All Questions
Tagged with logarithm expected-value
13
questions
3
votes
1
answer
66
views
Reducing Variance in Estimating the Exponential Average of Random Variables
Imagine we have a random variable called X, and the function form of the probability density for X is unknown. Now, I'm interested in finding the average value of the exponential of X, denoted as E[...
- 131
9
votes
2
answers
730
views
Mean of the log and variance of the log
I am struggling to derive the following identities:
$$
\mathbb{E}[\log Z]=2\log(\mathbb{E}[Z])-\frac12\log(\mathbb{E}[Z^2])
$$
$$
\mathrm{Var}[\log Z]=\log(\mathbb{E}[Z^2])-2\log(\mathbb{E}[Z])
$$
...
- 245
3
votes
1
answer
175
views
E(log(x)) to E(x) [duplicate]
Sorry if this is a straightforward question, but I have tried digging into econometrics book and cannot find anything about it. I worked on a model with ...
- 41
0
votes
1
answer
491
views
Expectation of log skew normal distribution
What is the expected value and expected variance of a log skew normal distribution?
In case I have the terminology wrong, I'm referring to data that is lognormal with some skew mild skew when it's log ...
- 372
3
votes
2
answers
2k
views
How to derive the expectation of $\ln \mu_j$ in Dirichlet distribution
I have derived the mean and variance of $\mu_j$ in Dirichlet distribution $\text{Dir}(\mu_1, \cdots, \mu_K|\alpha_1, \cdots, \alpha_k)$.
On https://en.wikipedia.org/wiki/Dirichlet_distribution, it ...
- 1,178
7
votes
1
answer
1k
views
What is the expected value of x log(x) of the gamma distribution?
Let $w(x) = x \log{x}$
$x \sim Gamma(\alpha = 3.7, \lambda = 1)$
Find $E[w(x)]$
I have set up the following integral:
$\int_0^{\infty} x\log{x} \frac{\lambda^{\alpha}}{\Gamma(\alpha)} x^{\alpha -1}...
- 73
3
votes
0
answers
349
views
Why do we use the log-derivative trick before Monte Carlo?
I still don't understand how we can approximate the gradient of an expected value... Indeed it's impossible to sample points and then to average the gradients of them as we have only samples... (How ...
- 399
5
votes
2
answers
591
views
Why is the expected gradient of a density not parallel to the expected gradient of the log density?
I'm confused by a seemingly counter-intuitive property of the interaction between distributions, log transforms, expectations and gradients.
Suppose I have some distribution over random variable $x$ ...
- 1,047
2
votes
2
answers
176
views
Is $\ln\{E[f(x)]\}$ equal to $E\{\ln[f(x)]\}$? [duplicate]
Is the logarithm of an expectation the same as the expectation of the logarithm?
3
votes
0
answers
574
views
Expectation of the log of a ratio of two lognormal random variables with additive constants
I have two independent lognormal random variables $X$ and $Y$ with known means and variances. I would like to know the expected value and the variance of $\ln\big((X+1)/(Y+1)\big)$. If there is no ...
- 41
1
vote
0
answers
65
views
Is this expectation known?
I'm wondering whether the following expectation has a known form:
$$\text{E}_{\beta}\left\{ \log \Phi(x^T \beta)\right\},$$ where $\Phi$ is the standard normal distribution function, $x$ is a fixed ...
- 405
10
votes
1
answer
508
views
If I have the expected value of the logarithm of a RV, can I obtain the expected value of the RV itself?
Assume that $\text{E}[\log(X)] $ is given, can I derive $\text{E}[X]$ in a closed form format?
- 407
34
votes
4
answers
77k
views
Expected value of a natural logarithm
I know $E(aX+b) = aE(X)+b$ with $a,b $ constants, so given $E(X)$, it's easy to solve. I also know that you can't apply that when its a nonlinear function, like in this case $E(1/X) \neq 1/E(X)$, and ...
- 341