Cholesky decomposition
We can write matrix $A=\left(\begin{array}{ccc}1 & 2 & 3 \\ 2 & 5 & 7 \\ 3 & 7 & 11\end{array}\right)$ as $A=L \times L^{T}$, where $L=\left(\begin{array}{lll}1 & 0 & 0 \\ 2 & 1 & 0 \\ a & b & 1\end{array}\right)$. Find $a+b$
This is a simple decomposition. Solve it straightforwardly. Get $a+b=4$.
Regression Coefficients
Let $X$ and $Y$ be random variables with $E(Y)=6, \operatorname{Var}(Y)=$ $3, E(X)=4, \operatorname{Var}(X)=5$, and $\operatorname{Cov}(X, Y)=0$
Suppose we do a linear regression of $X$ on $Y$, which gives us the resulting linear model $Y=a+b \times X$.
What is the value of $a+b ?$
Suppose we do a linear regression of $X$ on $Y$, which gives us the resulting linear model $Y=a+b \times X$.
What is the value of $a+b ?$
Covariance equal to 0, and thus $b=0$ for sure. The constant term is then chosen to be $6$ to meet the expectation of $Y$.
Fair value of a game
Blake flips a fair coin and stops flipping once he gets a head. He gets the minimum of $\$ 16$ and $\$ 2^{n}$, where $n$ is the total number of Blake's flips. For example, if Blake flips a head on the first try, then $n=1$.
What is the fair value of this game?
When $n\le4$, the payoff is just $2^n$. When $n\ge5$, the payoff is $16$. Calcualte the product of payoff and probability then sum them up. The final answer should be 5 (the infinite series when $n\ge 5$ sum to 1)What is the fair value of this game?
Unfair coin with distribution
A machine produces a weighted coin that lands on heads with an unknown probability $p$, and we know that $P(p \leq x)=x^{3}$.
You flip the coin 4 times, and every time it lands on heads.
What is the probability that the next flip will also land on heads?
(Enter your answer as a decimal).
You flip the coin 4 times, and every time it lands on heads.
What is the probability that the next flip will also land on heads?
(Enter your answer as a decimal).
Streaks of tosses
You have a coin which comes heads with probability $1 / 4$, and you toss it 500 times. If there are multiple coins in a row (could be just 1), we call them/it a streak. For example, if we get HTHTT, we have 4 streaks: first $H$, second $T$, third $H$, and last two $T$.
What is the expected number of streaks in 500 tosses?
What is the expected number of streaks in 500 tosses?
This is just a similar problem to the “Average Number of Runs” in probability problem 01. The difference is that the coin is not fair anymore. The probability that a pair (two tosses) are different is $2\times\frac{1}{4}\times\frac{3}{4}=\frac{3}{8}$. Regard this as $499$ pairs, then $1+499\times\frac{3}{8}=188.125$.
N-sphere
For any positive integer $n$, let $A_{n}$ denote the surface area of the unit ball in $\mathbb{R}^{n}$, and let $V_{n}$ denote the volume of the unit ball in $\mathbb{R}^{n}$.
Let $i$ be the positive integer such that $A_{i}>A_{k}$ for all $k$ not equal to $i$. Similarly let $j$ be the positive integer such that $V_{j} > V_{k}$ for all $k$ not equal to $j$. Find $j-i$
Use the reference here, https://en.wikipedia.org/wiki/N-sphere.
The n-sphere gets largest volume when $n=5$, and gets largest surface when $n=6$. Thus, the answer is $-1$.