Questions tagged [probability]
A branch of statistics that studies the likelihood of uncertain events occurring.
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When treating a relative, normalized utility function as a pmf, what is the interpretation of Shannon entropy or Shannon information?
Suppose $\Omega$ is a set of mutually exclusive outcomes of a discrete random variable and $f$ is a utility function where $0 < f(\omega) \leq 1$, $\sum_\Omega f(\omega) = 1$, etc.
When $f$ is ...
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Understanding the construction of stochastic processes
I've seen stochastic processes modeled/constructed in the following way.
Consider the
probability space $(\Omega, \mathcal F, Pr)$ and let $\mathbb S$ be the (measurable)
transformation $\...
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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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Intuition behind risk premium
In Lecture 20 of MIT's Microeconomics course, a situation is proposed where a 50/50 bet will either result in losing \$100 or gaining \$125 with a starting wealth of \$100. It is stated that a person ...
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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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Show that $W_t - \int_0^t \xi_s ds$ is forward-measure-Brownian
Definitions and stuff:
Consider a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ where
$$T > 0$$
$$\mathbb P = \tilde{\mathbb P}$$
This is risk-...
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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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Pricing a European call option while absence of arbitrage is violated
Assume that we have a general one-period market model consisting of d+1 assets and N states.
Using a replicating portfolio $\phi$, determine $\Pi(0;X)$, the price of a European call option, with ...
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Showing that a transformation is measure preserving
Note: This question is related to this question about the construction of stochastic processes. Specifically, it relates to the transformation $\mathbb S: \Omega \rightarrow \Omega$ that is mentioned. ...
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Proof that the diff-in-diff (wrt sample size) of the expectation of a first-order statistic is positive (Stigler 1961)
I'm trying to prove a claim made in Stigler (1961), "The Economics of Information." This claim has to do with showing that the marginal benefit
of making an additional search (e.g., searching an ...
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Conditional probability in Kaplan, Menzio (2014)
This is question about Kaplan and Menzio's shopping time model.
Pages 7,8: Unemployed search once or twice (for a seller).
$\psi_u$:probability of searching twice, searching once with prob $1-\...
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Normalizing comparisons of corporations and countries
When publicly-traded corporations reach record valuations, articles in the media often compare such valuations to the GDPs of countries throughout the world, typically in the form "Company X's ...
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Third price auction from Auction Theory by Krishna, Order statistics
Notation:
$Y_1$: Highest order statistics of $(N-1)$ players' valuation.
$F_n^M:$ The distribution function of the highest $n$th order statistics of $M$ players.
$f_n^M:$ The density of the highest $n$...
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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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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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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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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
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Deriving an equation in Banerjee "A Simple Model of Herd Behavior" (1992)
I am reading "A Simple Model of Herd Behavior" by Banerjee (1992). A short summary of the model is the following.
There is a probability $\alpha$ that each person receives a signal telling ...
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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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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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Resampling for a Probabilistic Model to Balance Outcomes
I wanted to construct a logit model for determining the probability a recession will be determined for any given month using the usual Macro indicators; however, I noticed that 90% of the months in my ...
4
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1
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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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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
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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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2
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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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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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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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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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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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Joint distribution from differential equations
I have the following problem -
Z is a random variable which can take any real value in the range [0,1]
a and b are independent variables drawn from uniform distribution in the interval [0,1].
Z is a ...
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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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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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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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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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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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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 ...
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Applying the Martingale central limit theorem to the score process of an autoregressive model
This question is a natural continuation of the following question: How do I construct the score process of a Markov model and verify that it is a Martingale? In this problem, we set up as follows:
...
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What does it mean when I say that CDF is bounded away from 1?
Suppose $\theta \in [\underline\theta, \bar\theta]$ is distributed with CDF F(.). What does it mean when I say that this F is bounded away from 1? Does it mean that F can never take the value 1 in ...
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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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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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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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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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Something Terrible is Happening - But When Is it Likely to have Happened?
We live in continuous time $t$ and something terrible is happening at a poisson rate of $r(t)$.
How can I compute the length $T$ such that with a probability of $P$ (for example, 0.99), at least one ...
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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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Augmented Filtrations and Martingales in the Martingale Representation Theorem
Note: This question is related to the following question about complete markets in continuous time. In the linked question, the answer mentions that complete markets in this setting is a result of the ...
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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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To bet or not to bet
Your utility from having $x$ dolars is $u(x)$.
There is a gamble in which the winnings in dollars are a random variable, $Y$. It is known that $E[u(Y)]>E[u(1)]$, so you prefer to bet than to get ...
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Estimating excess probability
I have three players A,B,C, where C ideally functions as a baseline. They play different games (not against each other) and I can observe their win probability, for example P(A wins in game I) = 90%, ...
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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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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