# Questions tagged [probability-theory]

Use this tag only if your question is 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.

26,896 questions
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### A question about the proof of Theorem 5.21 in Van der Vaar(1998)

If we know that $\hat\theta_n\overset{p}\to\theta_0$, how does the following equation \begin{equation} \sqrt{n}V_{\theta_0}\cdot(\theta_0-\hat\theta_n)+\sqrt{n}o_p(|\hat\theta_n-\theta_0|)=G_n\psi_{\...
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### “expectation of sum is sum of expectation”, is this claim true? if yes, how to justify this claim?

this post is saying linearity of expectation gives following equation $$\mathbb{E} [\sum_{j\neq i} Y_i Y_j] = \sum_{j\neq i} \mathbb{E} [Y_i Y_j]$$ per wiki, Linearity of Expected_value is ...
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### For a sequence of experiments where each $X$ is the number of trials until success with varying $p$, is each $X$ independent?

Assume that, every time you buy a box of Wheaties, you receive a picture of one of the $n$ baseball player. Let $X_k$ be the number of additional boxes you have to buy, after you have obtained $k-1$ ...
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### Expectation of an inner product in an infinite dimensional Hilbert space

Let $\mathcal{H}$ be a Hilbert space with the Borel $\sigma$-algebra. Let $(\Omega, \mathcal{F}, P)$ be a probability space and $x,y$ two $\mathcal{H}$-valued random variables, i.e. measurable maps ...
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### Expected value of the number of different numbers drawn in 37 rounds of roulette?

I need help with this problem. What is the expected value of the number of different numbers drawn in 37 rounds of roulette? Is this possible to interpret as the number of records? So the expected ...
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### Proving $h(X_n)\implies h(X)$ in distribution

Let $X_n$ be a sequence of random variables on $(\Omega,\mathscr{F},P)$such that $X_n\implies X$ (converges in distribution). Let $h:\mathbb{R}\to\mathbb{R}$ be a function whose points of ...
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### $X_i$, $i=1,2,..n$ independent R.Vs $P(X_i=1)=\frac{3}{4} ,\ P(X_i=-1)=\frac{1}{4}$. Prove $\sum_{i=0}^nX_i \to \infty$ a. s. as $n \to \infty$

I am asked to prove $X_1+X_2+X_3+...+X_n$ diverges almost surely as $n \to \infty$ Let $Y_n=X_1+X_2+...+X_n$ then what we want to prove is $P(Y_n=k)=1, \text{ as} (k,n) \to (\infty,\infty)$ Let us ...
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### Probabilistic PCA: Derive $\mathbb{E}[z_n \mid x_n]$

Consider the probalistic pca setting, where $x \in \mathbb{R^d}$ is an input vector drawn from $p(x)$, $z \in \mathbb{R^m}$ is an explicit latent variable with $p(z) = \mathcal{N}(0,\mathbb{I}_m)$ ...
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### How can we proof this implication in Corollary 4.8.7 in the book of Ethier and Kurtz?

I'm trying to understand the proof of the implication "(g) $\Rightarrow$ (f)" in Corollary 8.7 of Chapter 4 in the book Markov Processes: Characterization and Convergence by Stewart N. Ethier, Thomas ...
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### Distinct random variables in sampling

Let $X_1,X_2,\dots$ be i.i.d. random variables with $X_1 \sim U[0,1]$. Throwing out a null set all the variables are distinct. Can anyone explain this second sentence? What does he mean with "...
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### It would be possible to define an uniform distribution on $\Bbb N$ using infinitesimals?

In standard analysis it is clear that it is impossible to define an uniform probability distribution on $\Bbb N$ because there is no constant $c\in\Bbb R$ such that $\sum_{k=1}^\infty c=1$. Using ...
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### Strong Markov property and another stopping time

I'm trying to prove that given a regular continuous time Markov chain $X_t$ (pure jump process), its embedded chain given by $Y_n=X_{T_n}$ is a homogeneous Markov chain, where $T_n$ is the time of the ...
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### Tricky information inequality $I(X;Z) \geq H(T)$

I am wondering whether $I(X; Z) \geq H(T)$ when the following conditions hold: $H(T | X) = H(T)$ $H(T | Y) = H(T)$ $H(T | X, Y) = 0$ $H(Y | Z) = H(T | Z) = 0$ $X, Y, Z, T$ are discrete. I know first ...
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### Upper bound on number of cliques in a Vietoris-Rips complex

Does there exist an upper bound on the number of cliques of order $k$ in a Vietoris-Rips complex? I found this work --> https://arxiv.org/pdf/1104.0914.pdf I understand it makes the assumption of ...
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### Odd moment and the characteristic function of a random variable

Let $X$ be a random variable and $\phi_X(t)$ be its characteristic function. Let $n$ be a positive even integer. If $\phi_X(t)$ is $n$-times differentiable, then the $n$-th moment of $X$ exists and ...
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### What will be the probablilty in these cases? [on hold]

So we have a fair and unbiased dice, which is rolled thrice in a row. 1)What is the probability to get the sequence [1,2,3] in the three continuous trials? 2)What is the probability of getting the ...
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### Finding the expected number of a certain colored ball drawn from an urn in k draws

Suppose we have an urn containing c yellow balls and d green balls. We draw k balls, without replacement, from the urn. Find the expected number of yellow balls drawn. Hint: Write the number of ...
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### How is P(B) derived and why is $P(D_i)$ equal to 55/72 and not $(55/72)^i$

So this is a question with its solution below to which I don't understand 2 things. How is P(B) derived? And, why is $P(D_i)$=55/72 and not $(55/72)^i$. Since, for example, obtaining heads in the n ...
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### $P(T_2=T_{-3}), P(T_1<T_4<T_{-1})$ and $P(T_3<2)-P(T_{-3}<2)$

Let $W(t)$ be a Brownian motion and $T_x=\inf\{t:W(t)=x\}$. I need to calculate $P(T_2=T_{-3}), P(T_1<T_4<T_{-1})$ and $P(T_3<2)-P(T_{-3}<2)$. I'm not sure if I understand these ...
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