Linked Questions
15 questions linked to/from Mean and variance of the reciprocal of a random variable
1
vote
1
answer
937
views
The mean and variance of the inverse of a normal distribution [duplicate]
I would want to ask if I have a random variable $A \sim N(b,c)$ then what is the distribution of the inverse of $A$?
- 11
0
votes
1
answer
116
views
Expectation of the reciprocal of a standard normal random variable [duplicate]
If $\mathbf{X} \sim_{iid} \mathcal{N}(\mu, 1)$ then we know that the sample mean $\bar{X} \sim \mathcal{N}(\mu, 1/n)$, how would we show that $$\mathbf{E}\left(\frac{1}{\bar{X}}\right) = \infty $$ and ...
- 31
0
votes
0
answers
32
views
What is the standard deviation and mean of the reciprocal of normal distribution in terms of that of the normal distribution? [duplicate]
What is the standard deviation and mean of the reciprocal of normal distribution in terms of the standard deviation and mean of the normal distribution?
- 143
51
votes
4
answers
12k
views
Taking the expectation of Taylor series (especially the remainder)
My question concerns trying to justify a widely-used method, namely taking the expected value of Taylor Series. Assume we have a random variable $X$ with positive mean $\mu$ and variance $\sigma^2$. ...
- 695
35
votes
3
answers
4k
views
I've heard that ratios or inverses of random variables often are problematic, in not having expectations. Why is that?
The title is the question. I am told that ratios and inverses of random variables often are problematic. What is meant is that expectation often do not exist. Is there a simple, general explication of ...
- 82.8k
18
votes
2
answers
43k
views
Var(X) is known, how to calculate Var(1/X)?
If I have only $\mathrm{Var}(X)$, how can I calculate $\mathrm{Var}(\frac{1}{X})$?
I do not have any information about the distribution of $X$, so I cannot use transformation, or any other methods ...
- 690
3
votes
3
answers
3k
views
What's the distribution of $\bar{X}^{-1}$?
What's the distribution of $\bar{X}^{-1}$ with X being a continuous iid random variable that is uniformly distributed? Can I use the CLT here?
- 2,244
5
votes
1
answer
1k
views
Covariance of two variables that are products of shared random variables
How to analytically express cov(X,Y), when:
X=C*A/(A+B)
and
Y=C*B/(A+B)
Here C, A and B are independent variables with ...
- 151
3
votes
1
answer
480
views
Sampling Distribution of Reciprocal of Sample Mean
Given $X_1, X_2,..., X_n $ i.i.d. random variables.
$E[X_i] = \mu_1 \in \mathbb{R} $ $\&$ $ V[X_i] = \sigma_1^2 \in \mathbb{R}^+$ $\forall i \in \{1,2,3,...,n\}$.
The statistics $\bar{X} = \frac{...
user340483
3
votes
0
answers
726
views
Bound for the variance of 1/X, where X is a Gaussian RV
Consider the following problem:
Let $X \sim N(\mu, \sigma^2)$ and assume that $|\mu| \gg \sigma^2$. Then, we construct a new random variable $Y = 1/X$ with pdf
$$f_Y (y) = \frac{1}{\sqrt{2 \pi} \...
- 131
0
votes
0
answers
410
views
Non-central correlated normal ratio - distribution of the ratio of two dependent normally distributed variables
I am currently trying to solve a problem in the context of a Bayesian analysis that concerns normal distributions. The situation is as follows.
I have an equation that looks like this, where I know ...
- 1
2
votes
1
answer
91
views
A follow-up to 'The meaning of an analyt. result concerning the… mean of the square of a reciprocal of a norm. distrib. rand. variable'
This question concerns the same subject matter as this previous question of mine. However, a moderator felt that the questions I posed there are significantly different from the question I am about ...
- 171
0
votes
1
answer
181
views
Expectation of inverse of normal RV, given that it is below a certain value
I have a normal random variable $X$ with mean $\mu$ and variance $\sigma^2$. Any advice on how to compute the conditional expectation $E[\frac{1}{X}|X \leq T]$ where $T$ is a positive constant?
- 1
3
votes
1
answer
101
views
The meaning of an analytical result concerning the (formally nonexistent) mean of the square of a reciprocal of a normally distributed random variable
This question arose as I was writing this answer to this question.
Let $X$ be normally distributed with mean $\mu$ and standard deviation $\sigma$, and let $Y=1/X$.
First, note that the integral ...
- 171
0
votes
0
answers
35
views
How to find the variance for $\frac{\sum Y}{\sum X}$ when X and Y are both independent and normal random variables
The original problem is:
Given $Y_i = \beta X_i + \epsilon_i$, $i=1,2,...,n$, where $X \sim N(\mu, \tau^2)$ iid and $\epsilon \sim N(0, \sigma^2)$ iid, $X$ and $\epsilon$ are independent. What is the ...
- 1