Questions tagged [probability]
A branch of statistics that studies the likelihood of uncertain events occurring.
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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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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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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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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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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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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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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 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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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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2
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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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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 ...
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References about market sampling
Suppose the government wants to determine the efficient price of a certain commodity for which there is no competitive market. One way to do this is to take a sample of the potential buyers and ask ...
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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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Lack of historical data for calibration of probability of default
It is a known fact that default rates seem to exhibit cyclic behavior. Most probability of default models use one-year averages of default rates to calibrate the models. The one-year averages should ...
user6717
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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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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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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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Urn balls and probabilities
Think of the following balls as individuals of populations.
Say I have $U$ urns, and some balls. Both numbers are really large. So large, that authors like Blanchard and Diamond have approximated ...
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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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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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Is there a name for this type of problem?
I am having trouble formulating the concept I am thinking about. It has to do with looking at observed behavior of the sales of a particular product during each hour of the day, and trying to adjust ...
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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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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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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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Example of the change of measure proposed in Hansen (2012)
In this question, I'm continuing to explore the tools used/presented in Lars Hansen's Econometrica paper "Dynamic Valuation Decomposition within Stochastic Economies" (2012).
I'm trying to compute an ...
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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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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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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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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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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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