Articles of probability

Are there any random variables so that E and E exist but E doesn't?

Are there any random variables so that E[X] and E[Y] exist but E[XY] doesn’t?

Pick a card from a set with the help of a dice

Some days ago i was playing “Settlers of Catan”, an homemade version. In this game there is an event where a players pick, randomly, a card from another player. So here is the problem: can we use the dice (six faces) to pick the card randomly? Until the opposite player has less than seven cards, […]

What is the expected time to cover $m \leq N$ elements in a set by sampling uniformly and with replacement?

The expectation for covering a set of $N$ elements by sampling uniformly and with replacement is $N * H(N)$, where $H(N)$ is the $N$th harmonic number. Is there a similar expression using harmonic numbers for the expectation for the number of sampling events necessary to see any $m \leq N$ unique elements in the set? […]

Average number of trials until drawing $k$ red balls out of a box with $m$ blue and $n$ red balls

A box has $m$ blue balls and $n$ red balls. You are randomly drawing a ball from the box one by one until drawing $k$ red balls ($k < n$)? What would be the average number of trials needed? To me, the solution seems to be $$\sum_i i * \frac{\mbox{the number of cases where k-th […]

Is the Event a Conditional Probability or an Intersection?

My question is based on Example 1.9, p 22, *Introduction to Probability (1 Ed, 2002) by Bertsekas, Tsitsiklis. Define the event $A$ = {an aircraft is present} and $R$ = {the radar registers an aircraft presence}. Express the following events in terms of $A$, $R$, and/or their complements. $\begin{align} & \text{(i) The radar correctly registers […]

Conditional expectation of an uniformly distributed random variable

Suppose $U_1, \ldots, U_n$ are i.i.d. random variables with $U_1$ distributed uniformly on the interval $(-1, 1)$. Compute $\mathbb{E}(U_1 + \ldots + U_n |\max(U_1, \ldots, U_n) = t)$ for $t \in (-1, 1)$. So my attempt was to try to compute $\mathbb{E}(U_1 |\max(U_1, \ldots, U_n) = t)$, writing $U_1 = U_1 1_{\{U_1 < t\}} + […]

If $\mathbb E=0$ for all $G\in \mathcal G$, does $X=0$?

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 […]

Probability distribution for a three row matrix vector product

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 […]

Book suggestion for probability theory

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?

birthday problem – which solution for expected value of collisions is correct?

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 […]