Questions tagged [probability]

A branch of statistics that studies the likelihood of uncertain events occurring.

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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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3 votes
1 answer
74 views

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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3 votes
1 answer
109 views

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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1 vote
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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 answer
122 views

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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2 votes
1 answer
174 views

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 vote
1 answer
55 views

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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3 votes
0 answers
41 views

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 vote
1 answer
100 views

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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1 vote
0 answers
45 views

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 vote
1 answer
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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 vote
1 answer
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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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3 votes
1 answer
147 views

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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1 answer
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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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4 votes
2 answers
119 views

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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2 votes
0 answers
37 views

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 vote
1 answer
33 views

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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1 vote
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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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0 votes
1 answer
45 views

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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0 votes
1 answer
61 views

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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0 votes
1 answer
56 views

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 vote
1 answer
235 views

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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3 votes
2 answers
143 views

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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3 votes
2 answers
103 views

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 ...
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4 votes
1 answer
49 views

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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3 votes
0 answers
35 views

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 vote
1 answer
189 views

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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3 votes
1 answer
67 views

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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3 votes
1 answer
104 views

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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0 votes
1 answer
144 views

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 vote
1 answer
71 views

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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1 vote
0 answers
60 views

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 answers
45 views

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 vote
1 answer
54 views

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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6 votes
3 answers
6k views

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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3 votes
1 answer
309 views

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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5 votes
0 answers
189 views

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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1 vote
0 answers
99 views

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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2 votes
1 answer
149 views

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 ...
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1 vote
0 answers
16 views

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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3 votes
0 answers
67 views

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 ...
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0 votes
1 answer
23 views

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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  • 101
0 votes
2 answers
2k views

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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1 vote
0 answers
27 views

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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2 votes
2 answers
49 views

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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  • 393
2 votes
1 answer
37 views

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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5 votes
1 answer
194 views

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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  • 388
1 vote
0 answers
36 views

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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