Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [probability]

The tag has no usage guidance.

1
vote
1answer
43 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 ...
1
vote
0answers
12 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: $\...
1
vote
0answers
34 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 ...
0
votes
0answers
27 views

survival probability and integration by parts

I am trying to integrate by parts by using an indicator function. However, I am not really sure if it is a correct way to change the bounds of integral with indicator functions. I am trying to deal ...
0
votes
1answer
20 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 ...
0
votes
2answers
69 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.
1
vote
0answers
23 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 ...
2
votes
2answers
34 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^\...
2
votes
1answer
26 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 ...
3
votes
0answers
28 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 $\...
0
votes
0answers
43 views

How to prove that restricted OLS estimator's convergence in probability

I'm solving an econometrics problem about convergence in probability and I cannot prove it properly. The problem is below. Let $\tilde{\beta_R} $ be the restricted OLS estimator and $\tilde{\sigma^2} ...
0
votes
0answers
19 views

How to study the relationship (correlation) of the non-normal probability distribution function?

In my research, I am examining the relationship of one financial asset and a stock market. Since it's volatile we can say that the probability distribution function of the two is not Gaussian. Cross-...
1
vote
0answers
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 ...
5
votes
1answer
74 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 ...
0
votes
1answer
32 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 ...
6
votes
1answer
153 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 ...
2
votes
3answers
64 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 ...
1
vote
2answers
70 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 ...
9
votes
1answer
97 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 ...
3
votes
0answers
26 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} \...
1
vote
0answers
77 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 ...
3
votes
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 ...
2
votes
1answer
57 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 \...
4
votes
1answer
82 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 ...
0
votes
2answers
474 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 ...
5
votes
2answers
313 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 $P'$ $\implies$ $aP + (1-a)P''$ ...
1
vote
0answers
112 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 ...
4
votes
0answers
94 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 ...
1
vote
0answers
9 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 ...
9
votes
0answers
231 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-...
1
vote
0answers
46 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 ...
7
votes
1answer
140 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 ...
2
votes
1answer
70 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 ...
6
votes
2answers
56 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 ...
1
vote
3answers
169 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 ...
2
votes
1answer
74 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 ...
10
votes
3answers
635 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 ...
1
vote
1answer
58 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 ...
5
votes
2answers
95 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-\...
9
votes
2answers
2k 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
votes
1answer
309 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 ...
1
vote
0answers
53 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 ...
2
votes
0answers
38 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 $...
3
votes
0answers
98 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: ...
5
votes
1answer
91 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. ...
11
votes
3answers
650 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 $\...
2
votes
1answer
102 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 ...