# Tagged Questions

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### I have a question on conditional P with multiple events

I'm following this "Modeling and Reasoning with bayesian networks book's problems and Im stuck in this: ...
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### Probability and Sampling distribution

Would you please explain me the difference between Probability distribution and Sampling distribution easily ? Is that the difference : in probability distribution we have probability for every ...
63 views

### Maximum likelihood estimation of volume

There is a metric called as the Volume V Can somebody explain to me how to find the value of V, which is the volume of a dynamical system. I need to apply maximum likelihood estimation on V to find ...
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### I want to show $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process and I find mean and covariance functions

Let $B(t)$ be Brownian motion. Show that $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process. Find its mean and covariance functions. thanks .
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### I want to calculate $\int B(t)^2 dB(t)$ where $B(t)$ is Brownian motion

Let $B(t)$ be Brownian motion. I want to calculate $\int B(t)^2 dB(t)$. definition.A process $\{X(t),0\le t \le T \}$ is called a simple adapted process if there exist times ...
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### I want to show some properties for Brownian motion

Let $B(t)$ be a Brownian motion. Show that the following processes are Brownian motions on $[0,T]$ 1) $X(t)=-B(t)$; 2) $X(t)=B(T-t)-B(T)$, where $T\lt \infty$; 3) ...
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### Show that if $X\ge 0$ , $E(X)\le \sum_{n=0}^{\infty}P(X>n)$

If $X$ is a random variable and also let $X\ge 0$. I want to show $E(X)\le \sum_{n=0}^{\infty}P(X>n)$.
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### Inverse function for a non-decreasing CDF

For a CDF that is not strictly increasing, i.e. its inverse is not defined, define the quantile function $$F^{-1} (u) =\inf \{x: F(x) \geq u \},\quad 0<u<1.$$ Where U has a uniform $(0,1)$ ...
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### The joint pdf of two random variables defined as functions of two iid chi-square

Let $X$ and $Y$ be two i.i.d. chi-square distributed random variables with four degrees of freedom. How can we get the joint probability distribution function of the random variables $U=(X-Y)/(X+Y)$, ...
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### How to prove $(X_{n})_{n\in \mathbb N}$ and $(Y_{n})_{n\in \mathbb N}$ are supermartingale and $(Y_{n})_{n\in \mathbb N}$ is convergence to -7

Let $p \in [0 , \frac{1}{2}]$ and $\eta_{i}$ be i.i.d random variables and $P(\eta_{i}=1)=p$ and $P(\eta_{i}=-1)=1-p$ and $\mathcal F_{n}=\sigma(\eta_{1},\cdots,\eta_{n})$ and ...
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### I need to check if I'm correct, with conditional and bayes

I have this problem: ...
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### Conditional table, how to fill it

I found this question and since I'm learning probability I'm not sure how to go about it: ...
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### $X$ and $Y$ are independent if and only if $\sigma(X)$ and $\sigma(Y)$ are independent [closed]

Let $X$ , $Y$ be random variables on probability space $(\Omega , \mathcal B, P)$. show that $X$ and $Y$ are independent if and only if $\sigma(X)$ and $\sigma(Y)$ are independent.
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### Conditional probability problem - acceptance to two colleges

I'm doing this problem: ...
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### I have 2 problems with conditional probability, involving coins and dice

Hi I have these problems: ...
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### Probability to get equal cars

A famous car production company produces 3 different models of cars, available in 5 colors and 3 possible CC(1900 cc,2000 cc and 2200 cc). suppose that every combination of model, color and CC has ...
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### How does one prove $\int_{0}^{a}B(t)dt\sim \mathcal N(0,\frac{a^3}{3})$? [closed]

Let $B(t)$ is Brownian Motion. I want to prove the integral $\int_{0}^{a}B(t)dt$ has normal distribution , $\mathcal N(0,\frac{a^3}{3})$.
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### How to prove three properties of the moment generating function? [duplicate]

The moment generating function of a random variable $X$ is defined to be the function $$M_{X}(t)=E(e^{tX})=\sum_{n=0}^{\infty}\frac{E(X^n)}{n!}t^n.$$ Let $I=\{t\in\mathbb R:M_{X}(t)<\infty\}.$ I ...
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### Probability of failure

If the probability of failure of a certain component is $p = .0025$, how many backup components are needed to ensure the probability of a failure of all components is $1/1,000,000$?
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### I want to show $E(X)=\sum_{n=1}^{\infty}P(X\ge n)$

Let $X:\Omega \to \mathbb N$ be a random variable on probability space $(\Omega,\mathcal B,P)$ .show that $$E(X)=\sum_{n=1}^{\infty}P(X\ge n).$$ my definition from $E(X)$ is equal ...
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### Variance of sum of random number of random variables (Cambridge University Worksheet)

In the vein of my last question, I'm now at a roadblock on question 3 of this sheet: http://www.trin.cam.ac.uk/dpk10/IA/exsheet3.pdf (note: it's not my intention to ask every question I get stuck on ...
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### Show that $T=\min\{n:X_{n}\in B\}$ is an $\mathcal F_{n}$-stopping time

Let $X_{n}$ be an $\mathcal F_{n}$-martingale and let $B\in \mathcal B$. Show that $T=\min\{n:X_{n}\in B\}$ is an $\mathcal F_{n}$-stopping time. $\mathcal B$ is Borel $\sigma$-algebra and filtration ...
Example: Toss a coin twice. Letting $\mathbb P$ be a probability measure, suppose $\mathbb P(HH)=p^2,\mathbb P(HT)=\mathbb P(TH)=p(1-p), \mathbb P(TT)=(1-p)^2.$ I would like to answer the following ...