Articles of probability

One-to-One correspondence in Counting

I have a confusion on the one-to-one correspondence in combinatorics. Take the problem: In how many ways may five people be seated in a row of twenty chairs given that no two people may sit next to one another? Take the solution: Solution for Example: Consider some arrangement of the five people as specified, then […]

$L_p$ complete for $p<1$

It is rather straight forward to show that $L_p$ is complete for $p\geqslant 1$, but I am having trouble showing the same thing when $p<1$. For the former case I have shown that every absolutely convergent sequence converges by constructing a a function in $L_p$ but bigger than the series and used the dominated convergence […]

Probability all numbers will be selected?

I have 500 tries to draw 1 from 250 balls, all labelled (1 to 250), from an urn (with replacement). Each of the 250 balls are equally likely to be selected. What is the probability that I will, with the 500 tries, draw all 250 different balls at least once?

Do the pth powers of $p$-norms define the same partial ordering on the set of all probability distributions for all $p>1$?

Consider the $p$-th power of the Schatten $p$-norm $||q||_p$ of a probability distribution $q$ , ie, the function $\sum_j q_j^p$, where $\sum_j q_j = 1$ and $q_j \geq 0$. For fixed $q$ and $p>1$ this is a nonincreasing function of $p$. The question is: given $p_1, p_2$ both $>1$, is it true that: whenever $\left(||q||_{p_1} […]

Probability of cycle in random graph

I create a random directed graph, with N vertices and N edges, in the following process: A. Each vertex has a single outgoing edge. B. The target of that edge is selected at random from all N vertices (self loops are possible). What is the probability that a certain edge, selected at random, will be […]

Question related to Polya urn model

Just in case I will remind what is called Polya urn: Suppose you have an urn containing one red and one blue ball. You draw one at random. Then if the ball is red, put it back in the urn with an additional red ball, otherwise put it back and add a blue ball. Now […]

Convergence in distribution ( Two equivalent definitions)

I read that for convergence in distribution it is equivalent to have that either the characteristic functions of the random variables convergence pointwise or we have that $F_{X_n} \rightarrow F_{X}$ pointwise, where $F$(the distribution function) is continuous. I could not find a proof of this, so I was wondering how hard it is to show? […]

Definition: transient random walk

What exactly does a “transient random walk on a graph/binary tree” mean? Does it mean that we never return to the origin (assuming there is one as for the tree) or just any vertex of the graph or tree? Thanks.

Statistical Inference Question

From Statistical Inference Second Edition (George Casella, Roger L. Berger) “My telephone rings 12 times each week, the calls being randomly distributed among the 7 days. What is the probability that I get a least one call each day?” The answer is .2285, but I don’t know how they got it. My reasoning was as […]

$75$ students in three balanced groups: Probability, cardinality.

EDIT: more consistent attempt. Let say we have $n=75$ students, divided into 3 groups. What is the probability that the three groups are balanced and the probability that $Marc$ and $Gato$ are in the same group? Let $\Omega=\{(m_1,m_2,m_3)\in\{0,\cdots,75\}^3: m_1+m_2+m_3=75\}$ Let $\omega_1\in\Omega : \omega_1=(m_1,m_2,m_3)$ representes the number of studies in each group. Question: How can I […]