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Questions tagged [probability]

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

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51 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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1answer
59 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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58 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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38 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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1answer
43 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 ...
3
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1answer
858 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 ...
4
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1answer
138 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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49 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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52 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 ...
2
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1answer
73 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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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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40 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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1answer
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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2answers
264 views

Find all of the Pure and Mixed Strategy Nash Equilibria

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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26 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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2answers
39 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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1answer
28 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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1answer
47 views

One-step Binomial model's Radon-Nikodym derivative

In the one-step binomial model... For $\frac{d \mathbb Q}{d \mathbb P}$, 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 some asset with payoffs $\...
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35 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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1answer
105 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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1answer
33 views

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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1answer
184 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 ...
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3answers
70 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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2answers
73 views

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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1answer
117 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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30 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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82 views

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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1answer
62 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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1answer
62 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 \...
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1answer
83 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 ...
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2answers
566 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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2answers
422 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 $P'$ $\implies$ $aP + (1-a)P''$ ...
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0answers
123 views

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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0answers
137 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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11 views

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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0answers
260 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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0answers
47 views

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 ...
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1answer
160 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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1answer
71 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 ...
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2answers
57 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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3answers
189 views

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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1answer
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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3answers
781 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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1answer
59 views

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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2answers
102 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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2answers
3k 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 ...
5
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1answer
323 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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0answers
61 views

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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0answers
45 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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0answers
119 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: ...