3 votes

Proof for Particular Fair Shuffle Algorithm

One way of showing this is to consider the set of possible paths through the code. Note that at step $i$, a random number between $0$ and $i$ is chosen to determine where to insert the next card. This ...
Steven Stadnicki's user avatar
1 vote

Verifying Property of Constructed Brownian Motion using Haar Basis

This is shown in The Haar functions and the Brownian motion and Construction of Brownian Motion using Haar wavelets. For $t>s$ we have $$W_t-W_s=\sum_{k-1}^\infty G_k(t)-G_k(s)=\sum_{k-1}^\infty \...
Thomas Kojar's user avatar
  • 2,902
1 vote
Accepted

Expected radius of throwing a dart at a dartboard

Your answer doesn't agree with the official solution because you produce the median of the distribution, rather than its mean. In general, they are computed differently, and they are not guaranteed to ...
Brian Tung's user avatar
1 vote

Proof for Particular Fair Shuffle Algorithm

... I don't understand why it works. It's just inserting at random positions within the current shuffled deck isn't it? Imagine this. You have a deck of cards kept in front of you. You spread out one ...
Haris's user avatar
  • 2,667
1 vote

Discontinuous process with same marginals as Brownian Motion

The reason is that for any finite set on which you're taking marginals, the uniform random variable will almost surely never be on one of those points - this is true for a simpler example too: For $t \...
George's user avatar
  • 717
1 vote
Accepted

Is my informal understanding of probability definitions correct?

For your first question, I think it's mostly definitional, with possible confusion from the product measure on $\mathbb R^2$ compared to integrating along each variable. For your $(X,Y)$ random ...
George's user avatar
  • 717
1 vote

The difference between convergence in probability and convergence in observed value

I asked a similar question a while ago on MO: https://mathoverflow.net/questions/459782/definition-of-weak-conditional-convergence-of-random-variables We want to show that $$Z_n := \frac{\frac{1}{\...
Syd Amerikaner's user avatar
1 vote

Basketball scores - solving 2 variable recurrence relation

Here I explained how the two-dimensional recurrence can be solved directly without knowing the answer. Let us first fix $x=1$. Then, we have the following recurrence: $$P(1,n) = P(1,n-1)\left(\frac{n-...
Amir's user avatar
  • 2,387
1 vote

Probability of infinite coin toss

As pointed out in the comment, your formula involves double counting. Sketch: Let $S_n$ be the number of sequences with $n$ tails and $2n$ heads. Let $T_n$ be the number of sequences with $n$ tails ...
leonbloy's user avatar
  • 63k
1 vote

How many natural numbers $a\le100$ are there such that $a=[\frac a2]+[\frac a3]+[\frac a5]$, where [.] represents the greatest integer function?

Let $a=6q+r$. Then we need $r =\left[\dfrac{r}{2} \right]+\left[\dfrac{r}{3} \right]+\left[\dfrac{q+r}{5} \right]$ When $r=0$, $q$ can be $1,2,3,4$ For $r=1,2,3,4,5$ it is easy to see that $q$ has $5$ ...
Hari Shankar's user avatar
  • 3,548
1 vote
Accepted

Upper bound on $P(X \geq 100)$ if $g_X(10)=10$

From the OP, I guess you assumed that $X$ is non-negative, which implies $$10^X-1 \ge 0.$$ Now, from the Markov's inequality we have $$\mathbb P (X \geq 100)=\mathbb P (10^X-1 \geq 10^{100}-1) \le \...
Amir's user avatar
  • 2,387
1 vote

Why Doesn't the St Petersburg Paradox Happen All the Time?

As others have noted, your main confusion seems to be that an infinite sum can have either a finite answer or an infinite answer depending on the specifics. Finite example: If you make $n$ dollars for ...
Eric's user avatar
  • 5,734

Only top scored, non community-wiki answers of a minimum length are eligible