Questions tagged [probability]
A branch of statistics that studies the likelihood of uncertain events occurring.
84
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Bayes’ rule in "The sources of capital misallocation"
I am reading a paper titled "The sources of capital misallocation". In the model, firms are facing incomplete information about their future productivities. In particular, the productivity ...
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1
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A question about conditional expectation involving independence
If the vector $(u,v)$ is independent of the vector $x$, then I would like to show that
$$E(u|x,v)= E(u|v)$$
The only thing I can derive from the definitions is that if $(u,v)$ is independent of $x$, ...
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1
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What is the intuition behind Blackwell's Equivalence Theorem on Information Structures?
Let us suppose that we have a Bayesian game where the information structure is defined to be as $P^X=\{(X,\mathcal{X},P_\theta)\}_{\theta\in\Theta}$ where a signal generated by the information ...
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0
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Finding the risk attitude parameter in a CPT Risk Elicitation Model
I'm working with this article by Bauermeister et al. that compares the risk attitude parameters found using two different risk elicitation models. The models each use a series of gambling options to ...
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1
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Negative certainty equivalent
Let us consider an agent of initial wealth $w_0$ whose utility function is $u(x)=\sqrt{x}$. This individual faces a risk of loss $Z$ which occurs with probability $p$.
It is assumed that $w_0=60000$, $...
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Understanding the properties of extensive form games
In Heller et al, they use the Osborne and Rubinstein formal definition for the extensive form games with public information. To some point they refer to the following two properties
$P$ is a mapping ...
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1
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Maximizing profit with a simple probabilistic production function (basic practice problem)
A restaurant finds that less orders for their soup of the day are placed on warmer days so they discount the usual 7USD price to 5USD on warmer days. The cost of making the soup is given by
$$
C = 0.1{...
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Model the uncertain impact of a proposed policy by expected utility or other probabilistic approach
The impact of a proposed policy is often uncertain and subjected to randomness. As such, it seems natural to use probabilistic models. How to model the policy impact using the expected utility ...
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1
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Expected payoff calculation difficulty in the Bayesian environment of Bergemman and Morris
Suppose that we have two states of the world equally likely to occur, and say $\psi$ is the common prior of the state $\theta\in\Theta=\{G,B\}$. The types of the players are given by the following ...
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Bayesian update in the beliefs about the signals
Suppose that we have tow states of the world $\omega_1$ and $\omega_2$, where $p(\omega_1)=p(\omega_1)=1/2$ and there are three different signals, $s_H,s_M,s_L$ that are equally likely to occur in ...
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The most used probability distributions in moral hazard models?
I'm engaged in making a variation of the canonical moral hazard model, but I need some examples of probability distributions to make some simulations and graphs. What are the probability distributions ...
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Questions about the definition of a protocol
The following definition is from this paper about protocols of communications (definition $2.4$ in the paper). You can check the paper for more details and previous of my questions.
$\textit{...
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1
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How to compare investments with different risk and expected return?
Supposing I can choose to invest money in several different investments, each having
risk $\sigma_i$, for example, calculated as standard deviation
and expected return $r_i$
let's assume they have ...
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1
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Mechanisms of communication in game theory
In the spirit of the previous question that I have done, here considering the paper here I am trying to make the matching definition $2.2$ here.
I will give two definitions and I would like to clarify ...
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Difficulty in understanding the notation related from probability theory with game theory
The question that I have is a little technical and it has to do with the notation and the combination between some mathamatical properties in the probability theory of information economics.
Say $\...
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What is the meaning of the support set in game theory?
What is the meaning of the support set in game theory? I have seen it, in many papers, however none there explains how did they find it or why did the define it in a specific way. I understand that ...
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2
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Olivier Gossner - Secure Protocols or How Communication Generates Correlation
The paper of Olivier Gossner in Security Protocols in 1998 has some definitions that confuse me too much. I will cite here these definitions and my questions and I hope someone is familiar with these ...
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In economics, what variables do we usually assume to follow an exponential distribution?
In economics, what variables do we usually assume to follow an exponential distribution? I would like as many examples as possible, and it would be great if you have a rationale(economic reason or ...
