1
vote
0answers
16 views

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 ...
0
votes
2answers
34 views

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 ...
1
vote
1answer
60 views

Probability of event in normal distribution

Let $X$ be a random variable that is normally distributed and $X_1,\ldots,X_n$ be (independet) copies of $X$, then we can estimate this probability by using a simple Monte-Carlo estimator: $p := P (X ...
0
votes
0answers
25 views

Change of variables in calculating the integral of multivariable differential entropy

I know that for one dimensional differential entropy of a density function $p(x)$, one has the following formula by change of variables: $$H(p)=\int ...
0
votes
1answer
29 views

Is this a Markov chain property

For $A,B$ measurable sets and $(X_n)_n$ a Markov chain. Do any of the following properties hold? $$P(X_2 \in B | X_1=x_1,X_0 \in A) = P(X_2 \in B|X_1=x_1)$$ or $$P(X_2 \in B|X_1 \in A,X_0=x_0) = ...
0
votes
0answers
23 views

Solution to a certain moment problem

I'm looking for a function $f$ that satisfies $f(x)\geq0$ $\int f(x) \mathrm{d}x=1$ $\int xf(x) \mathrm{d}x=0$ $\int x^2f(x)\mathrm{d}x=1$ $\int x^4f(x)\mathrm{d}x=\delta$ $\int ...
1
vote
3answers
34 views

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 ...
0
votes
1answer
36 views

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 ...
5
votes
1answer
136 views

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 ...
1
vote
0answers
25 views

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, ...
0
votes
0answers
20 views

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 ...
0
votes
1answer
40 views

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 ...
2
votes
1answer
30 views

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 $\mathbb{P}(X_T>M_\epsilon)<\epsilon$ $\forall T$. ...
1
vote
1answer
32 views

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? $X-Y$ ...
0
votes
1answer
37 views

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 ...
0
votes
1answer
68 views

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 ...
2
votes
1answer
131 views

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 $F_{X_n} \rightarrow F_{X}$ ...
3
votes
1answer
87 views

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 $X$ ...
0
votes
1answer
32 views

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 ...
0
votes
2answers
125 views

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?
1
vote
1answer
81 views

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 ...
1
vote
1answer
46 views

locally linearize a CDF

I have a sequence of discrete CDF's that converge to continuous CDF. Assume we call it $F_n(x)$. If say at some point, say $R$, $F_n$ is differentiable, then we can write $F_n(R+\xi) \approx ...
1
vote
1answer
81 views

Variance of this probability density

I have the function $\rho(x) = \frac{sin^2(x)}{x^2}$ and I want to calculate its variance on $\mathbb{R}$. Does anybody know how to do this? Cause afaik the integral does not converge.
2
votes
0answers
86 views

Quantifying the “flatness” of functions which are the Fourier transforms of positive functions

I have a question which I admit is a little cumbersome for me to try to state succinctly, and which I fear may not have a simple answer, but I figured I'd give it a shot. In broad terms, I'm trying to ...
1
vote
1answer
77 views

Limiting distribution for $X_A$~ Beta distribution on [0, A] as A $\rightarrow \infty$ holding $E[X_A], Var[X_A]$ Constant

I am trying to determine the limiting form of a beta distribution as its range expands under isoparametric constraints on its first two moments. For reference $X_A \sim Beta(0,A,\alpha,\beta) = ...
0
votes
1answer
43 views

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 ...
0
votes
2answers
161 views

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 $F_{X,Y}(x,y)$?
2
votes
1answer
78 views

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, ...
2
votes
0answers
106 views

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 $\mu(\{0\})=2/5, \mu(\{1\})=1/5, \mu(\{2\})=2/5$. ...
2
votes
1answer
77 views

Prove expectation inequality

Any ideas on how I could prove the veracity or falseness of the following inequality? Let $X:\Omega \to \mathbb{R}$ a random variable such that the expressions under are well-defined. Then $$E[e^X] ...
3
votes
2answers
205 views

Expectation of composition of functions with density as R-N derivative

In prior probability courses, I've always seen and used the fact that, for a continuous random variable X and a function $\phi$, $E[\phi(X)]=\int_{ \mathbb{R}}\phi(x) f_X(x)dx,$ but I cannot find a ...
0
votes
1answer
222 views

The space of probability measures or probability distributions

I am noticing some probability theory notions like the space of Borel probability measures on some specific metric spaces. But then I cannot quite understand the reason of defining different kinds of ...
1
vote
0answers
72 views

Integrals of derivatives of normal distribution multiplied by polynomial?

