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

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3
votes
1answer
118 views

Distributional equality

Let $(W_t)_{t\geq0}$ be a standard Brownian motion. I have to show that the following equality holds in distribution. Does someone has a good hint to show this? $\sup_{t \geq 0}( |W_t| -t) = \sup_{t \...
-1
votes
0answers
57 views

Simple Markov property on stopping times [on hold]

Suppose $(S_n)_{n\geq1}$ is a Markov chain on the two dimensional lattice of the integers. Then define the stopping time $\tau_A'=\inf\{n\geq1:S_n\notin A'\}$ and consider the following for $A\subset ...
1
vote
1answer
29 views

Transformation technique to find PDF

Consider two random variables with the following joint PDF: $$ f_{X,Y}(x,y) = \begin{cases} 2, & x > 0, y > 0, x + y < 1 \\ 0, & \text{otherwise} \end{cases} $$ I need to find ...
0
votes
3answers
43 views

Generating samples from a Beta(2,2) distribution

I'm looking for a convenient way to generate $\text{Beta}(2,2)$ random variables, using independent $\text{Uniform}(0,1)$ random variables and elementary functions. I'd prefer to avoid rejection or ...
0
votes
2answers
25 views

Conditional Probability: Birth rank of children in randomly chosen families

(BH 4.7) A certain small town, whose population consists of 100 families, has 30 families with 1 child, 50 families with 2 children, and 20 families with 3 children. The birth rank of one of these ...
0
votes
1answer
58 views

Probability for a leading candidate to eventually win

Two candidates contest a close election. Each of the $n$ voters votes independently with probability $\frac12$ each way. Fix $\alpha \in (0,1)$. Show that, for large $n$, the probability that the ...
3
votes
1answer
31 views

Bell numbers and the Moments of expected number of fixed points

Let $X_N$ be the random variable corresponding to the number of fixed points (1-cycles) in a permutation chosen uniformly at random from $S_N$. Then, the $m^{\text{th}}$ moment, when $m < N$, is ...
2
votes
1answer
32 views

Probability distribution of order statistics

Let $X_1$, $X_2$ and $X_3$ be independent random variable with continuous distribution $$f(x;\theta)=\frac{1}{\theta}I_{(0,\theta]}(x), \ \theta \gt 0$$ I need to find distribution of $Z=\frac{X_{(...
0
votes
1answer
35 views

I am trying to find answer to this bivariate normal problem. Can anyone help. [closed]

The distribution of the heights of husband-wife pairs in a particular population is modelled by a bivariate normal distribution. The mean height of the women is 165cm and the mean height of the men is ...
1
vote
1answer
26 views

20 identical balls to be distributed in 3 identical boxes with MAX & MIN balls in each box?

As the title suggests, In how many ways can 20 identical balls be distributed in 3 identical boxes with at most 8 balls in each box and minimum 1 ball in each box ?
0
votes
0answers
19 views

If $(W_t)_{t\ge 0}$ is a $L^2(D)$-valued Wiener process, then $W_t(x)$ is normally distributed

Let $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $D\subseteq\mathbb R^d$ be a domain $U:=L^2(D)$ and $\langle\;\cdot\;,\;\cdot\;\rangle:=\langle\;\cdot\;,\;\cdot\;\rangle_U$...
0
votes
1answer
13 views

Distribution of binomial with poisson-distributed trials

Assume there is an underlying process, $K$, governed by a Poisson distribution, $K\sim Poisson(\lambda)$. We observe realizations this process, but with an imperfect count detection device with ...
-1
votes
2answers
43 views

Probability of even sum of $n$ integers with uniform distribution from $\{1,2,\dots, 2n\}$.

Choosing with Uniform distribution $n$ numbers from $\{1,\dots,2n\}$ with returns and the order is important. What is the probability that the sum of these number will be even? Thanks.
1
vote
0answers
38 views

Equivalent definitions for weak/distribution convergence

We let $X$ be a compact metric space and consider $C(X)$ to be the space of all continuous functions on $X$. The dual space of $C(X)$ can be seen as the set of all signed borel measure on $X$. My ...
1
vote
0answers
22 views

find the conditional function (continuous- discrete)

Let $X_1$ be a continuous random variable with pdf $f(x)=\frac{2}{x^3},~~~~x>1$. $0 $ otherwise. Additionally the random variable $X_2$ is defined as $X_2=I_{[1,2]}(x)$. ($I$ := Indicator function) ...
3
votes
1answer
72 views

How to evaluate this Fourier Transform $A\int_{-\infty}^{\infty} \frac{e^{ikx}}{(1+x^{2})^{\frac{\nu+1}{2}}}dx$

This is basically the Fourier transform of a Student´s T pdf. How do we compute it? $$A\int_{-\infty}^{\infty} \frac{e^{ikx}}{(1+x^{2})^{\frac{\nu+1}{2}}}dx$$ for $\nu$ any number greater than zero ...
2
votes
1answer
122 views

Calculation of $\ln\left( \frac{S_{1}(t)}{S_{2}(t)}\right)$ where $S$ are stocks

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration of an incomplete finance market with stocks $S_j(t)$ for $...
2
votes
0answers
54 views

