Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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0
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0answers
8 views

Consider a random walk where $p \neq 1/2$, where the starting point is random and has a binom distn. Find the probability of absorption at $N$.

Consider a random walk $\{0,1, ... , N\}$ with up probability $p$ and down probability of $p-1$ where $p \neq 1/2$. Suppose there are absorbing barriers at $0$ and $N$ and that the starting point, ...
0
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1answer
8 views

Joint cumulative density function of two independent Gaussian random variables

Assume we have two independent random variables $\theta_1$ and $\theta_2$ which each have separate Gaussian distribution functions $D_{\theta_1}$ and $D_{\theta_2}$. $\theta_1$ describes a threshold ...
4
votes
2answers
45 views

Can some probability triple give rise to any probability distribution?

Suppose we have a probability triple $(\Omega,\mathcal{F},P)$ and random variable $X:\Omega\to(\mathbb{R},\mathcal{B})$ with $\mathcal{B}$ denoting the Borel $\sigma$-algebra. Then, the distribution ...
0
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0answers
17 views

Find limiting distribution

The question is like this: $X_i$ are i.i.d with $P(X_i\leq x)=1-e^{-x}$. $S_n=X_1+\cdots+X_n$. Find the limiting distribution of $\sum_{i=1}^nP(X_iS_n>1)$. It seems that the problem is related ...
0
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3answers
41 views

If $X$ has a Poisson distribution with $E[X]=\lambda$, does $Var[X^2]=4\lambda^3+6\lambda^2+\lambda$?

Suppose $X$ has a Poisson distribution with mean (and therefore variance) $\lambda$. Using Excel to explore properties of the distribution of $X^2$ with some small integer values of $\lambda$ I ...
1
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0answers
13 views

What is the pdf of $X$, where $dX_t = -aX_t + d N_t, N_t$ is a compound Poisson process?

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where (I'm not sure this notation makes sense, I'm not very familiar with the stochastic calculus ...
5
votes
1answer
20 views

Density of stochastic integral

I am working on finding the PDF of $X_t^2$, where $X_t = \int_0^t A(u) \,dW_u$, a Wiener integral, i.e., $W_t$ is Brownian motion and $A(t)$ is a deterministic function. Here, would like to ask that ...
0
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0answers
28 views

Monotonocity of ratios of normal CDFs

I am solving a problem in decision theory under uncertainty and need to establish whether $\frac{\Phi(x)-\Phi(x-\varepsilon)}{\Phi(x+\varepsilon)-\Phi(x-\varepsilon)}$ $(\ast)$ is monotonically ...
2
votes
2answers
32 views

Binomial distribution central moment calculation

If for a binomial distribution the mean is $4$ and variance is $3$, find th $3^{\text{rd}}$ central moment. I understand that the first and second central moments are mean and variance ...
0
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0answers
22 views

Support lemma - Game theory

Let α be $a$ mixed strategy profile, $a_i ∈ supp(\alpha _i), a_i \notin B_i(\alpha _{−i}), a_i' ∈ B_i(\alpha _{−i})$ and $a_i'$ defined by $\alpha_i'(a_i)=0$, $\alpha_i'(a_i')=\alpha _i ...
1
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1answer
16 views

Computing expectation of a function of two random variables

I have two arrays $X$ and $Y$ of length $N$ each. In array $X$, I have random numbers $x_1$, $x_2,\ldots,x_N$, whose sum is $S_x$. Similarly in array $Y$, I have random numbers $y_1$, ...
0
votes
1answer
17 views

uniform angular distribution-change of origin

Given a variable which is uniformly distributed for $0<\theta<\pi$ on, let's say, a circle around the origin $O$ with radius $R$($\theta$ starting on the positive x-axis and turning ...
-2
votes
0answers
18 views

What is the p-value of this problem? [on hold]

Over a 7 year period, an event happens 126 times during 154 opportunities for this kind of event to happen. Over the next 8 years, the same event happens 142 times during 169 opportunities for this ...
0
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0answers
19 views

Sum of Gaussian and Binomial distribution

I need to calculate the probability of sum of two probability variable, each of which is distributed as binomial distribution and Gaussian respectively. I mean how to calculate the probability of ...
0
votes
1answer
23 views

Probability of winning a simple game

Consider two players, A and B start with 8 and 6 stones respectively. A rolls a six-sided die to determine how many stones to take from B. B performs the same task to determine how many stones to ...
2
votes
1answer
26 views

How to represent $Prob(X_1+X_2 \leq a, X_2+X_3 \leq b, X_3 +X_4 > c)$ with mutually independent random variables?

There are four mutually independent random variables: $$X_i : \Omega \to \mathbb R$$ for $i= 1,2,3,4$ The cumulative distribution function of them is given as $F_i(x_i)$. How to represent ...
0
votes
1answer
25 views

Finding distribution function of the ratio of two continuous uniform random variables where the denominator random variable is squared.