Is there anything in the literature related to obtaining bounds of integrals of the form: $$\int_{\mathbb{R}} |P^{(k)}(t,z-z_0)|dz\leq \mbox{some function of t and }z_0$$ where $P(t,z)$ the density of ...
1
vote
1answer
105 views

Non-centered Gaussian moments

I would like to find a (closed nice) expression for the non-centered Gaussian moments with mean $\mu$ and variance $\sigma$. In found something in wikipedia: ...
2
votes
0answers
51 views

Two dimensional distribution

Let $F$ be a two variable continous function that satisfy: if $x_1\leq x_2$ and $y_1\leq y_2$ then \begin{equation} F(x_2,y_2)-F(x_2,y_1)-F(x_1,y_2)+F(x_1,y_1)\geq 0. \end{equation} Define ...
4
votes
0answers
240 views

An absolutely continuous cumulative distribution function that fails to have a Riemann-integrable pdf.

We know that if a real-valued random variable $ X $ on a probability space has an absolutely continuous cumulative distribution function (cdf) $ F $, then $ X $ possesses a probability density ...
0
votes
1answer
32 views

Extrema of the Gauß function

Let $\mu\in\mathbb R$ and $\sigma >0$. Determine the extrema of the Gauß function $\varphi$ and explain whether they are minima or maxima where $$ \varphi:\mathbb R\to \mathbb R,\qquad ...
3
votes
2answers
99 views

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 $f(\mu) = \frac{1}{\Gamma(a)b^a}\mu^{a-1}e^{-\mu/b}$ ...
1
vote
2answers
81 views

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 ...
3
votes
1answer
495 views

Conditional Moment Generating Function With A Twist

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 ...
1
vote
0answers
58 views

Equation Involving Bilateral Laplace Transform

Assume that $f(x,y)$ is a compactly supported, joint probability density function on $\mathbb{R}^2$ and nice enough for the following function to make sense: $$P_t(y):=e^{ty}-\int_{-\infty}^\infty ...
1
vote
0answers
57 views

A correction to confidence interval.

I have set of random values with the same distribution $y_1, \ldots, y_N$ , $N = mN_1$. $ m \ge 4$, $N_1$ is big enougth( $\approx 1000$ ). I want to to estimeat $E(x)$. How I do it: I make $m$ ...
7
votes
1answer
977 views

Limit using Poisson distribution [duplicate]

Show using the Poisson distribution that $$\lim_{n \to +\infty} e^{-n} \sum_{k=1}^{n}\frac{n^k}{k!} = \frac {1}{2}$$
3
votes
2answers
81 views

Unique continued fraction

If $x$ is a uniformly random number in $[0,1]$, what distribution should the $n$-th term in its continued fraction expansion follow? What is the expected vale of $a_n$ in $[a_0;a_1,a_2,\dots]$? Here ...
1
vote
1answer
229 views

Convergence of second/higher central differences to derivative?

If I only know, that the second derivative $f''(x)$ of f at some point x exists (but not necessarily anywhere else) and nothing more (except what follows directly from that, or is implied such ...
2
votes
2answers
2k views

How Binomial and Normal distributions approximate Poisson distribution respectively?

From Wikipedia: In some cases, the cdf of the Poisson distribution is the limit of the cdf of the binomial distribution: The Poisson distribution can be derived as a limiting case to the ...
4
votes
0answers
212 views

Equivalence of two sequences

I'm having some trouble showing that two things I really want to be the same are in fact the same. I want to show that these two sequences are, in fact, the same thing: $$a_0=1,a_1=-1, ...