Integral involving the von Mises-Fisher distribution

I'm going quickly through the VonMises-Fisher distribution $M$ on $\mathbb S^{d-1}$ and its properties. Its probability density function is: $$f(x; \kappa,\mu)= c(\kappa)\exp(\kappa x^T\mu)$$ where $...
1
vote
1answer
39 views

Fisher information for exponential family: Regularity conditions

for the Fisher-Information to be defined certain regularity conditions have to be fulfilled (like in Lemma 5.3. in Theory of Point Estimation by E.L. Lehmann or on slide 2 here: http://www.stat.nus....
0
votes
0answers
36 views

Function of random variable: Two ways to find the pdf

Suppose $X$ is a r.v with pdf $f_X(x)$. Let $Y = g(X)$. To find the pdf of $Y$ - $f_Y(y)$. I use one of two ways and I assume g to be a monotonically increasing function. Method I first using the ...
4
votes
1answer
43 views

Why is the domain of the error function scaled by $\sqrt{2}$

The normal distribution function $\Phi(z)$ has the definition $\Phi(z) \equiv \frac{1}{\sqrt{2 \pi}} \int_0^z e^{\frac{-x^2}{2}} \, dx$. However the error function is conventionally defined such that ...
0
votes
1answer
29 views

Expected value and variance of discrete random variable

Let $Y$ be a discrete random variable with density function: $$p(y;\theta)=\left(\frac{\theta}{2}\right)^{\lvert y\rvert}(1-\theta)^{1-\lvert y\rvert}$$ where $y\in\{-1,0,1\}$ and $\theta \in[0,1]$...
0
votes
0answers
17 views

Indicator variable syntax

Are these syntax equivalent? $$f(x,\lambda)=\lambda e^{-\lambda x}I_{(0,+\infty)}(x),\ \lambda > 0$$ $$f(x,\lambda) = \left \{ \begin{array}{cl} \lambda e^{-\lambda x} & x \gt 0 ...
0
votes
0answers
13 views

Determine eigenvalue distribution support

I am working on a project regarding random matrix spectra and I need some help with the following: let us assume we are looking at some particular family of NxN random matrices in the limit of N -> ...
3
votes
0answers
49 views

How to approximate the cumulative distribution function of the normal by a product of functions?

Suppose, there are $n$ vectors $\mathbf{X}_1$, $\mathbf{X}_2 \ldots \mathbf{X}_n$ of unequal lengths which can be combined to a new vector as $$ \mathbf{X} = \begin{bmatrix} \mathbf{X}_1 & \mathbf{...
0
votes
2answers
82 views

A fair die is rolled 100 times. Which of the following has a probability of at least 95%?

A fair die is rolled 100 times. Which of the following has a probability of at least 95%? $ $ 1.) Sum of the rolls is greater than 322 2.) Sum of the rolls is less than 392 3.) Number of rolls ...
0
votes
1answer
25 views

Moment generating function (MGF) of the ratio distribution $\displaystyle\frac{X}{Y}$

If we know the moment generating functions (MGFs) of the random variables $X$ and $Y$ to be $M_{X}(s)$ and $M_{Y}(s)$, respectively. The MGF of the sum $X+Y$ will $M_{X}(s) \cdot M_{Y}(s)$. So what ...
0
votes
0answers
40 views

Central Limit Theorem for gambling return ratio

Consider a single bet with odds $o$ and thereby implied probability $1/o$. Assume that the real probability $p$ is known. Let $I$ be the stake, and $y$ the return from the bet. Then, $\mathbb{E}(y) ...
0
votes
0answers
15 views

Queue depth to keep workers busy

I'm trying to find a probability of keeping w workers busy with a q queue depth feeding those w workers. When the queue has at least one item in it the item can be taken and the item was randomly ...
2
votes
1answer
27 views

Understanding the flat (uniform) Dirichlet distribution density over a simplex

This should be really straightforward from the formula, but somehow I'm having trouble understanding the density of a Dirichlet distribution with $\alpha = [1, 1, ... 1] \in R^k$, which is a uniform ...
1
vote
0answers
10 views

Finding $P(S<0)$ with standard Normal Cumulative Distribution function

I know I'm supposed to use the the Standard Normal Cumulative distribution function. But I can't seem to get everything I need. Let $X$ be a random variable with $P(X=-1)=P(X=0)=0.25$ and $P(X=1)=0.5$...
2
votes
1answer
187 views

Find the almost sure limit of $X_n/n$, where each random variable $X_n$ has a Poisson distribution with parameter $n$

$X_{n}$ independent and $X_n \sim \mathcal{P}(n) $ meaning that $X_{n}$ has Poisson distributions with parameter $n$. What is the $\lim\limits_{n\to \infty} \frac{X_{n}}{n}$ almost surely ? I ...
1
vote
1answer
27 views

Joint density function of $T_1,T_2$ and expectation of $E[T_1 ^2 +T_2 ^2 ]$

Given that $T_1,T_2$ are random variables representing the useful life (in hours) of two electrical appliance. The joint probability function of two variables distributed uniformly in the domain ...
2
votes
1answer
28 views

