# Tagged Questions

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### What is the probability of going bankrupt in roulette?

Imagine that the bank has the money $M_1$ and the player has the money $M_2$. The rules are the following: You win with a chance of $\frac{17}{36}$ and lose with $\frac{19}{36}$ each round. Now you ...
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### derivation law from the call option formula

i am reading a article about the option pricing. and i got stuck with some typical statement. $C(K)=\int (x-K)^+\mu(dx)$ is given. here, $\mu$ is implied law of asset price and C(K) is the price ...
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### Reliability Probability problem

What is the Probability that at least one close path is formed from A to B where each switch has a Probability of close = p and each switch acts independent of the other Proposed Solution Let ...
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### Defining the scale function of a diffusion process

My question has to do with correctly calculating the scale function of a diffusion process, but ultimately might only have to do with calculus. I'll briefly set-up my calculations, so you can quickly ...
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### Identically distributed and same characteristic function

If $X,Y$ are identically distributed random variables, then I know that their characteristic functions $\phi_X$ and $\phi_Y$ are the same. Does the converse also hold?
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### Mean of cumulative distribution function

Suppose a CDF given by: $F(x)=0$ if $x < -2$, $0.6$ if $-2 <= x <= 1$, $1$ if $x > 1$. How can I then calculate $E[X]$ and $E[X^2]$? I found that $E[X]=0$ since there are only jumps, so ...
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### Derivation 9.97 in Jaynes' Probability Theory

In page 298 of Jaynes' Probability Theory: the Logic of Science, equation (9.97), Jaynes says: We expect that, if hypothesis $H$ is true, then $n_k$ will be close to $np_k$, in the sense that the ...
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### Almost sure convergence of generalized random harmonic series

Problem. Prove that $$\sum_{n=1}^\infty \frac{1}{n} \cos(2 \pi \cdot2^{n^2} x)$$ converges for almost every $x \in [0, 1]$ (with Lebesgue measure). (Credits to B. Tsirelson.) My partial solution. I ...
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### Probability Density Function from: F(x)=x$, for$0\leq x\leq \frac12

Probability Density Function from: F(x)=begin{cases}0 & \text{if }x<0\ x & \text{if }0\leq x\leq\frac{1}{2}\ 1 & \text{if }x>\frac{1}{2} \end{cases}. Do somebody know how to determine ...
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$$\int_0^{\infty} P(y > z) \, dz = \int_0^{\infty} \int_z^{\infty} h(y) \, dy \, dz = \int_0^{\infty} \int_0^y \, dz \, h(y) \, dy$$ Why do we have the last equality? I used Fubini and derived the ...
How do I prove that $\int_{0}^{+\infty}\text{exp}(-x)\cdot\text{log}(1+\frac{1}{x})dx$ is finite? (if it is) I tried through simulation and it seems finite for large intervals. But I don't know ...
### Other way to express $e^{|x|+|y|}$
I have a joint PDF with $e^{|x|+|y|}$. I know I can separate the function in two functions, $e^{|x|}$ and $e^{|y|}$. The limits for $x$ and $y$ are from $-\infty$ to $\infty$. Can I integrate from $0$ ...