Tagged Questions

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Putting a bound on some probability inequality

Assume that we have the following polynomial: $$ax^2 + bx =c$$ and a, b, c are i.i.d uniform random variables in [0, 1]. I'm trying to calculate the probability that the root is real, and that ...
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Integrating with indicator functions

I want to evaluate $$\int_{-\infty}^{\infty}(A_1e^{-\beta_1(b-x-y)}+B_1e^{-\beta_2(b-x-y)})(pn_1e^{-n_1y}1_{\{y\geq0\}}+qn_2e^{n_2y}1_{\{y<0\}})dy,$$ $b>x, \beta_1<n<\beta_2$. I am trying ...
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Prove the existence of exactly two maxima for a positive $L^1$ function

I have a function $f:\mathbb R \rightarrow \mathbb R^+$ with the following properties it is $L^1$ and $C^2$ it has one single extremum (maximum) at $x=0$ it is symmetric: $f(x)=f(-x)$. it is ...
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Methods to distinguish continuous probability distributions

I read in the Wikipedia article for Variance The variance is one of several descriptors of a probability distribution. In particular, the variance is one of the moments of a distribution. In that ...
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Multivariate normal distribution independet iff uncorrelated

I found a few threads about this but none of them answered my question. I am supposed to show that if you have random variables $X_1$,$X_2$ that are gaussian distributed and they fulfill that ...
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Continuous probability distribution with no first moment but the characteristic function is differentiable

I am looking for an example of a continuous distribution function where the first moment does not exist but the characteristic function is differentiable everywhere. Cauchy distributions do not ...
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Properties of this set of functionals (mixed pairings)

(from the 4th page of http://www.math.toronto.edu/mccann/papers/econ.pdf) Let $X$ be a compact Hausdorff space, and let $\omega$ be a Borel probability measure on $X$. A Borel probability measure, ...
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Understanding certain parts of the proof of Helly's Selection Theorem

I have read through the following proof of Helly's Selection Theorem. There are just two parts, which I have highlighted, that are left for the reader to fill in, and I would like to know how to prove ...
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Average of IID Cauchy RVs

Suppose that $X_i$'s are iid Cauchy RV's with pdf $f_u (x) = \frac{1}{\pi} \frac{u}{u^2+x^2}$. I am aware that the RV $Y:=\frac{1}{N}\sum_{k=1}^N X_k$ has the same density as the $X_i$'s. I am trying ...
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Big O notation preserved under convex functions?

Suppose that the random variable $X_T$ is $O_p(1)$ as $T \rightarrow \infty$, i.e. $\forall \epsilon>0$, $\exists M_\epsilon>0$ such that $\mathbb{P}(X_T>M_\epsilon)<\epsilon$ $\forall T$. ...
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Addition corresponds to convolution and subtraction?

We know that if two random variables have proper densities, than the density of the sum of them is given by the convolution. But what can we say about the difference of two random variables? $X-Y$ ...
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Continuity of probability measure

Sorry, I just wanted to know whether I understand this correct. Let $(x_n)$ be an increasing sequence such that $x_n \rightarrow a$, then we have for the probability measure on an arbitrary ...
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Distribution of ceiling function and absolute value of random variable

Given a distribution function $f_X$, where $X$ is some random variable. I want to get the distribution functions of $|X|$ and $\lceil X \rceil$( the last one may only have an easy form if $X$ is ...
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Convergence in distribution ( Two equivalent definitions)

I read that for convergence in distribution it is equivalent to have that either the characteristic functions of the random variables convergence pointwise or we have that $F_{X_n} \rightarrow F_{X}$ ...
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Random variable exponentially distributed?

I just want to be sure about this: If I read the phrase ' a random variable is exponentially distributed'( which is often said in probability theory and then it is never explictely stated what $X$ ...
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Probability qual problem about Polya's criterion (I guess)

Suppose $\mu$ is non negative, $\sigma$-finite measure on $(0,\infty)$ so that $$c:=\int_0^\infty x\mu (dx)\in(0,\infty)$$ Let $$\phi(u):=\exp\left(\int(e^{iux}-1)\mu (dx)\right)$$ Prove that there ...
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When does equality in Markov's inequality occur?

Markov's inequality states that given any nonnegative random variable and $a>0$ then we have: $$P(X \geq a) \leq \frac{E(X)}{a}$$ At which $a$ is equality supposed to hold?
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Lebesgue integral of a bounded random variable

Given a random variable $X$, if we take a measurable and bounded function $f(X)$ then can we say that $f$ is Lebesgue integrable wrt a probability measure on $\mathbb R$? In Real Analyses book by ...
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Where is the error? Expectation, independent random variables

Let $X,Z$ be two correlated variables and $Y,Z\sim N(0,1)$ where Y is independent of $X,Z$. Consider the expectation: $$E[f(X,Y)Z].$$ If $f(X,Y)$ and $Z$ are independent then clearly ...
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Conditional Expectation given joint distribution

Given 2 random variables $X,Y$, is it possible to write conditional expectation $\mathbb{E}[X|Y]$ in terms of their joint distributional function $F_{X,Y}(x,y)$?
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A variance-mixture model

So I've tried to make a probability distribution which has a tunable degree of kurtosis and which becomes Gaussian if the control-parameter goes to 0. Now Levy-distributions are out of the question, ...
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Differentiability of a continuous (not absolutely continuous) CDF

Similarly to Cantors Distribution define a Cumulative Distribution Function $F$ on $[0,1]$ as follows: Let $\mu$ be the measure on $\{0,1,2\}$ with $\mu(\{0\})=2/5, \mu(\{1\})=1/5, \mu(\{2\})=2/5$. ...
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Is an average of power-law curves a power-law curve?

If I look at an exponential function, $p(t) = e^{-\mu t}$ where the parameter $\mu$ varies over a gamma distribution given by the density function $f(\mu) = \frac{1}{\Gamma(a)b^a}\mu^{a-1}e^{-\mu/b}$ ...
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A probability question on sum

Let $X_{1}$, $X_{2}$, $X_{3}$, $X_{4}$, $X_{5}$ and $X_{6}$ be real-valued random variables that have the same probability distribution with finite moments, and they are independent. Does anyone know ...
Let $X$, $X'$ be identically distributed (not necessarily iid) random variables with compact support, on the same probability space. Define $G_t(x):=\mathbb{E}[e^{t(X'-X)} | X=x]$ In other words a ...