All Questions
Tagged with expected-value moments
71
questions
0
votes
0
answers
16
views
Second moment of weighted average of random variables
I stumbled upon problem 254 from the SOA Exam P list in
https://www.soa.org/globalassets/assets/Files/Edu/edu-exam-p-sample-quest.pdf
for which I am puzzled by the solution described in
https://www....
- 1
1
vote
1
answer
80
views
properties of a expectation for a non-negative random variable
Say I have a non-negative discrete random variable $X$ (values of $X$ can be mapped to integers $(0, 2^n -1)$ for $n \in \mathbb{Z}$) and an associated distribution $P(X)$. Given a non-negative scalar ...
- 13
5
votes
1
answer
198
views
expectation value, distribution function and the central limit theorem
The problem goes thus:
${\{X_n\}}$ is an $iid$ sequence of random variables with mean 0 and variance $\sigma^2$. If the third moment is finite, show that $$\lim_{n \to \infty} \mathbb{E} \left(\left(...
- 153
0
votes
0
answers
17
views
Taking expectation of higher moments using original distribution [duplicate]
If we want to calculate $\mathbb{E}[X]$, we know this is $\int_{-\infty}^{\infty} x f_X(x) dx$.
When we want to calculate $\mathbb{E}[X^n]$ (for, say, positive integer $n$), we also do this with ...
- 111
2
votes
1
answer
116
views
Moments of sum of squares of independent gaussians $X_i \sim \mathcal{N}(\mu_i,\sigma^2_i)$, or $||X||^2$
Say that we have $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$. Is there some formula to calculate analytically the expected value of the sum $S = \sum_i^n X_i^2$?. This is equivalent to computing $\...
- 1,328
3
votes
1
answer
89
views
What is the fourth moment of a Euclidean Norm?
Let $X=\lVert M^\top p\rVert_2$, where $M$ is an $n\times n$ non-random matrix and $p\sim N(0,I_{n\times n})$ is an $n\times 1$ vector, and$\lVert \cdot\rVert_2$ is the Euclidean norm.
Using some ...
- 1,226
0
votes
0
answers
39
views
Given two rvs $X$ and $Y$, if $X Y = Z$, is it possible to change the mean and sd of $X$ without changing the mean and sd of $Y$ and $Z$
I have two lognormal rvs $X$ and $Y$, and a third rv $Z$ which is the product of the former two. I know the mean and standard deviation of the three.
Is it possible to calculate an alternative pair of ...
- 113
0
votes
0
answers
54
views
Variance of powers of a standard normal random variable
To predict growth of money in a stock market I try to calculate expected return over a longer timeframe (e.g. 30 years) with a confidence interval. The simple math of taking an average stock market ...
- 139
0
votes
0
answers
30
views
Expectations with respect to affine transformation of a log-normal distribution
Let $X$ be a log-normal distribution and consider $Y=aX+b$ for some $a,b>0$.
I would like to know if one can compute
$$\mathbb{E}[\log(Y)]$$
This would be very easy if it was $b=0$, since in this ...
1
vote
1
answer
60
views
Convergence of moment of functional of random variable
Define $X_n$ a continuous random variable that converges in distribution to $X$. Morever, we know that $E[|X_n|^p] \rightarrow E[|X|^p]$ for some $p > 0$.
Then, could we prove that for any ...
- 329
6
votes
2
answers
494
views
Absolute third central moment for standard distributions reference
I have to write an R function that computes the absolute third central moment (i.e. $\mathbb{E}[|X-\mathbb{E}[X]|^3]$) in the cases that you are given the name of ...
- 163
2
votes
2
answers
58
views
Possible typo in discussion of moments of a random variable
I'm struggling to understand some notation in this excerpt from Larsen & Marx. Under "Comment" j is defined as ...
- 171
4
votes
2
answers
453
views
Expected squared dot product between IID Gaussian vectors?
Suppose $x,y$ are IID samples from a Gaussian distribution in $\mathbb{R}^d$. The following seems true:
$$2\ \mathbb{E}\left[\langle x, y\rangle^2\right] = \mathbb{E}\left[\|x\|^4\right]-\mathbb{E}\...
- 6,307
2
votes
2
answers
163
views
What is $E\left[\frac{X^2}{X^2+Y^2}\right]$ if $X$ and $Y$ are normally distributed but not iid.?
I assume that X and Y are normally distributed with individual mean and variance. So far, I have found that an analytic expression exists for $E[X^2+Y^2]$, $E[X^2*Y^2]$ and $E[X^2*(X^2+Y^2)]$, all ...
- 310
3
votes
1
answer
237
views
Moments of inverse of a non-central chi distributed variable
I have a non-central chi variable $r$ with the distribution,
\begin{align}
p(r) = \frac{r^3\lambda}{(\lambda r)^{3/2}}\exp\left[-0.5(r^2 + \lambda^2)\right]I_{1/2}(r\lambda)
\end{align}
I'm looking ...
