Questions tagged [expected-value]
The expected value of a random variable is a weighted average of all possible values a random variable can take on, with the weights equal to the probability of taking on that value.
2,344
questions
72
votes
7
answers
234k
views
Find expected value using CDF
I'm trying to solve the following problem:
Let $X$ have the CDF $F(x) = 1 - x^{-\alpha}, x\ge1$.
Find $E(X)$ for those values of $\alpha$ for which $E(X)$ exists.
How can I determine which values of ...
- 947
4
votes
1
answer
270
views
Meaning of this expectation equation?
I was actually looking at this problem on slide 12. I will write it here briefly:
Problem:
Unknown number of people arriving in a fixed time period and my goal is to maximize my probability of ...
- 4,542
4
votes
2
answers
228
views
Systems for symbolically solving expectations?
Do there exist any systems for symbolically solving expectations?
This is sort of a follow-up to my question List of Tricks for Solving Messy Expectations? Basically, I'm looking for ways to solve a ...
- 2,147
5
votes
2
answers
543
views
Calculating expected return
Assume I have a trading system that I'm evaluating over a three-year period. The returns are 25%, -40% and 25%. Empirically, I can see that this system loses because at the end of three years, I have ...
- 702
5
votes
2
answers
2k
views
A list of tricks for calculating expectations?
Does anyone know of a good resource listing known tricks (with examples?) for calculating closed form expressions from messy expectations? (e.g., moment generating function, law of iterated ...
- 2,147
6
votes
2
answers
4k
views
Expected value of certain exponential transformation of standard normal variable
This is in reference to the Girsanov theorem however question is general. If $X$ is a standard normal variable $N(0,1)$, why is expectation of $e^{-\mu X - \mu^2/2}$ equal to 1?
Shouldn't it be $e^{-\...
- 2,759
12
votes
3
answers
3k
views
Using MCMC to evaluate the expected value of a high-dimensional function
I am working on a research project that is related to optimization and recently had an idea to use MCMC in this setting. Unfortunately, I am fairly new to MCMC methods so I had several questions. I'll ...
- 5,075
3
votes
1
answer
310
views
What is the expected dot product of two evolving vectors?
Suppose we have the following random process. We start with two vectors $a_1=(0)$ and $b_1=(0)$. In going from $i$ to $i+1$, we will a perturbation to $a_i$ and $b_i$. With probability $p$, we ...
- 7,651
17
votes
1
answer
1k
views
What is the expected value of modified Dirichlet distribution? (integration problem)
It is easy to produce a random variable with Dirichlet distribution using Gamma variables with the same scale parameter. If:
$ X_i \sim \text{Gamma}(\alpha_i, \beta) $
Then:
$ \left(\frac{X_1}{\...
- 1,412
5
votes
3
answers
396
views
Increasing Exam Expected Mark
You are in an exam, and are presented with the following question:
Write down what mark do you expect to
take in this exam... If you get it
right in range of +/-10 % then you
will take 10% ...
- 303
3
votes
1
answer
479
views
Expected values for chi-squared test on binned paired counts
I want to do a chi-squared test on data that looks like this:
A B
0 0
1 0
0 1
1 1
8 0
3 4
...
You can think of each pair as one trial with two participants. In ...
- 5,189
8
votes
3
answers
2k
views
Expected number of uniques in a non-uniformly distributed population
I am wondering if there is any reasonably simple way of calculating the following problem:
Drawing, with replacement, $n$ balls from a bin of $N$ different colored balls, with a known probability of ...
- 183
11
votes
3
answers
18k
views
Expectation of product of Gaussian random variables
Say we have two Gaussian random vectors $p(x_1) = N(0,\Sigma_1), p(x_2) = N(0,\Sigma_2)$, is there a well known result for the expectation of their product $E[x_1x_2^T]$ without assuming independence?
asd123
9
votes
1
answer
2k
views
Questions about antithetic variate method
Suppose we are to estimate a expectation problem $E(f(X))$, where $X$ is a random variable with known distribution, by simulation and Large Law of numbers estimator. Antithetic method is a way to ...
- 19.6k
4
votes
2
answers
247
views
Calculating $\operatorname{Var}\left\{(\hat{m}-m)^2\right\}$ for a univariate normal distribution
Suppose $\hat{m} = \frac{1}{N}\sum_{i=1}^{N}(X_i)$ where $X_i \sim N(m,\sigma)$.
Are the following steps correct?
\begin{align}\operatorname{Var}\left\{(\hat{m}-m)^2\right\} &= \mathrm E\left\{(\...
- 1,003