, which humans can do with $$98$$% accuracy. Our machine learning algorithm has $$95$$% accuracy on the training set and $$70$$% accuracy on the test set. What is our machine learning algorithm likely suffering from and how can we surmount this issue?

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Amadej Kristjan K.
Answer:

The algorithm is likely suffering from overfitting. This means that the algorithm 'memorises' a lot of irrelevant information that is specific to the set it's training on, but fails to generalise the patterns to the general data distribution. When faced with a yet unseen test set, it performs much worse. In such a scenario, what we can do several of the following: - we can increase the amount of training data (this way the algorithm is faced with a slightly more representative representation of the general data distribution), - we can reduce the representational capacity of the model (making the model simpler, to force the algorithm to only encode the most crucial, generalizable information into the model), - we can incorporate regularisation techniques to reduce overfitting (add a penalising term to the loss function as done in Ridge and Lasso regression or in case of using neural networks we can add dropout regularisation).

Subject: Discrete Math

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Question:

Prove that there are infinitely many prime numbers.

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Amadej Kristjan K.
Answer:

Suppose for the sake of contradiction that there are only finitely many prime numbers. Then we can list them as follows: $$ p_1,p_2,...,p_n$$. Now consider the number $$P = p_1p_2...p_n + 1$$. It is not divisible by either of the prime numbers listed, as it gives a remainder of $$1$$ when divided by any of them. Therefore it is only divisible by $$1$$ and itself, having only $$2$$ positive integer divisors, which makes $$P$$ a prime number. But we assumed the above list of primes contained all prime numbers. Since we found a prime outside of the list with supposedly all primes, we arrived at a contradiction. Hence, there are infinitely many prime numbers.

Subject: Calculus

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Question:

Let $$R$$ be a rectangle with side lengths $$a$$ and $$b$$. If we keep its perimeter $$P$$ fixed, at what $$a:b$$ ratio will the area $$A$$ of the rectangle be maximised?

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Amadej Kristjan K.
Answer:

The (fixed) perimeter $P$ of the rectangle is: $$P = 2(a+b)$$ The area is: $$ A = ab = a(\frac{P}{2}-a) = \frac{Pa}{2}-a^2$$ Thus, we can view the area of $$R$$ as a function of the variable $$a$$ and see at which value of $$a$$ the area is maximised subject to our given perimeter $$P$$. We can obtain the maximum by figuring out where the derivative of this function is $$0$$. $$ \frac{dA}{da} = \frac{P}{2}-2a$$ so we need to solve $$0 = \frac{P}{2}-2a$$, giving $$a = \frac{P}{4}$$. Now substituting this into $$P = 2(a+b)$$ gives $$b=\frac{P}{4}$$, meaning that for maximising the area we need $$a = b$$.

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