Articles of probability

Probability that no two consecutive heads occur?

A fair coin is tossed $10$ times. What is the probability that no two consecutive tosses are heads? Possibilities are (dont mind the number of terms): $H TTTTTTH$, $HTHTHTHTHTHTHT$. But except for those, let $y(n)$ be the number of sequences that start with $T$ $T _$, there are two options, $T$ and $H$ so, $y(n) […]

moment generating function

can anyone help me to inform the name of any book where I can get the following theorem, or give some detailed hint to solve this one: Let $X$ and $Y$ be two random variables, if the moment generating functions of $X$ and $Y$ are equal then the probability distributions of $X$ and $Y$ will […]

Is there anything special about a transforming a random variable according to its density/mass function?

Lets say that $X\sim p$, where $p:x\mapsto p(x)$ is either a pmf or a pdf. Does the following random variable possess any unique properties: $$Y:=p(X)$$ It seems like $E[Y]=\int f^2(x)dx$ is similar to the Entropy of $Y$, which is: $$ H(Y):=E[-\log(Y)]$$ It seems like we can always make this transformation due to the way random […]

Analogue of the Schwartz–Zippel lemma for subspaces

Let $f : \mathbb{R}^n \to \mathbb{R}$ be a nonzero multivariate polynomial of total degree $d$ over the reals, and $S \subset \mathbb{R}$ be finite. Pick a positive integer $k$, choose $y_1, \ldots, y_k$ randomly and uniformly from $S^n$, and consider the $k$-variable polynomial $$g(t_1, \ldots, t_k) = f(t_1 y_1 + \cdots + t_k y_k)$$ Question: […]

What tactics could help with this probability questions

I’m not too sure if this question is solvable (I sort of just thought of it yesterday) but when I brute force numerical answers on my computer they seem to show a pattern, so I believe it to be solvable. The question: I have $N$ people in a stadium, and there are $5$ different coloured […]

probability of picking a specific card from a deck

Question: Assume you have a deck with with $52$ cards ($4$ suites of $13$ cards: numbers $1\ldots 9$, and faces J,Q,K). What is the probability you draw jack of hearts in a hand of $5$? My way of thinking is the following: $$\frac{\left(\dfrac{1\cdot51\cdot50\cdot49\cdot48}{4!}\right)}{ \left(\dfrac{52\cdot51\cdot50\cdot49\cdot48}{5!}\right)}$$ $1$ is for the jack of hearts being drawn, and then […]

Random walk in the plane

A particles moves in $\mathbb{R^2}$ started at the origin. At each stage $i (i = 1, 2, …)$, the particles would move, independently of all the stages before, one of the four directions North, East, South and West 1 unit, with probability $1 \over 4 $ each.Let $T_n$ be the distance from the origin just […]

Modified gambler's ruin problem: quit when going bankruptcy or losing $k$ dollars in all

In each round, the gambler either wins and earns 1 dollar, or loses 1 dollar. The winning probability in each round is $p<1/2$. The gambler initially has $a$ dollars. He quits the game when he has no money, or he has lost $k>a$ rounds in all by this time, no matter how many rounds he […]

How many random samples needed to pick all elements of set?

This question already has an answer here: Expected time to roll all 1 through 6 on a die 2 answers

Minimizing the variance of weighted sum of two random variables with respect to the weights

Suppose $X$ and $Y$ are two random variables. I would like to see if the solution to $$ \min_w \quad \mathrm{Var}(wX+(1-w)Y) $$ can be negative. I know that \begin{align*} &\mathrm{Var}(wX+(1-w)Y) \\ &= w^2 \mathrm{Var} X + 2w(1-w)\mathrm{Cov}(X,Y) + (1-w)^2 \mathrm{Var}Y \\&= w^2 (\mathrm{Var} X – 2\mathrm{Cov}(X,Y) + \mathrm{Var}Y) + 2w(\mathrm{Cov}(X,Y) – \mathrm{Var}Y) + \mathrm{Var}Y \end{align*} […]