Linked Questions

2
votes
1answer
31 views

Expected value of E(exp(1/X)) where X~N(0,sigma_x) [duplicate]

I would like to obtain the expected value of $ \mathbb{E}(\exp(1/X))$ where $X$ ~ $N(0,\sigma_x)$
1
vote
0answers
40 views

For some RV's $X$ with continuous density on $[0,1]$, is $\mathbb{E}|1/X|=\infty$? [duplicate]

Recently I tried to give an argument for why, if $Z \sim N(0,1)$ denotes the standard normal, $\mathbb{E}|\frac{1}{Z}| = \infty$. However, in hindsight this argument does not seem to rely on many ...
11
votes
3answers
4k views

Mean of inverse exponential distribution

Given a random variable $Y = Exp(\lambda)$, what is the mean and variance of $G=\dfrac{1}{Y}$ ? I look at the Inverse Gamma Distribution, but the mean and variance are only defined for $\alpha>1$ ...
2
votes
1answer
4k views

Expected value of product of dependent random variables

$X_1$ and $X_2$ are two independent random variables whose expected values are known. I am trying to find the expected value of $(X_1/(X_1+X_2))$. Since $X_1$ and $X_1+X_2$ are dependent, I tried to ...
2
votes
2answers
455 views

Variance of the reciprocal of a strictly positive random variable

In this post it is stated that due to Jensen's inequality the expected value of the reciprocal of a strictly postive random variable $X$ will satisfy: $$\mathbb{E}\left[\frac{1}{X}\right] \geq \frac{...
2
votes
2answers
360 views

What is the ratio of a N[0,1] and U[-1/2,1/2]?

I have come across a problem where I can reasonably assume that the numerator is a uniform distribution of the type U[-a,a], i.e., centered on zero, and the denominator is N[0,b]. This seems to be ...
2
votes
1answer
1k views

Expectation of a ratio of random variables

Suppose $X$~$N(\mu,\sigma^2)$. Is the following statement correct: $\mathbb{E}\left(\frac{X}{X^2}\right)$ $\ne$ $\frac{\mathbb{E}(X)}{\mathbb{E}(X^2)}$?
6
votes
1answer
920 views

Analysing ratios of variables

Ratios (X/Y, e.g. body mass index) are variables with odd distributions. They have no means or moments (e.g. variances, skewness or kurtosis). Thus, my questions are: How to compare between (or among)...
12
votes
2answers
355 views

Ratio of sum of Normal to sum of cubes of Normal

Please help me to find the limiting distribution (as $n \rightarrow \infty$) of the following: $$ U_n = \frac{X_1 + X_2 + \ldots + X_n}{X_1^3 + X_2^3 + \ldots X_n^3},$$ where $X_i$ are iid $N(0,1)$.
6
votes
2answers
320 views

Expectation of inverse of sum of positive iid variables

Let $(X_i)_{i}$ be a sequence of iid positive variables of mean 1 and variance $\sigma^2$. Let $\bar{X}_n = \frac{\sum_{i=1}^n X_i}{n}$. My question is: Can we can bound $\mathbb{E}(1/\bar{X}_n)$ as ...
1
vote
1answer
221 views

Show that Y/Z does not have finite expectation

The unit interval (0, 1) is divided into two sub-intervals by picking a point at random from inside the interval. Denoting by Y and Z the lengths of the longer and the shorter sub-intervals ...
1
vote
1answer
365 views

Calculating the covariance between 2 ratios (random variables)

I am a little stuck with my project. In the calculations of my project, I need to calculate the spread of some random variables. Up to now, there was no special difficulty to analytically calculate ...
3
votes
2answers
138 views

How to compute the variances of quotients of normal variables?

$x$ and $y$ are independent normal random variables. $z1=x/(x+y)$ and $z2=y/(x+y).$ How to obtain the variances for $z1$ and $z2$? I understand through the delta method, $\operatorname{Var}(z1)$ ...
7
votes
1answer
97 views

Expected root of quadratic random polynomial

Suppose $A,B,C$ are i.i.d. random variables with uniform distribution on $[-1,1]$. I'm interested in the expected roots of the polynomial $Ax^2 + Bx + C$, which are complex random variables given by $$...
3
votes
1answer
57 views

Expression for $\mathbb{E}(\exp( ab X Y))$ when $X \sim \text{N}(0, \sigma_X^2)$ and $Y \sim \text{N}(0, \sigma_Y^2)$

Let $X \sim \text{N}(0, \sigma_X^2)$ and $Y \sim \text{N}(0, \sigma_Y^2)$ be independent normal random variables with zero mean, but (possibly) different variances. Given some constants $a$ and $b$, ...

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