﻿ oipapio

### expectation - where's the error of this calculation?

Let $X=2e^{-2\theta}$, ask to calculate $E[X^2]$. Here's the right way to calculate it$$E[X^2]=var[X]+(E[X])^2={1\over 2^2} + ({1\over 2})^2={1\over 2}$$but if I do it in another way$$E[X^2]=E[(2e^{-2\theta})^2] = E[4e^{-4\theta}]= {1\over 4}$$Then it ends up $1\over 4$, what's wrong with the second calculation?...Read more

### Hypergeometric Random Variable Expectation

In a binomial experiment we know that every trial is is independent and that the probability of success, $p$ is the same in every trial. This also means that the expected value of any individual trial is $p$. So if we have a sample of size $n$, by the linearity property of the expectation, the expected value of the same is just $n \cdot p$. This is all intuitive.When the population size is finite and when we don't replace the items after every trial, we can't use the binomial distribution to get the probability of $k$ successes in a sample of s...Read more

### expectation - Using posterior expected loss to make a decision between two pdfs

I am trying to solve a question from a past exam paper. Suppose you have a single observation $X$ from a continuous distribution for which the probability density function (pdf) is either $f_0$ or $f_1$, where $$f_0(x) = \begin{cases} 1, & \text{for 0} < x < 1 \\ 0, & \text{otherwise} \end{cases}$$ $$f_1(x) = \begin{cases} 4x^3, & \text{for 0} < x < 1 \\ 0, & \text{otherwise} \end{cases}$$ On the basis of one observation, we would like to determine whether $f_0$ or $f_1$ is the ...Read more

### expectation - Calculating Expected Utility with Beta-distributed RV - which way is correct?

I have an agent with a utility function of the form $u(x) = -(x-g)^2$, with $g$ being a constant. $x \sim Beta (a,b, \alpha, \beta)$, with $[a,b]$ being the support, and $\alpha, \beta$ being the parameters. I would like to calculate the expected utility of the agent. Now, I have tried to approach this in several ways, including Taylor series approximations, and would always get a different result. I am sure I am making a mistake somewhere, would you be able to point it out?Here are the 2 main approaches I have tried and I have no ide...Read more