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1
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Interpretation of multinominal logit regression (Stata)
I have a few questions about mlogit. I have a set of independent variables and a categorical, but ordered, dependent variable with three categories (Disagree, Neutral and Agree). The assumptions for ...
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0
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Question on the choice of boundary in the CUSUM test when we make some resampling
Question on the choice of boundary in the CUSUM test when we make some resampling
We are considering to make a CUSUM test for some economical time series $𝑋=(𝑥_1,..,x_n)$. Suppose 𝑋 contains many ...
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Expected value of order statistics for uniform distribution
I have $X$ ~ $U(0,1)$ interval.
Let n=2, i.e. $X_1 < X_2$
I have to calculate the expected value of ${X_2}^{m/(1-m)}$.
Where, $0≤m≤1$
I want to confirm if I have calculated it correctly?
$$\int_0^1 ...
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1
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Integration by parts with CDF
I am told that the following equality follows from integration by parts:
$$\int_{R-k}^{1}(\theta-R)dG(\theta)-G(R-k)k=\int_{R-k}^{1}(1-G(\theta))d\theta-k$$ Where $R>k>0$ and $G$ is the CDF of $\...
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Is it worth betting on this case?
Let's imagine a coin-flip game, which uses an unbiased coin.
Starting with X dollars, your total increases 50% every time you flip heads. But if the coin lands on tails, you lose 40% of your total. ...
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1
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Show that conditional variance of error in linear probability model is heteroskedastic?
I have a problem that asks me the following:
" Consider the linear probability model, in which we specify the
regression equation to be linear in X,
E(Y |X = x) = Pr(Y = 1|X = x) = x'β
We can ...
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2
answers
143
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How to average CDFs of one variable across years
I have wealth-to-income data for 10 years. I computed the cdf of this variable in each year.
Now I'm trying to average the cdfs across years. In each each, the number of observations is different.
...
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2
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103
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How to approach rigorous probability theory from an economics background?
I am attempting to read around the theory of probability theory from the ground up, coming from a background of economics I have little experience in set/measure theory, whilst I am not new to ...
user23720
4
votes
1
answer
49
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Average ability conditioning on having accepted an offer
There is a continuum of workers between 0 and 1. These have ability $\alpha\sim U[0,2]$. A firm offers them a salary $v$ and has profits
$$
\pi = (\rho \alpha-v) n(v)
$$
where $n(v)$ is the fraction ...
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0
answers
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Estimating probability of Central Bank's interest rate changes
Recently, I came across this article, which offers a simple model for estimating the probabilities of interest rate cut/hike from a central bank. This is done by using market data, especially normal ...
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1
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Risk neutral probability for each of 3 states
I need help to find the risk-neutral probability for states 1,2 and 3
I have two stocks: A and B.
The price of A today is 180 and in a year it will be worth 288 (S1), 180 (S2) or 120 (S3);
The ...
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1
answer
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Modelling involving sum of random variables: Simple CDF?
This question emerges from a project in microeconomic modeling.
I have $n$ agents receiving noisy i.i.d signals $s$.
In my model, a situation of interest occurs when the average signal across $n$ ...
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1
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Intertemporal choice with possibility of death
Here is the setup:
Suppose that there is an individual who lives up to two periods. He lives with absolute certainty during period $1$, and during this period his sub-utility function is given by:
$$...
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1
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Expected Utility and Jensen's Inequality
Consider two random variables (costs and valuations) distributed $v\backsim G(.)$ and
$c \backsim F(.)$ with pdfs $g(.)$ and $f(.)$. Let the supports of $c$ and
$v$ be $[x,y]$. Let $x<a=E(v)<b&...
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1
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Is the set of optimal strategies convex in a single-agent decision choice problem?
EDITED with insights from the comment below.
Consider a decision maker who has to choose an action among $\mathcal{Y}\equiv \{1,2,...,L\}$. The payoff from choosing action $y\in \mathcal{Y}$ depends ...
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Assessing risk in a decision problem with repeated toss
The problem starts at time t0.
At each time step, the participant can choose to opt out and claim a loser's reward Rl.
At each time step, the participant has a probability p to win a winner's reward ...