Let $X_{1}$ and $X_{2}$ be independent and uniformly distributed between 0 and 1. I want to find the distribution function of $X_{3}=\dfrac{X_{2}}{X_{1}^{2}}$. Denote this distibution function by ...
0
votes
1answer
18 views

Expected Value: how to understand this expression?

So I have come across a question asked by my peers. Define: $$g:=\sqrt{E[|y_r(t)|^2]}$$ Given that $$y_r(t)=\sqrt{t}\cdot h+b+k+c,$$ where $h$, $b$, $k$, and $c$ are independent random variables. ...
3
votes
1answer
20 views

Independence of random variables and covariance in the limit.

Consider two sequences of random variables $(X_n)$ and $(Y_n)$ which converge in distribution to $X$ and $Y$ respectively, where $X$ and $Y$ are independent, but each pair $(X_n, Y_n)$ is not ...
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0answers
16 views

Copula theory on discrete random variables [on hold]

How can I find the joint pmf on two discrete random variables using the copula theory
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0answers
13 views

Posterior probability estimation in MAP model

I have a question about probability. I am using Bayes rule to determine which class the $x$ belong to. According to Bayesian formula, the MAP estimation is equivalently found by $$p(x \in \Omega_i|x)= ...
3
votes
0answers
12 views
+50

Estimates for the normal approximation of the binomial distribution

I'm interested in estimates of the normal approximation for binomial distributions, i.e. in estimates of $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - ...
-2
votes
0answers
32 views

Two random variables have the same p.d.f [on hold]

We know that Probability density function of two random variables X and Y are equal for all values.In other words, $f_X(t)=f_Y(t) $ for all real values of $t$ . I know that in general, mentioned ...
-1
votes
1answer
26 views

What is the distribution of $B(t_1)+B(t_2)+…+B(t_n)$ [on hold]

$\{ B(t), t\ge 0\}$ is a standard Browian Motion Process. What is the distribution of $B(t_1)+B(t_2)+...+B(t_n)$ ?
0
votes
1answer
20 views

Conditional Expectation of joint/(composite function?)

I'm prepping for an exam and looking through previous exams I came across this question: Let $Z$ be a Poisson distributed stochastic variable with parameter $Λ$. In turn, $Λ$ is a Poisson distributed ...
0
votes
2answers
34 views

Having two random generated natural numbers between 1 and 255, and generate out of it natural number between 1 and 256

Let's say you have two cube with 255 sides and you have to use them to simulate a single cube with 256 sides, how can I do it? $f(n)$ and $g(n)$ returns random number between 1 and 255. I thought ...
0
votes
1answer
38 views

chi square distribution probability

I am having a problem with this. Suppose a stock's returns are normally distributed with mean $m$ and variance $\alpha^2$ and we compute the sample variance from a sample of $41$ periods and find ...
0
votes
1answer
39 views

Proof for Mean of Geometric Distribution

I am studying the proof for the mean of the Geometric Distribution http://www.math.uah.edu/stat/bernoulli/Geometric.html (The first arrow on Point No. 8 on the first page). It seems to be an arithco ...
0
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0answers
32 views

How can two random variables are continuous infers that their jointly random variable is continuous?

We assume that $\forall a,b$ such that $a2+b2>0$, $aX+bY $ is continuous random variable. But we don't assume that $X$ and $Y$ are independent. My question is the following: Under which ...
1
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0answers
29 views

Integration of gaussian divided by square root of -log(1-x) - does the Meijer G function help me?

After some modelling of my data I came to the following integral: $$ \int_0^{1}\dfrac{exp{\left(-\dfrac{\left(x-\mu\right)^2}{2\,\sigma^2}\right)}}{\sqrt{-\log{(1-x)}}} $$ I cannot solve it, and ...
1
vote
1answer
9 views

When the domain of a continuous distribution exceeds feasible values, what should I do?

Now I need a (maybe approximated) model for this distribution: $$X=(x_1, x_2, …, x_n)$$ where $x_i$ is a real number between $0.0$ and $1.0$, and the sum of $x_i$ equals $1.0$. Now, I want to use ...
0
votes
1answer
23 views

When limit distribution of $\min(\xi_1,\dots,\xi_n) - a_n$ is non-trivial?

