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.
44,522
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Comparisons of Conditional Expectations [closed]
Suppose $X_1,X_2,...,X_m$ are $m$ independent random variables. Let $L(k)$ be the $k$ largest random variables among them. Do we always have $$E[X_i\mid X_i \in L(k)]\ge E[X_i\mid X_i \not \in L(k)]?$$...
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63
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Quadratic variation and elementary processes
The context here is the construction of the stochastic integral. In this setting, one defines elementary processes that can be written as:
$$
K_t=\sum_{i=0}^{k}X_{a_i}\mathbb{1}_{t \in (a_i,a_{i+1}]}
$...
2
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1
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51
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What is $q$ in $\mathbb {E}_q$ in KL divergence?
I am reading about $D_{KL}$ and I faced these equations which are confusing for me. Can someone explain what $q$ is doing as the index of expectation?
$$\log p(x)=\log \int_zp(x,z)=\log \int_z p(x,z) \...
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How to prove / deduce that moment of this expression is equal to moments of normal random variable.
Consider the following paragraph in notes of number theory:1
Can you please explain what are moments of $\frac{ \omega(n) -\log \log x}{ \sqrt{\log \log x}}$?
and how they become equal to moments of ...
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2
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83
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Probability that no two gloves next to each other are matching [duplicate]
Suppose you have k different pairs of gloves (k left gloves and k right gloves, for 2k gloves in total) in your chest. You take the gloves out of the chest one by one without looking and lay them out ...
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2
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63
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Exponential distribution related questions [closed]
Suppose that the inter-arrival times of male customers entering the bank are iid exponential random variables with $\frac{1}{\lambda_1}$ and for those females are iid exponential random variables with ...
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126
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Conditional expectation - a functional analytic proof of existence?
$
\newcommand{\E}{\mathbf{E}}
\newcommand{\pp}{\mathbb{P}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\scrF}{\mathscr{F}}
\newcommand{\scrG}{\mathscr{G}}
$Let $(\Omega,\scrF,\pp)$ be a probability space ...
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39
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How to calculate iterative independent event?
Say you iteratively go through an event that has 1/800 success rate and 799/800 failure rate. What's the probability that you draw successfully within the first 100 attempts, or generally x attempts? ...
2
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1
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240
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Alternative CLT for non-centered random variables
The central limit theorem for the i.i.d case states that for $X_k \sim \text{i.i.d.}(\mu, \sigma^2)$, we have that:
$$ \dfrac{\sum_{k=1}^{n} X_k - n \mu}{\sigma\sqrt n } \xrightarrow{d} \mathcal{N}(0,...
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Why is it necessary for density functions to be absolutely continuous with respect to a measure in order for the cross entropy to be defined?
In the Wikipedia page describing cross entropy, the following expression is written down to denote the cross entropy $H$ between two densities $p(x)$ and $q(x)$:
$H(p,q) = - \int_\mathcal{X}p(x)\log q(...
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Apply the Girsanov theorem, determine the stochastic dynamics of $S^{(1)}$ and determine the risk-neutral price of $X$
I'm not sure if I'm applying Change of Numeraire and Girsanov correctly in part c) and d). Also with the information I got, I don't know how to get a result for e).
Consider a financial market with 2 ...
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1
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233
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Dirac measures closed in Radon measures?
Let $M(X)$ be the space of Radon probability measures on a compact space $X$. Consider the subset $Y:= \{\delta_x: x \in X\}$ of point-measures (Dirac measures), explicitely defined by
$$\delta_x(A) = ...
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Almost everywhere convergence and $L^p$ probability
Let $(\Omega, \mathcal F, P)$ be a probability space. If $X_n \leq C$ for each $n\geq 1$ and $X_n \to X$ almost everywhere, prove that $X_n \to X$ in $L^p$.
So far I have just showed that $X, X_n \in ...
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82
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Infer mean of multivariate r.v. by observing biased subset of points
Say you have a multivariate normal r.v. $X$ with unknown mean $\mu_X \in \mathbb{R}^n$ and known covariance matrix $\Sigma_x \in\mathbb{R}^{n\times n}$. You want to estimate the mean $\mu_X$ but you ...
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688
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What is a deterministic random variable?
Let $X$ be a (real-valued) random variable on a probability space $(\Omega, \mathcal{F}, P)$ with
\begin{equation}
\text{Var}X = 0.
\end{equation}
Is it correct to call X deterministic?
I ...
4
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2
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Donsker and Varadhan inequality proof without absolute continuity assumption
I've been attempting to understand the proof of the Donsker-Varadhan dual form of the Kullback-Liebler divergence, as defined by
$$
\operatorname{KL}(\mu \| \lambda)
= \begin{cases}
\int_X \log\left(\...
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1
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75
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What am I doing wrong in the classing breaking the stick in 3 portions problem?
So I'm trying to solve the classic stick breaking problem, where it is given that we've a stick of length $9 units$ which is broken into 3 small sticks by putting two break points randomly.
What is ...
