# Tagged Questions

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### Show that if E is measurable set and f is continuous on E, then f(E) is measurable set

Please tell me how to prove or disprove it ! Show that if E is measurable set and f is continuous on E, then f(E) is measurable set
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### Convergence in Probability of a Sequence of Exponential Random Variables

If $X$ is an exponential random variable with $\lambda = 3$ and $Y_n = \frac{X^n}{n}$, I am trying to prove whether or not $Y_n$ converges in probability. My original approach was the following: ...
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### Convergence of eigenvectors/eigenvalues for infinite, non-negative, irreducible matrices with row sums less than $1$

I actually had two questions but decided to put them in separate posts for clarity. Suppose you have a matrix $M$ with infinitely many rows and columns, and a sequence of matrices $M_m$ with the ...
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### Properties of logarithmic mean.

I have been studying the logarithmic mean for the last few days now. Could someone please help me with the following two questions? 1) We know that the log mean is in between the geometric mean and ...
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### Linearization of exponential function with expectation

I ran into the following formula in my class: $$E[\exp(X)]\approx \text{constant}+E[X].$$ Since the linearization takes place at $X=0$, I don't understand why the constant in the formula is not ...
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### Set Difference Probability [duplicate]

Here is the question: Prove that for every $\epsilon>0$ and every set $A\in\mathcal{B}(\mathbb{R}^{n})$ there is a compact set $K\subset A$ such that $P(A\setminus K)\leq\epsilon$. --I have ...
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### The probability distribution function of uniform random variables is as given

Given $U_1, U_2, \dots, U_n$ where each $U_i \sim U[0,1]$, then use uniqueness theorem to show probability distribution function of $X = U_1 + U_2 + \ldots +U_n$ (sum of independent uniform random ...
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### $\mathbb L^1 +$ a.s. convergence of sequence $(X_n)$ does not imply $\sup(x_n)$ is integrable

I'm looking for a counterexample. The setting is this: Given an probability space $(\Omega,\mathbb{F},\mathbb{P})$, I look for sequence of random variables $(X_n)_n$ and a random variable $X$, all in ...
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### how to solve this integral in survival analysis

Let $T$ be a positive random variable, $S(t)=P(T\geq t)$. Prove that $$E[T]=\int^\infty_0 S(t)dt.$$ I have tried this unsuccessfully.
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### A naive question about “random” probability distributions

So I've come up with this idea, which may be mathematically unprecise, but it goes as follows. Example: Suppose you have a random variable $X$ taking values in the interval $[-1,1]$. Then, say, ...
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### Too stupid to understand random variable questions?

I have two excercises: 1.) Let $X_1,X_2,X_3$ be independent uniformly distributed random variables on $[0,1]$. What is the density function of $X_1+X_2+X_3$? 2.) Let $X_1,...,X_4$ be independet ...
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### Why would $f_n(x) = (\lfloor 2^nf(x)\rfloor/2^n)\wedge n$ converge to $f(x)$?

Why would $$f_n(x)=\frac{\lfloor 2^nf(x)\rfloor}{2^n}\land n$$ converge to $f(x)$? I saw this step in the proof of change of variable formula in Rick Durrett's Probability Theory and Examples.
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### Find $c$ such that $\text{P}(\limsup\limits_{n\to \infty} X_n/\sqrt{\log n}=c)= 1$

Need to find $c$ such that $\text{P}(\limsup\limits_{n\to \infty} X_n/\sqrt{\log n}=c)= 1$, where $X_n$ are a sequence of independent random variables such that $X_k\sim\mathcal{N}(0,1)$ I need to ...
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### Gap distribution independence proof

I have a question bout the proof of the independence of gap RVs. Given the independent exponentially distributed random variables $\xi_1$, $\xi_2$ ~ $\text{Exp}(\lambda)$, and a corresponding order ...
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### Proof of Khintchine's inequality

I'm trying to understand the proof of Khintchine's inequality in these lecture notes: http://www.math.ubc.ca/~ilaba/wolff/notes_march2002.pdf On page 27, second display-style equation after (51), the ...
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### Derive the probability density function of the distance

I was and still wondering how to derive the probability density function of $d^2$, where $d$ is the distance between two points that are inside a circle of radius $R$. The two points are uniformly ...
I was wondering how to derive the probability density function for the sum of $n$ independent iid distributed random variables on the interval $[0,1]$. A formula for that is given on ...