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4

Suppose that I have a set $S$ of white balls. The calculation $$\binom{n}k\binom{n-k}\ell$$ counts the ways to perform the following operation: pick $k$ of the balls in $S$ and paint them red, then pick $\ell$ of the remaining white balls and color them blue. I could get the same results by first picking any $k+\ell$ balls, then choosing $k$ of those to ...


3

Let me answer questions 2) and 3) before dealing with 1). For second question: Martingales are tools which we can use to reduce complicated computations (example computing conditional probabilities), into appealing to a plethora of results about Martingales which you learnt for example in Stochastic process courses. In the research world, probabilists try ...


3

For brevity of notation, set $$g(x) := \frac{\int_F f(x,y) \, dy}{\int f(x,y) \, dy}. \tag{1}$$ To prove $$\mathbb{P}(Y \in F \mid X) = g(X)$$ it suffices to show $$\int_A g(X) \, d\mathbb{P} = \int_A \mathbb{P}(Y \in F \mid X) \, d\mathbb{P}$$ for any $A \in \sigma(X)$. Recall that any $A \in \sigma(X)$ can be written as $A= X^{-1}(B)$ for some Borel set ...


2

The way to understand this must include noticing that not only would the "variance" be negative if $a\notin[0,1]$, but one of the "probabilities" would be negative if $a<0$ and the other if $a>1$. This is actually a probability distribution only if $a\in[0,1]$.


1

Recall the reflection principle: For any Brownian motion $(W_t)_{t \geq 0}$ it holds that $$- \min_{0 \leq s \leq t} W_s \sim |W_t|.$$ Applying this to the Brownian motion $$W_t := B_{q^k+t}-B_{q^k}$$ yields $$\begin{align*} \mathbb{P}(\Omega_k) &= \mathbb{P} \left( \min_{t \leq q^{k+1}-q^k} W_t \geq - \sqrt{q^k} \right) \\ &= \mathbb{P} \left( ...



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