All Questions
Tagged with expected-value integral
64
questions
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51
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Find $E[Y]$ when $f(x,y) = \frac{x}{3}e^{-xy}$
Truth be told, I don't really have an issue with this problem in general, but in it's calculation. Let me explain.
We need to find $E[Y]$ when $f(x,y) = \frac{x}{3}e^{-xy}$, $1<x<4$ and $y>0$...
- 195
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0
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20
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Question on the proof step in the theorem 1 of the Gap statistic paper
From the Gap statistic paper, during the proof for the theorem 1, we can see the below equality (p. 422),
$\begin{aligned} \operatorname{var}(X) & =\frac{1}{2} \int_{-\infty}^{\infty} \int_{-\...
- 303
4
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1
answer
238
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Rewriting the expectation of f(x) by means of its derivative
I have a question regarding this proposition.
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an a.e. differentiable function so that $\int \frac{\left|f^{\prime}(x)\right|}{(1+|x|)^s} d x<\infty$ ($...
- 329
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61
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How to numerically get expectation of a non-linear function of a normally distributed random variable
I'm trying to calculate the expectation of the following numerically:
$$\mathbb{E}[V(\theta)]$$ where $\theta\sim N(\mu,\sigma^2)$ and $V(\theta)$ is strictly increasing.
I'm struggling to understand ...
- 213
5
votes
1
answer
173
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How to deduce $ \mathbb{E}(\sqrt{X}) < \infty \implies\int_{\mathbb{R}^+} (1 - F(x))^2 dx < \infty,~X$ being a non-negative integrable rv?
Let $X$ be non-negative random variable and $F$ be its distribution function. Prove the following implications:
$$
\mathbb{E}(X) < \infty \Longrightarrow \mathbb{E}(\sqrt{X}) < \infty \...
- 185
1
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0
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48
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Expectation of a Compound Poisson Distribution
I am trying to understand the proof of Theorem 16.14 of Probability Theory by A. Klenke (3rd version) about the Levy-Khinchin formula.
I would like to know how to prove this:
$$E[X]=\int x e^{-v(\...
- 211
1
vote
1
answer
63
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How could I evaluate $A = \int_0^1 \log\left(\theta^s(1-\theta)^{n-s}\right)p(\theta)d\theta$?
Suppose that I want to evaluate the following integral:
$$A = \int_0^1 \log\left(\theta^s(1-\theta)^{n-s}\right)p(\theta)d\theta,$$
where $p(\theta)\equiv$ Beta$(ws+1, w(n-s)+1)$ and $n$, $w$, and $s$ ...
- 2,103
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50
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Expectation of squared integral over distribution
Right now, I have some function $g(x,\theta)$ where the expectation this function at a given $x$ evaluated over $\theta$ $E_\theta[g(x,\theta)]$ is known, and I want to upper bound (or compute if at ...
- 131
3
votes
1
answer
161
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Calculate expected values E(x) & E(y) & variance of x & y of joint PDF, which was previously transformed from Polar to Cartesian
Given two independently uniform distributed random variables angle $\theta \in [0,2\pi]$ and radius $r \in [0,1]$.
I obtain for the joint density function with polar coordinates: $$ f_{r,\theta}(r,\...
- 33
1
vote
0
answers
127
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Compute Expectation of Product Using Joint Survival Function
I know that, for a nonnegative random variable $X$,
$$ E[X] = \int x dF(x) = \int S(x) dx$$
where $F(x)$ and $S(x)$ are the CDF and survival function of $X$, respectively.
This was derived using ...
- 225
5
votes
1
answer
143
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For a general multivariate normally distributed $\boldsymbol{X}$, what is the expectation of $1/(\boldsymbol{X}^T \boldsymbol{X})$
For $\boldsymbol{X} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$, where $\boldsymbol{\mu} \in \mathbb{R}^N$, $\boldsymbol{\Sigma} \in \mathbb{R}^{N \times N}$ is positive definite, how to ...
- 203
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35
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Expected value of 1/(1−X) of a Gamma distribution [duplicate]
I was interested in calculating $E\left(\dfrac{1}{1-X}\right)$ where $X\sim$ Gamma ($n,\lambda$), but I wasn't able to solve the associated integral using standard integration techniques.
$$E\left(\...
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1
answer
55
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A proof related to an expected value (revised)
I uploaded a question asking how to proof an equation.
But, I felt that I made some confusions, and I will ask the question in a more tidy form with details.
Suppose that $X \sim N(0,c)$. That is, $X$ ...
- 935
0
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1
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211
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Integrating with considering two indicator function
Consider exponential random variables $X$, $Y$, and $Z$ with $\lambda_x$, $\lambda_y$, and $\lambda_z$, respectively. Now I want to calculate the following integration:
$$E[X1_{\{X<Y\}}1_{\{X<Z\}...
- 123
3
votes
1
answer
67
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$\int_{-\infty}^{\infty} x^3 f(x) dx < \infty$ then $Cov(X,X)<\infty$ ? TRUE OR FALSE
$x \in R$ is a continuous random variable.
Is the statement : IF $\int_{-\infty}^{\infty} x^3 f(x) dx < \infty$ then: $Cov(X,X)<\infty$ .TRUE?
My thought was that Var(x)=Cov(x,x) , so $Var(x)=...
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