Let $\xi_1,\xi_2\dots$ independent and identically distributed uniformly on $[0,1]$ and $\zeta_n = \min(\xi_1,\dots,\xi_n)$. Find such constants $a_n$ such that limit distribution of $\zeta_n - a_n$ ...
0
votes
1answer
24 views

Find limit distribution i.i.d $\xi_1,\xi_2\dots$ uniformly on $[0,1]$

Let $\xi_1,\xi_2\dots$ independent and identically distributed uniformly on $[0,1]$ and $\zeta_n = \min(\xi_1,\dots,\xi_n)$. Find limit distribution $n^{\gamma}\zeta_n$, $\gamma\in R$. My try. ...
0
votes
1answer
11 views

Distribution of exponential(X/c)

Suppose $X \sim Exponential(\lambda)$. That is, the PDF for $X$ is $f_X(x)=\lambda \cdot e^{-\lambda x}$, $x\ge 0$, and the CDF of $X$ is $F_X (x)=\int_{-\infty}^x f_X(x)=1-e^{-\lambda x}$, $x\ge ...
-2
votes
0answers
21 views

The mean deviation from mean in a normal distribution is equal to $4\sigma/5$ [on hold]

Show that the mean deviation from mean in a normal distribution is equal to $4\sigma/5$. Progress. I have tried going by the usual definitions of deviation and mean deviation but am stuck. Tried ...
-1
votes
1answer
23 views

If you have 50 envelopes and only 3 envelopes contain a symbol what is the probability of picking all 3? [on hold]

If you have 50 envelopes and only 3 envelopes contain a symbol. the person picks only 3 envelopes out of the 50. What is the probability that they will pick 1 symbol? Two symbols? All 3 symbols?
0
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0answers
10 views

Given a sample determine using Chi-squared test whether these values fit in an EXPONENTIAL distribution

Here I've got such a problem. I was given $n = 20$ values for time of good functioning of a robot between two consecutive defects. 1200, 1432, 1502, 1100, 3286, 4235, 1149, 5236, 2234, ...
0
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0answers
3 views

Limit distribution of absolute value maximum of stationary non-differentiable Gaussian process

Consider a real-valued stationary Gaussian Process $\{ X(t) \colon t \geq 0 \}$ with zero mean and unit variance and covariance function $r$ satisfying $r(t) = 1 - C|t|^{\alpha} + o(|t|^{\alpha}), ...
0
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0answers
8 views

Density of the Absorbed Process

The curiosity arose while reading the Ch.18 of Arbitrage Theory in Continuous Time 3/ed, dedicated to pricing Barrier Options. Definition 18.1 For any $y\in R$, the hitting time of y, $\tau(X,y)$, ...
0
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1answer
11 views

Relation between joint probability and marginals for two dependent random variables?

Consider two continuous real valued random variables $X$ and $Y$. Let $f(X,Y)$ be their joint probability distribution and $f_X (X),f_Y(Y)$ their marginals. Suppose that $X$ and $Y$ are dependent. Is ...
0
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0answers
19 views

How to compute the average power of an ergodic process?

Rxx(0)=3 is the average power and if i take limit as t goes to infinity i will get the (E[x])^2 to get variance you subtract 3-2 = 1 is this correct ? and can someone tell the difference ...
0
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0answers
13 views

How to compute a probability expression (for a transition matrix of a Markov Decision Process)? (part 2)

I am creating a transition matrix (for a Markov Decision Process) and I am computing it using a Matlab script, which I am currently writing. My probability expression (for certain cases) looks like ...
0
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0answers
19 views

How to compute a probability expression (for a transition matrix of a Markov Decision Process)? (part 1)

I am quite new in the world of statistics, hence I am quite unsure when working with probabilities. I am creating a transition matrix (for a Markov Decision Process) and I am computing it using a ...
0
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0answers
9 views

Chernoff type bounds for negative binomial distribution

If I recall correctly I remember reading that we cannot get Chernoff type results for the negative binomial distribution because of something regarding lebesque measure. I don't quite know all the ...
1
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1answer
25 views

Expectancy of a joint density

A machine consists of two components, whose life times have the joint density function $ f(x,y)= \begin{cases} 1/50, & \text{for }x>0,y>0,x+y<10 \\ 0, & \text{otherwise} \end{cases} ...
0
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0answers
9 views

Decay time distribution with uniformly distributed source

Consider a kind of particle (source) that can decay into some other particle (product) with decay constant $\lambda$, i.e. the p.d.f is $f(t)=\lambda e^{-\lambda t}$, and the source is uniformly ...
1
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0answers
25 views

Survival probability (1D Brownian Particle)

Here is an interesting article from Wikipedia: First-hitting-time model I am particularly interested in how the following density is derived: $$p\left(x,t;x_0,x_c\right)=\frac{1}{\sqrt{4 \pi D ...
3
votes
2answers
29 views

Notation $E[t^X]$ where $X$ is a random variable

I have a quick question which occured in the context of probability-generating functions but maybe the issue is more basic. For a random variable $X$, the probability-generating function is given as ...
1
vote
2answers
25 views

Probability distribution of number of columns that has two even numbers in a chart

We distribute numbers $\{1,2,...,10\}$ in random to the following chart: Let $X$ be the number of columns that has two even numbers. What is the distribution of $X$? My attempt: ...
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0answers
29 views

What is the right answer among the following four options? [closed]

probability calculus works upon joint occurance of the event under consideration alternative occurance of $d$ events under consideration both joint n alternative occurance of $d$ events under ...