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0
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Rate of convergence and asymptotic dominance in $\Vert x \Vert \gg \Vert(\hat\beta-\beta)\cdot u\Vert $
Let $\Vert A \Vert$ denote the spectral norm of a random matrix. Let $x$ and $u_k$ be N$\times$T matrices. Denote $\beta \cdot u = \sum_{k=1}^K\beta_ku_k $, where $\beta$ is a K-vector and $\beta_k$ a ...
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1
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Probability of the event knowing that I received no informations
First I want to thank you if you pay attention to my post. I apologize if it seems elementary to you, note that I searched a lot an answer before posting.
I have a particular informational framework ...
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3
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Type - I Error & Type - II Error: Pregnancy test analogy - is it legit?
I found this picture in my stats book but I'm now confused to what 'positive' and 'negative' is referring to.
As seen in the table below, Type 1 error is the error that its H0 is actually true but ...
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Is First Order Stochastic Dominance (FOSD) relation convex?
A convex relation is that $x\succeq y$ implies $\alpha x+(1-\alpha)y\succeq y$.
Let $>_{FOSD}$ be $\succ$, is the FOSD convex? Intuitively it seems convex.
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How to interpret the proof that information cascades will form?
I am reading the 1992 paper of Bikchandani, Hirshleifer and Welch on information cascades. They claim and prove that, given an environment of sequential decision making, an information cascade will ...
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Global games: How to derive posterior with uniform prior and signal
I have access to some lecture notes on Global games (following the model of Carlsson and van Damme (1993)) showing how to derive the players posterior beliefs. But I don't really grasp how players ...
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Best-responding to a stochastically higher distribution of bids
In Auction Theory, Krishna writes that:
a bidder who faces a stochastically higher distribution of bids–in the
sense of reverse hazard rate dominance–will bid higher
(This follows the proof of ...
user17900
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0
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Expectation conditional on a sum of random variables
The setting is a simple OLS regression where the true model has regressor $x$ and error term $u$, but we can only measure $\bar{x}=x+v$ where $v$ is iid with mean 0.
According to the textbook:
$\...
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Mean Field/Differential Game and Measurability
Consider the following scenario. There is a continuum of players in a population, with population measure normalized to $1$. Each player has a type $\theta \in [0,1]$ and we suppose that $\theta$ is ...
user11305
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1
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When to invest into additional products?
This is a very applied question so I hope it's the correct adress here for it:
I'm running a small entertaining business for virtual reality experiences. Investment was about 120 k. I now build it ...
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2
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Find all of the Pure and Mixed Strategy Nash Equilibria [closed]
When I do the basic calculations for mixed probability, I get that the Column player always plays B. However, I am getting a negative probability for the row. Any help is appreciated.
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How to determine the default probability of a county in a bond that is not in its native currency?
Consider the following case:
Country P uses the currency Euro and gives p percent interest on a one year bond issued in Euro.
Country Q uses the currency TL and gives q percent interest on a one ...
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understanding the proof of stochastic dominance.
$\int_a^b u(x)dF(x)$ (1)$ = u(t)F(t)|_a^b - \int_a^b F(t)u^\prime(t)dt$ (2)$ = u(b)-\int_a^b F(t)u^\prime(t)dt$
$= u(b)-(\Phi(t)u^\prime(t)|_a^b-\int_a^b \Phi(t)u^{\prime\prime}(t)dt=u(b)-\Phi(b)u^\...
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Dimensional analysis for the qdf/quantile function corresponding to the pdf/CDF for the size distribution of income
I have previously posted a very similar question on Stackoverflow, but based on responses there I have decided that the real nub of my question is economic. I will give a longish introduction, mainly ...
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What exactly is/How exactly do we interpret the binomial model's Radon-Nikodym derivative?
Related: Lewis' triviality result?
As I recall the one-step binomial model goes like this:
The time periods are now $t=0$ and later $t=1$.
We have
2.1. a stock that pays off $u$ for going up or $d$...
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0
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Marital status determinants
I am looking for the researches that have studied factors influencing marital status probability. I need them for citation purposes. After looking for a long time I have not found anything. So I need ...
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