Confusion in Calculating Conditional Probability mass function

Question: If $X_1$ and $X_2$ are independent binomial random variables with respective parameters $(n_1,p)$ and $(n_2,p)$, calculate the conditional probability mass function of $...
0
votes
0answers
47 views

Integral evaluation - Gamma distribution

I have a sequence of independent random variables which are $\chi^2(1)$ distributed, $(X_i)_{i=1}^n$, $X_i\sim\chi^2(1)$. If I consider the sum $\frac{t}{n}\sum_{i=1}^n{X_i}$ this should be $\sim\text{...
2
votes
1answer
43 views

Distribution of $\lceil X \rceil - X$ where $X$ has an exponential distribution

Suppose $X$ is a random variable with exponential distribution of parameter $\lambda > 0$. That is, $X$ has density $f(x) = \lambda e^{-\lambda x} \mathcal{1}_{\mathbb{[0,\infty [}}$. The question ...
2
votes
2answers
40 views

How to calculate probability distribution of a function of two independent Poisson random variables?

I can't figure out how to determine the probability distribution function of $$aX + bY,$$ where $X$ and $Y$ are independent Poisson random variable. Basically, I want to check whether $aX+ bY$ ...
2
votes
1answer
36 views

Expected number of duplicates

Suppose I have $m$ bins and throw $n\ll m$ balls into the bins uniformly at random. (In my application $n\sim m/\log m.$) What is the expected number of duplicates? In other words, if there are $k_i$ ...
3
votes
1answer
28 views

Proving that a positive-integer valued random variable has the lack of memory property iff it has a geometric distribution.

Suppose that $X$ is a positive-integer valued random variable with the lack of memory property which states: Given that $X>n$, then $\mathbb{P}(X=n+k) = \mathbb{P}(X=k)$. Consider the case ...
1
vote
0answers
32 views

Moment Generating Function for $r$th central moment

When using moment generating functions, to find the $n$th raw moment ("$n$th moment about the origin"), you take the $n$th derivative of the MGF and evaluate at $t=0$. To find the $m$th central ...
1
vote
1answer
57 views

Expected value of an infinite sum of random variables

For k=1,2... let Xk be independent and identically-distributed random variables with E(Xk)= $\mu$ and V(Xk)= $\sigma^2$ and let N be independent of the Xk with mean $\lambda$ and variance $\lambda^2$. ...
7
votes
1answer
200 views

Is there any probability model for multi-stage motion of an object such as this.

I have this following case (please refer to attached pic below) where a particle is resting on the ground and it needs a minimum amount of force (Fmin) to reach from one level to the next level. But ...
4
votes
1answer
70 views

Calculate a limit $\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \Big\{\frac{k}{\sqrt{3}}\Big\} $

The problem is to calculate a limit $$ \lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \Big\{\frac{k}{\sqrt{3}}\Big\} $$ where {$\cdot$} is a fractional part. I believe that this limit is equal to $\...
2
votes
2answers
57 views

operations on probability distributions

I've found that you can do certain arithmetic operations on random variables such as : multiply or divide two log-normal distributed variables add or divide two gamma distributed variables I've ...
2
votes
1answer
42 views

Probabilistic constraint implying deterministic constraint?

Suppose $X$ is an $N$-dimensional random variable $X := [X_1 \; X_2 \; \cdots \; X_N]$ such that all entries can either be 0 or 1 while satisfying the following: (i) $\mathbb{P}(X_i = 1) = p_i \; \; ,...
0
votes
0answers
25 views

Distribution of sum of weighted geometric random variables

Take $g_i$ to be a geometric random variable with parameter $1/2$, such that $$P(g_i = k) = \frac{1}{2^{k+1}}$$ for any integer $i$. I'm surprised at how much more difficult it is to evaluate this ...
3
votes
2answers
169 views

Calculating integral $\int_{0}^{\infty}x^2 \frac{f'(x)^2}{f(x)}dx$

This is a follow up question for this question: How can I calculate or simplify the following integral $$\int_{0}^{\infty}x^2 \frac{f'(x)^2}{f(x)}dx$$ If I know f(x) is a probability density ...
0
votes
0answers
16 views

Given probability generating function (PGF) for $X$, find PGF for $Y=2x+1$

PGF for $X$ is given by $G_{X}(t)=t^2e^{t-1}$, where $t\in\mathbb{R}$. Find PGF for $Y=2X+1$. I started it doing this: $G_Y(t)=\mathbb{E}t^{2X+1}=t\cdot\mathbb{E}t^{2X}=t\sum_{k=0}^{\infty} t^{2k}\...
0
votes
1answer
49 views

Distribution of a exponetial Random Variable

i have a stopping time $T$ of an Poisson Process $N$ with rate $\lambda$. Somehow this stopping time is exponential distributed. So we have $ T \sim exp(\lambda)$. I want to know the distribution of ...
1
vote
2answers
60 views

Proofs related to chi-squared distribution for k degrees of freedom

I was reading a proof related to chi-squared distribution for k degrees of freedom from wiki. https://en.wikipedia.org/wiki/Proofs_related_to_chi-squared_distribution I think I might understand the ...