24 Birthday Problems
Q1
Given that $A$ and $B$ are sets like so:
a)
Problem
Give an expression for the conditional probability $P(A|B)$
b)
Problem
Hence or otherwise derive Bayes' (first) rule for conditional probability:
\[P(A|B) = P(B|A) \times \frac{P(A)}{P(B)}\]
c)
Problem
Prove that the symmetric difference $A\Delta B = (A - B) \cup (B - A)$ is the same as $(A\cup B) - (A\cap B)$.
Q2
Monty Hall Problem:
a
Problem
normal version
b
Problem
hard version
Q3
a
Problem
Prove that there are equally as many natural numbers as there are integers, i.e. that \(|\mathbb{Z}| = |\mathbb{N}|\).
b
Problem
Prove that there are equally as many integers as there are rational numbers, i.e. that \(|\mathbb{Q}| = |\mathbb{R}|\).
maybe alpha lists:
- stuff
- more stuff
- trying stuff
c
Problem
Prove that the real numbers are uncountable; i.e. that \(|\mathbb{R}| \neq |\mathbb{N}|\).
Q4
a
Problem
What does the sequence \(1,\tfrac14,\tfrac19,\tfrac1{16},\ldots\) converge to? (If at all)
b
Problem
What does the sequence \(1,\tfrac12,\tfrac13,\tfrac1{14},\ldots\) converge to? (If at all)
Q5
Problem
Which is larger asymptotically as $n \rightarrow \infty$? \(2^n\) or \(n!\)? Give a proof by induction.
Q6
Problem
Rewrite this problem in terms of the Wolfe dual. Hence solve it.
Q7
Problem
pick a function spaces. use norms on it. see your analysis assignments.
Q8
a
Problem
4.3b, 4.4 find the eigenspaces
b
Problem
4.5, 4.6, determine diagonalisability.
Q9
Problem
MML: 6.1
Q10
Problem
MML: 6.11
Q11
Problem
State the Moment Generating Function (or pdf, not sure which is required for this) for the following distributions:
- Poisson
- Bernoulli
- Normal
Q12
Problem
Derive the mean and variance of the following distributions.
- Poisson
- Bernoulli
Q13
Problem
Distance between an arbitrary point and hyperplane given by $wx + b = 0$
Q14
Problem
Optimisation steepest descent method. ask about the number of iterations.
Q15
a
Problem
Prove that
Proposition
\[|a| \leq b \iff -b \leq a \leq b\]
Proof
b
Problem
Prove that
Proposition
\[\forall a, b \in \mathbb{R}, \quad |a|\cdot |b| = |a\cdot b |\]
Proof
c
Problem
Complete the following
Proposition
Two real numbers, $a$ and $b$ are equal if and only if, for every $\epsilon > 0$, $|a-b| <\epsilon$.
Proof
Q16
Problem
Complete the following
Proposition
Let $\mathcal{A}\subseteq \mathbb{R}$ and $m\in \mathcal{A}$ be the minimum of $\mathcal{A}$, then $\inf \mathcal{A} = m$.
Proof
Q17
a
Problem
State and prove the Triangle Inequality
Proposition
Proof
b
Problem
Hence, or otherwise, state and prove Cauchy-Schwarz inequality.
Proposition
Proof
Q18
Problem
Complete the following
Proposition
Every $\epsilon$-ball in a metric space is open.
Proof
Q19
a
Problem
Define mathematically what it means for a set to be convex.
b
Problem
Determine if the following set is convex:
c
Problem
Hence, or otherwise, find the maxima of this function.
Q20
a
Problem
Give the definition of a Hilbert space
Definition
(Hilbert Space)
b
Problem
not sure which problem to ask here
Q21
Problem
topology question. discrete, coarse, fine, etc.
Q22
Problem
convergence of a function
Q23
Problem
convergence of the central limit theorem.
Q24
Problem
What is the maximum straight-line distance in an N-dimensional hyper-cube?