Questions tagged [probability]
A branch of statistics that studies the likelihood of uncertain events occurring.
64
questions
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15 views
Probability transition matrix as a function of the variance in Matlab?
I am working on Probability transition Matrix on Matlab. I Have say 5 points (states) of discretized productivity grid. I would like to have two matrix of probability transition subject to the ...
0
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1answer
49 views
Integration by parts with CDF
I am told that the following equality follows from integration by parts:
$$\int_{R-k}^{1}(\theta-R)dG(\theta)-G(R-k)k=\int_{R-k}^{1}(1-G(\theta))d\theta-k$$ Where $R>k>0$ and $G$ is the CDF of $\...
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1answer
47 views
Is it worth betting on this case?
Let's imagine a coin-flip game, which uses an unbiased coin.
Starting with X dollars, your total increases 50% every time you flip heads. But if the coin lands on tails, you lose 40% of your total. ...
1
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1answer
61 views
Show that conditional variance of error in linear probability model is heteroskedastic?
I have a problem that asks me the following:
" Consider the linear probability model, in which we specify the
regression equation to be linear in X,
E(Y |X = x) = Pr(Y = 1|X = x) = x'β
We can ...
3
votes
2answers
83 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.
...
3
votes
2answers
81 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 ...
4
votes
1answer
49 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 ...
3
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0answers
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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1answer
56 views
Risk neutral probability for each of 3 states
I need help to find the risk-neutral probability for states 1,2 and 3
I have two stocks: A and B.
The price of A today is 180 and in a year it will be worth 288 (S1), 180 (S2) or 120 (S3);
The ...
3
votes
1answer
66 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
1answer
83 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:
$$...
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1answer
126 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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vote
1answer
63 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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0answers
60 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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43 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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vote
1answer
47 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 ...
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votes
3answers
4k 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 ...
3
votes
1answer
262 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.
5
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0answers
64 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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0answers
72 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
108 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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0answers
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:
$\...
3
votes
0answers
51 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
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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 ...
0
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2answers
907 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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0answers
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 ...
2
votes
2answers
44 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
35 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 ...
4
votes
1answer
112 views
One-step Binomial model's Radon-Nikodym derivative
In the one-step binomial model...
Question 1: What exactly is a '$\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 ...
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0answers
36 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
150 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
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 ...
9
votes
1answer
230 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
76 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
74 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 ...
11
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1answer
173 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 ...
2
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0answers
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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0answers
87 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
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 ...
2
votes
1answer
69 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 ...
3
votes
1answer
90 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
733 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
...
4
votes
2answers
481 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 ...
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0answers
135 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 ...
3
votes
0answers
155 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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0answers
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 ...
8
votes
1answer
363 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
51 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 ...
8
votes
1answer
197 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
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 ...
