Questions tagged [probability]

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

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13 votes
3 answers
1k views

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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13 votes
3 answers
1k views

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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11 votes
1 answer
212 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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10 votes
2 answers
4k views

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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9 votes
1 answer
239 views

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 ...
8 votes
1 answer
425 views

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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7 votes
1 answer
216 views

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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6 votes
3 answers
7k 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 ...
6 votes
1 answer
110 views

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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6 votes
2 answers
61 views

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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5 votes
2 answers
106 views

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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5 votes
1 answer
427 views

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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5 votes
1 answer
214 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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5 votes
1 answer
207 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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5 votes
0 answers
190 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 ...
4 votes
2 answers
504 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 ...
4 votes
1 answer
31 views

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 votes
1 answer
50 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 ...
4 votes
2 answers
121 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 ...
3 votes
2 answers
153 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
1 answer
317 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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3 votes
1 answer
75 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
152 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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3 votes
1 answer
136 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 ...
3 votes
2 answers
119 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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3 votes
1 answer
68 views

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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3 votes
1 answer
98 views

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 ...
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 $n$ ...
3 votes
1 answer
108 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: $$...
3 votes
1 answer
60 views

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 ...
3 votes
0 answers
42 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 ...
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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3 votes
0 answers
70 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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3 votes
0 answers
160 views

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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3 votes
0 answers
147 views

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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2 votes
2 answers
50 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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2 votes
3 answers
80 views

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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2 votes
1 answer
175 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 ...
2 votes
1 answer
166 views

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 ...
2 votes
1 answer
75 views

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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2 votes
1 answer
79 views

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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2 votes
1 answer
178 views

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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2 votes
1 answer
39 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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2 votes
1 answer
72 views

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 ...
2 votes
0 answers
40 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 ...
2 votes
1 answer
154 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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2 votes
0 answers
35 views

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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2 votes
0 answers
100 views

How to use Girsanov theorem to prove $\hat{W_t}$ is $\hat{\mathbb P}$-Brownian motion?

Assumptions: 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 $...
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2 votes
0 answers
77 views

How can I construct a process for cumulative returns that is riskless?

This question is a little more specific than the title. Here I use the same notation that is set forth in this other question about cumulative returns (the sum of return observations). That is, let $...
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1 vote
1 answer
26 views

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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