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A particular homogenous travel accident insurance portfolio consists of $10,000$ policies
A particular homogenous travel accident insurance portfolio consists of $10,000$ policies issued within a period of one year. The probability that a randomly chosen insured will require the ...
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Insurance statistics book
Am am looking for a book on insurance statistics.
The problem is that there are a lot of googleable variants that it is difficult to decide which one is good and which ones are better in good ones. ...
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0
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existence of sequence $\{a_n\}$ that make the tail probability decrease at a certain rate
This appeared in example (b), Section 8.8 of An Introduction to Probability Theory and Its Applications, Vol 2, by Feller.
Let $F(x)$ be the cumulative distribution function of some random variable.
...
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Bounding probability measure on arbitrary but finite intervals [closed]
Let $\mu$ be a probability measure supported on $\mathbb R$.
My question is the following:
For a given $\varepsilon \in (0,1)$, does there exist a finite number $F > 0$ such that $\mu([-F, F]) >...
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Word problem involving random variables confusion
Suppose a university is composed of $55\%$ female students and $45\%$ male students. A student is selected to complete a questionnaire. There are $25$ questions on the questionnaire administered to ...
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2
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74
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Why the almost sure convergence of the SLLN doesn't hold here
Let $X$ be a random variable that takes value $1$ with probability $\frac12$ and value $-1$ with probability $\frac12$. For every $i>1$, define the random variable $X_i=X$ with $\mathbb E\:X_i=0$ ...
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Is the product of two independent random distributions commutative?
Based on the formula in:
https://en.wikipedia.org/wiki/Product_distribution
It seems that Z = XY = YX. However, if X is a bounded uniform distribution, one could have either
$f_Y(z/x)$ or $f_Y(z)$, ...
0
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1
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42
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Expectation of squared error
Let $X \in \mathbb{R}^{m \times n}$ an observation data matrix, $A \in \mathbb{R}^{m \times k}, B \in \mathbb{R}^{k \times n}$ two random variable matrices
I want to calculate:
$$E[(X_{ij} - A_iB_j)^2]...
4
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1
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393
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Proof of Sampling/Importance Resampling (Weighted Bootstrap) technique
From Casella Berger exercise 5.65: Let us have $X \sim f$. Then, assume we produce $m$ i.i.d. random variables $Y_1,...,Y_m$ from another distribution $g$.
Let us have
$$q_i = \frac{\frac{f(Y_i)}{g(...
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$A\geq 0$ a.s., $E[A^a]<1,E[A^{a+d}]<\infty$. Show: $\exists e>0,g\in(0,1): \forall z\in [-e,e]:\mathbb{E}[(A/g)^{a+z}]<1$.
Assume a random variable $A\geq 0$ a.s., $\exists \: \alpha,\delta>0: \mathbb{E}[A^\alpha]<1, \: \mathbb{E}[A^{\alpha+\delta}]<\infty$. Show: $\exists \: \epsilon>0: \: \exists \: \gamma\...
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Reference for time-derivatives of measure-valued "processes"
The notion of the weak derivative of a process of probability measures $(\mu_t)_{t\geq 0}$ in optimal transport theory seems to be so basic that, e.g., in "Optimal Transport, old and new" ...
2
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242
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Hoeffding's inequality proof
I was reading proof of Hoeffding's inequality, I couldn't understand the last step. How does last step follows from proceeding one? I use that value of $s$ obtained but I couldn't reach the outcome ...
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1
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Probability of drawing the minimum within a group [closed]
Suppose there are n individuals each drawing a value from some distribution G(.). What is the probability of a given individual drawing the lowest value among the n people?
Note: I am not asking about ...
0
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1
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317
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Proving (or disproving) that $\mu_{X,Y}$ is a probability measure on $\mathbb{R}^2$
For context, this is an attempt of mine to find a solid definition of joint probability distribution using one measure space and two random variables with real values.
Let $(\Omega,\mathcal{A},\mu)$ ...
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108
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Understanding how to compute $P(W = w)$, given that $W = X + Y$
Suppose that a bowl contains $100$ chips: $30$ are labelled $1$, $20$ are labelled $2$, and $50$ are labelled $3$. The chips are thoroughly mixed, a chip is drawn, and the number $X$ on the chip is ...
4
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265
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The stability of a gradient flow (discrete JKO scheme, proximal point)
Define a free energy functional on the space of probability densities (on $\mathbb{R}^d$, denoted $\mathcal{P}(\mathbb{R}^n)$)
$$E(\rho):=\int_{\mathbb{R}^d} f(x) \rho(x) dx+\int_{\mathbb{R}^d} \rho(x)...
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4
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592
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Let $X∼U[0,1]$ and $Y=X^2$ then prove that $\mathbb E(Y)=\frac13$
Let $X∼U[0,1]$ and $Y=X^2$ then prove that $\mathbb E(Y)=\frac13$ directly from the definition of expectation, here is the definition of expectation,
If X only takes finitely many values $x_1,\cdots,...
