Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and $\mathcal G$ a subfield of $\mathcal F$. I have that $\mathbb E[X\boldsymbol 1_G] = 0$ for all $G\in \mathcal G$. Do we have that $X=0$ ? I proved that if $X\geq 0$ a.s. then $X=0$ a.s. but if $X$ is just measurable, then I have that […]

Consider a fixed (non-random) $3$ by $n$ matrix $M$ whose elements are chosen from $\{-1,1\}$. Assume $n$ is even. I am trying work out what the probability mass function of $Mx$ is when $x$ is a random vector with elements chosen independently and uniformly at random from $\{-1,1\}$. Each of the three elements of $y […]

I need a good rigorous book to learn probability theory. So far, I’ve been suggested Gnedenko’s Theory of Probability; Shiyayev’s Probability; Feller’s An Introduction to Probability Theory and Its Applications. . Which one would you reccomend the most and why? Are there other books worth mentioning?

I am trying to understand the difference of the two solutions for the expected value of collisions for the birthday problem: https://math.stackexchange.com/a/35798/254705 derives the following solution: … so the expected number of people who share birthdays with somebody is $n\left(1-(1-1/N)^{n-1}\right)$. whereas https://math.stackexchange.com/a/952272/254705 gives This leads to an expectation value $\lambda$ (date collisions in terms of […]

Given a joint distribution $P(A,B,C)$, we can compute various marginal distributions. Now suppose: \begin{align} P1(A,B,C) &= P(A) P(B) P(C) \\ P2(A,B,C) &= P(A,B) P(C) \\ P3(A,B,C) &= P(A,B,C) \end{align} Is it true that $d(P1,P3) \geq d(P2,P3)$ where d is the total variation distance? In other words, is it provable that $P(A,B) P(C)$ is a better […]

If someone have $4$ red bricks and $8$ blue bricks and arranges them randomly in a circle, what is the probability that two red bricks are not side by side? Is the sample space $\frac{12!}{4!\,8!} \,=\, 495$ ? and the answer $P(\text{Reds non-adjacent}) \:=\:\frac{70}{495} \:=\:\frac{14}{99}$ ?

Suppose $X_1, \dots, X_n$ are truncated standard normal variables, truncated so that $X_i \geq 0$ (that is, $X_i$ is drawn as a standard normal, conditional on $X_i \geq 0$) Let $c_1, \dots, c_n$ be non-negative coefficients. What does the distribution of $\sum_i c_i Y_i$ look like? Does it have, or approximately have, a standard distribution, […]

If I place $k$ balls in $n$ cells randomly with uniform probability, then for large enough $n$ and $k/n$ small enough, I expect about $n e^{-k/n}$ cells to be empty (using the Poisson approximation to the Binomial.) However, I cannot find the limit of the variance of the number of empty cells? One can compute […]

Sent to me by a friend, I’m stumped. We have an urn. It is first filled with balls of $C$ different colors, ${N_1,N_2,…N_c}$ of each of the colors. Lastly, a ball with a distinct marking is added. We are not given $C$ or any of the $N$ values. A machine removes the balls without replacement […]

Let $v_1=(x_1,y_1),…,v_n=(x_n,y_n)$ be $n$ two dimensional vectors where each $x_i$ and each $y_i$ is an integer whose absolute value does not exceed $2^{n/2}/(100 \sqrt{n})$. Show that there are two disjoint sets $I,J \subset \{ 1,2,…,n\}\ $ such that $ \ \ \sum \limits_{i \in I} v_i = \sum \limits_{j \in J} v_j$ I tried to […]

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