# Questions tagged [probability-theory]

For questions solely about the modern theoretical footing for probability, for example, probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use [tag:probability-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.

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### Can I think of probabilities as proportions instead?

I am new to probability theory, so bare with me if I do not nail all of the terminology (I will still try my best)! Also, I gave a short "What is my question" sentence, but I invite you to ...
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### Let $(\Omega,F)$ be a measurable space, $F=\{\varnothing,\Omega\}$, prove that $X:\Omega\to\Bbb R$ is a random variable iff $X$ is constant.

Let $(\Omega,F)$ be a measurable space, $F=\{\varnothing,\Omega\}$ prove that $X:\Omega\to\Bbb R$ is a random variable if and only if $X$ is constant. I was thinking this is not necessarily true as if ...
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### How can I show that a Poisson process with my definition below has stationary and independent increments?

We had the following definition: Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_t, \Bbb{P})$ be a filtered probability space. An $(\mathcal{F}_t)_t$ Poisson process $(N_t)_{t\geq 0}$ is a right ...
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### Calculating a probability with the hypergeometric distribution

I'm studying the hypergeometric distribution with the probability mass function $$Pr(X=k) = \frac{{K \choose k}{(N-K) \choose (n-k)}}{{N \choose n}},$$ where $N$ is the population size, $K$ is the ...
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### Trouble understanding order statistics

Order statistics were introduced in my text as follows: I am trying to understand what this means. $X_1 , \dots , X_n$ is a random sample, i.e. an independent and identically distributed sequence of ...
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### How does $\Omega$ figure in stochastic processes?

So I read this page for clarification on trajectories and $X(\omega, \cdot): T\to \mathbb R$ maps while going through lectures on stochastic processes. I still have doubts which are described as ...
1 vote
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### Is Memorylessness a necessary assumption in Shannon's Proof of the Channel Coding Theorem?

It is often said that the achievability proof for Shannon's coding theorem relies on the channel being discrete and memoryless. At the same time, following the classical proof (using random coding and ...
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1 vote
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### Prove that X is not always random variable in case that X^2 is random variable [closed]

Could someone give me an idea how to prove it?
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### . Suppose 𝐴 and 𝐵 are two events. Prove that 𝑃(𝐴) + 𝑃(𝐵) − 1 ≤ 𝑃(𝐴 ∪ 𝐵) ≤ 𝑃(𝐴) + 𝑃(B)

I'm totally confused on this since the only available proof is 𝑃(𝐴) + 𝑃(𝐵) − 1 ≤ 𝑃(𝐴 ∪ 𝐵) only explained in online.
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### Find distribution of $\sum_{i=1}^{n}\frac{X_{i}^{2}}{\sigma^{2}}$

Let $X_{1},...,X_{n}\sim f(x)=Kx^{2}e^{\frac{−x^{2}}{2σ^{2}}}$ i.i.d. I need find the distribution of $\sum_{i=1}^{n}\frac{X_{i}^{2}}{\sigma^{2}}$. To do this, calculate the normalizing constant, K, ...
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1 vote
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+100

### Independence is preserved by joint weak convergence

Suppose a sequence of random vectors $(X_n,Y_n)$ converges jointly to some $(X,Y)$ in the weak topology. Question: If $X_n$ and $Y_n$ are independent for all $n$, are also $X$ and $Y$ independent? ...
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### Find this expected value [closed]

Let $\xi_1,\xi_2,\cdots, \xi_n, \cdots$ be a sequence of identically distributed continuously randomly variables and $\nu = \text{min}\{k: \xi_{k-1} > \xi_k\}$. Find $\mathbb{E}(\nu)$ I'm ...
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### Brownian motion and Holder-$\frac{1}{2}$-continuity
Let B be a Brownian motion. For every $K>0$, we have $$P[\inf \left \{ t>0: B_t\geq K t^{1/2} \right \} =0]=1 \quad\quad\quad(1)$$ To prove this in Example 21.16 of Probability Theory (3rd ...