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491
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Convolution of random variables - Bernoulli and Binomial
For n ∈ N, p ∈ (0, 1) let X ∼ Binom(n, p) and Y ∼ Ber(p) be two independent random variables.
a) Determine the values P(X + Y = k) for k ∈ N. What is the distribution of X + Y ? Does
this intuitively ...
3
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2
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292
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A problem about the Borel-Cantelli lemma in Feller's Introduction to Probability
The problem is from Chapter 8 of the book, and it states the following.
"In a sequence of Bernoulli trials let $A_n$ be the event that a run of $n$ consecuitive successes ocurrs between the $2^n$...
3
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0
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Cumulative distribution function leftside limit
Define $F(a-)=\lim _{x\uparrow a}F(x)$. Then, if $F$ is nondecreasing, $F(a-)=\lim _{n\rightarrow\infty} F(a-1/n)$. Use (1.1) to show that if a random variable $X$ has cumulative distribution function ...
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Why non-zero finite variance in chebyshev inequality.
I read Chebyshev's inequality from certain places. The statement is:
Let X be a random variable with a finite mean denoted as $\mu$ and a finite non-zero variance, which is denoted as $\sigma^2$, for ...
6
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1
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Why is the drift of the stock price not important for options pricing?
This question is motivated by MSE question 4199364: Bachelier model option pricing.
There, one considers the price of a stock depending on time $t$, given by the family of random variables $(S_t)_{t\...
1
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1
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49
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Understanding and generalizing $P(X \in B)$ for any subset $B$ of $\mathbb{R}$
Suppose we flip three fair coins, and let $X$ be the number of heads showing. Write a formula for $P(X \in B)$, for any subset $B$ of the real numbers.
I first started by computing $P(X = x)$ for any $...
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1
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486
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Joint and conditional of a distribution with a discrete and continuous random variable.
Problem Setup
Suppose we have two random variables $X\sim D_X$ and $Y\sim D_Y$ where $X$ is continuous with domain [0,1] and $Y$ is discrete with domain $\{0, 1\}$. Further suppose these variables ...
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1
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If X is a random variable that only takes the values 0 or 1, does that guarantee X is an indicator function?
If we suppose that $X$ is a random variable that takes only the values $0$ or $1$. Must $X$ be an indicator function?
Would it be accurate to say that the answer to this question be a yes since this ...
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0
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Almost sure convergence of arithmetic mean of scaled sums of random variables
Let $S_n := \sum_{i=1}^n X_i$, where $X_i$ are mean $0$ variance $1$ i.i.d random variables. By Donsker's theorem, we know that
$$ \frac{1}{N} \sum_{n=1}^N f \left( \frac{S_n}{\sqrt{n}} \right) \to \...
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0
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45
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If the characteristic function of a distribution is twice differentiable, then will the density function be continuous?
This is from a probability subject.
I'm trying to relate how the differentiability of the characteristic function of a random variable X relates to the density being continuous.
The question ...
1
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0
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184
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Optimization: max to softmax for convexity?
Assume we have the following optimization problem: for a family of $m$ vectors $\{x_i\}\in \mathbb{R}^n$, a family of $l$ vectors $\{c_i\}\in \mathbb{R}^n$ with $l\ll m$ and for a family of $l$ ...
0
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1
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25
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Functions of discrete and continuous RV's
I was just wondering, in the context of analytical probability, if we have a discrete RV X and any function f (which maps from R to R), will Y = f(X) be a discrete RV as well?
How about the continuous ...
1
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0
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36
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Finding the probability of an element given the fact that $P([0, \frac{8}{4+n}]) = \frac{2+e^{-n}}{6}$
How can I go about finding the probability of the element $\{0\}$ (i.e., $P(\{0\})$ ) being that given $P([0, \frac{8}{4+n}]) = \frac{2+e^{-n}}{6}$ for all $n \in 1,2, 3, ....$ using the continuity of ...
0
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0
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53
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Indifference with utility function
Let u be the utility function $u(x)=-\frac{x^{-\eta}-1}{\eta},$ with $x,\eta>0$
Assume that an investor is indifferent between an investment with riskless outcome of 101.005 and a stochastic ...
0
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1
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47
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Proving with Cauchy Schwarz that $\sum_{j=1}^{\infty}w_j\phi_j(t)$ converges if $\sum_j w^2_j/\lambda_j<\infty$
Suppose that the covariance kernel $K$ of the Gaussian process W={W_t:t\in T}, defined on the probability space $(\Omega,\mathscr{U},P)$, can be written in the form $$K(s,t)=\sum_{j=1}^{\infty}\...
-1
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2
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278
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$\int_0^{+ \infty} \dfrac{1}{ x^{ 1 + \alpha}} e^{ - \lambda x} = \lambda^{\alpha} \Gamma( - \alpha)$? [closed]
$\lambda >0$
$\alpha <0$
$ h(x)= \dfrac{1}{ x^{ 1 + \alpha}} e^{ - \lambda x}$
We want to compute $\int_0^{+ \infty} h(x)$
The result is of interest for the study of a particular Poisson ...