Questions tagged [probability]

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

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Multinominal logit regressions - marginal effects

I am running a multinominal logit regression with five dependent variable categories: "I disagree fully", "I disagree", "Neutral", "I agree" and "I agree ...
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29 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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Take it or Leave it Offer [closed]

[![I am having a bad time solving second price auctions and lotteries with resale with credit constrains. Does someone know how to answer these?][1]][1] I have to solve some microeconomic bonus ...
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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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Probability transition matrix as a function of the variance in Matlab?

I am working on Probability transition Matrix on Matlab. I Have say 5 points (states) of discretized productivity grid. I would like to have two matrix of probability transition subject to the ...
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50 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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47 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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78 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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90 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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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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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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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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72 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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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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83 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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127 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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63 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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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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43 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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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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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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266 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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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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115 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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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 ...
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22 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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975 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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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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1answer
35 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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115 views

One-step Binomial model's Radon-Nikodym derivative

In the one-step binomial model... Question 1: What exactly is a '$\frac{d \mathbb Q}{d \mathbb P}$'? I think it's $\frac{d \mathbb Q}{d \mathbb P} = \frac{q_u}{p_u}1_u + \frac{q_d}{p_d}1_d$, so it's ...
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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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155 views

Higher order beliefs and coherency in game theory

I am reading about the higher order beliefs. Before getting into the formal definitions, I will define some common terminology which I will need for the formal definitions. If $X$ and $Y$ are two ...
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Is lowered probability of spending equals savings? [closed]

This a really basic one and logic says: Yes given enough iterations. But I am looking for validation Situation: There is a 7% probability of expenses of 1000 occurring. I have a tool that then ...
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Why is it possible to calibrate your subjective probabilities?

Humans tend to be overconfident in their predictions; when most people say that there's a 95% chance that something will happen, they're usually wrong far more than 5% of the time. Whereas what ought ...
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Investment and probability

Being a mathematician, I am familiar with probability calculations, but I need to ask a question related to investments and probability, and how this is handled seen from an economics view. Given a ...
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If an item is rarer than others, is it that much more valuable?

Say I am selling a pack of trading cards, and there are 5 cards inside. Four of these cards are basic cards, but the last card has a 1:10 chance of being a special insert card. Does the fact that the ...
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174 views

second order stochastic dominance without the same mean

Let $F$ and $G$ be two distributions with the same mean. $F$ is said to second order stochastically dominate (SOSD) $G$ if $$\int u(x)\mathrm dF(x)\ge \int u(x)\mathrm dG(x)\tag{1}$$ for all ...
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Is maximal utility conditional on information linear in convex combinations of priors?

This is related to a Mathematica question here - https://math.stackexchange.com/q/1952779/374929 Is a (maximal expected utility) function of the form $U(\mu, X) \equiv \int_\Theta \int_\mathcal{X} \...
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How to use the Girsanov theorem to prove $\hat{W_t}$ is a $\hat{\mathbb P}$-Brownian motion?

Let $T > 0$, and let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathbb P = \tilde{\mathbb P}$ (risk-neutral measure) and $\mathscr F_t ...
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Negative probabilities - Can we have negative payments in bonds?

In Half of a Coin: Negative Probabilities, the author mentions bond duration. Suppose we have payments at times $t = 1,2,...,n$ denoted respectively by $R_1, R_2, ..., R_n$ and the discount factor is ...
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Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

The problem: Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in ...
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Effort setting game - no idea where to start

I have been working on this problem for a few days but I am completely lost on how to start. Any suggestions, comments, hints are greatly appreciated. Here is a scenario: Participants are competing ...
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746 views

How to have same utility function for two persons?

I have a question regarding utility functions: Utility can be defined as follows: $U=1+e^{\frac{x}{RT}}$ U:Utility x: What we want to find the utility for (Certain equivalent) RT: Risk tolerance ...
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485 views

Does the Independence Axiom Require Statistical Independence?

First: Given this definition of the Independence Axiom, If for all $P$, $P'$, $P''$ in the set of lotteries over outcome space $X$, when: $P$ preferred to $P'$ $\implies$ $aP + (1-a)P''$ preferred to ...
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Can't solve this matrix for Nash Equilibrium?

So, I have the following 9 by 9 probability matrix. I want to solve it for a nash equilibrium. https://docs.google.com/spreadsheets/d/16Y1FqxRIAHsHpgEz1ckxDt2sEOInOG3zz_wU8kBHvB4/edit?usp=sharing For ...
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Job-finding rate in an urn-ball model with types

Setup Say you have two types of workers, high and low. The share of low-types among the unemployed population is $P$. I want to find the job-finding rate for these types. Matching Matching is ...