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

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0
votes
0answers
7 views

Find limiting distribution

The question is like this: $X_i$ are i.i.d with $P(X_i\leq x)=e^{-x}$. $S_n=X_1+\cdots+X_n$. Find the limiting distribution of $\sum_{i=1}^nP(X_iS_n>1)$. It seems that the problem is related to ...
0
votes
3answers
29 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
vote
0answers
10 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 ...
4
votes
1answer
18 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
votes
0answers
26 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
30 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
votes
0answers
21 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
vote
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
votes
0answers
18 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
24 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 ...
-2
votes
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
0
votes
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
11 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
votes
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
vote
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
23 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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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
vote
0answers
24 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: ...
-2
votes
0answers
29 views

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

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 ...
0
votes
0answers
7 views

specific examples of random variables satisfying a given condition.

Theorems such as the central limit theorem only says random variables satisfying certain conditions have some properties. Now, what I am curious about is the existence of such random variables. For ...
-1
votes
1answer
27 views

Birth and Death process, CTMC, how is the solution here derived? [on hold]

My question is about how the solution is reached, as I am completely lost on how. Any thoughts? Consider a birth and death process with birth rates $λ_i = (i+1)λ \;\;, \;\; i≥0$, and death rates ...
1
vote
0answers
19 views

How to distribute two independent rows of bits

Consider two independent rows of 100 bits. The bits are mutually independent and have an equal chance to be 0 or 1. The first row is being read and during that process there is a chance $\epsilon$ ...