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

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65 views

Show the following random variables in $\mathbb{R}^2$ have the same distritbution

$X_1, X_2, \cdots, X_n$ are independent Gaussian $N(0,1)$ random variables. I need to show that the following random variables in $\mathbb{R}^2$ have the same distribution: $\displaystyle ...
2
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1answer
60 views

simplify expectation definition Hidden Markov Model

I am reading Rabiner's paper entitled "A tutorial on hidden markov models and selected applications in speech recognition". There is a very simple example where he simplifies the calculation of an ...
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2answers
348 views

Transformation of beta distribution into gamma distribution

How can I convert a Beta Distribution to a Gamma Distribution? Strictly speaking, I want to transform parameters of a Beta Distribution to parameters of the corresponding Gamma Distribution. I have ...
2
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2answers
412 views

Joint distribution of integer and fractional part of random variable

Here's the set up: Let the random variable $X$ have the following distribution $$f(x)= \begin{cases} e^{-x} & \quad \text{if}\hspace{2mm} 0<x<\infty\\ 0 & \quad ...
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2answers
193 views

Probability distribution function of the length of an interval taken from a uniform probabilty distribution.

This is a consequence of my suggested solution to this question. Consider the probability distribution function that is uniform over the interval $[-a,a]$: $$F(x)=\begin{cases} 0 & x \leq -a\\ ...
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1answer
110 views

geometric distribution throwing a die

The problem says as follows: We throw a die repeatedly. $X$ and $Y$ denote, respectively, the number of rolls until we reach a $5$ and $6$. Then the question is to compute $E[X\mid Y=1]$ and $E[X\mid ...
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0answers
51 views

Deconvoluting linear combinations of unknown distributions

Summary I am trying to deconvolute the distribution $T(x)$ of a population's $x$ parameter into sub-distributions ($P(x)$, $Q(x)$, $R(x)$ ...), of which I don't know the form (only that they have ...
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1answer
382 views

Transformation of probability density function

Having two independent random variables $X$ with pdf $f_X$ and $Y$ with pdf $f_Y$ what is the correct way to derive the formula for $f_Z$ where $Z = X/Y$ and $f_W$ where $W = XY$ ? I know that the ...
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1answer
507 views

How to fit the Cauchy distribution to the data

I have data on financial returns, and I want to fit the Cauchy distribution and student distribution to that data. Furthermore I want to check the goodness of fit in both cases. Where should I start ...
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4answers
483 views

How can I interpret $\max(X,Y)$?

My textbook says: Let $X$ and $Y$ be two stochastically independent, equally distributed random variables with distribution function F. Define $Z = \max (X, Y)$. I don't understand what is ...
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1answer
81 views

Joint Distribution Function

Hi Guys, Just need help understanding how to go about doing this question. I know how to convert single distributions but I'm unsure about how to do the joint ones. I usually draw a diagram ...
2
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1answer
225 views

Average distance between random points inside a cube

Let $U_1, U_2, \ldots, U_n$ be a set of random variable uniformly distributed over some box in $\mathbb{R}^3$ and let \begin{equation} R = \frac{1}{2 n(n-1)}\sum\limits_{i,j< i} |U_i - U_j| ...
3
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3answers
463 views

Expected number of overlaps between intervals

Suppose $N$ intervals of length $\delta$ are positioned in $[0,1]$. The starting point $l_i$ of each interval is drawn from an uniform distribution, i.e., $l_i \in [0, 1-\delta]$, thus it will ...
2
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2answers
3k views

Finding the probability density function of $Y=e^X$, where $X$ is standard normal

Let the random variable $X$ have the $N(0,1)$ distribution for which the probability function is: $$ f(x)= \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x^2}{2}\right), -\infty< x <\infty $$ Let ...
2
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2answers
117 views

Discrete probability problem

Problem: Assume the number of cars passing a road crossing during an hour satisfies a Poisson distribution with parameter $\mu$, and that the number of passengers in each car satisfies a binomial ...
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2answers
150 views

Are waiting times always going to be exponentially distributed?

I'm studying for CAS/SOA Exam P/1 and a question I have here is: We have a portfolio of $20$ insurance policies. The number of claims per policy in a $3$-month period has a Poisson distribution ...
8
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5answers
550 views

Successful approaches to the modelization of ''randomness''

If you pick a number $x$ randomly from $[0,100]$, we would naturally say that the probability of $x>50$ is $1/2$, right? This is because we assumed that randomly meant that the experiment was to ...
2
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1answer
99 views

The probability distribution $\alpha^{\beta^x}$

I am interested in exploring the probability distribution given by:- $$\mathbb{P}(X\ge x)=F(x)=\alpha^{\beta^x}$$ with probability density function, ...
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1answer
154 views

Is Lottery probability really the same for all combos?

http://justwebware.com/uklotto/uklotto.html Test run quickpick Test run 1,2,3,4,5,6 Test run (single digit,teens,twenties,twenties,thirties,forties) 1000 times or more each cycle for as many ...
0
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1answer
383 views

Gamma distribution from the sum of independent exponential distributions

From a paper I'm currently reading: In the simplest setting, the average waiting time (or equivalently the departure rate) in each stage is assumed to be equal: the overall infectious period is ...
1
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2answers
90 views

Conditional distribution question

When we talk about conditional distribution we mean $F_{Y|X}(y|x) = \mathbb{P}(Y\leq y|X=x)$. Does there exists the following object: $F_{Y|X}(Y|X)$? I'm refering to conditional expectations when we ...
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1answer
377 views

Maximum order statistic for Binomial distribution

Let $X_i$, $1\le i\le t$, be $t$ independent random variables with Binomial distribution $B(n,\frac1t)$. I would like to find the distribution of $X_{Max}=\max_{i=1}^t(X_i)$ Note that this is the ...
9
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1answer
199 views

$P(x,n) = \frac{x(n-1)!}{n^{x}(n-x)! }$ — What is the name of this probability distribution?

$$ P(x,n) = \frac{x(n-1)!}{n^{x}(n-x)! } $$ I'm having a really tough time describing what this distribution does, but it's simple in code. So if you know code, then read on: ...
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2answers
7k views

Multiplication of a random variable with constant

Suppose $X$ is a random variable which follows standard normal distribution then how is $KX$ ($K$ is constant) defined. Why does it follow a normal distribution with mean $0$ and variance $K^2$. ...
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1answer
53 views

Independence of unknown Random variables

Suppose X and Y are two unknown random variables with Pdf's f(x) and g(x) then by looking at their graphs can we say any thing about their independence.
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1answer
219 views

Find the marginal density $f_Y(y)$ of a random vector $(X,Y)$.

Given a (continuous) bivariate random vector $(X,Y)$ with a probability density: $$f_{XY}(x,y)=\left\{ \begin{array}{l}\tfrac{1}{y}e^{-(y+\tfrac{x}{y})} & \text{if }x>0 \text{ and } y>0 \\ ...
2
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0answers
288 views

Joint pdf, conditional density and correlation between max and min of two independant exponetially distributed variables

I am trying to solve this problem where instead of uniform distribution, x and y are distributed EXP(1/b): Suppose X and Y are two independent random variables, both uniformly distributed on (0,1). ...
2
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2answers
155 views

Approximation of discrete distribution

I have a discrete random variable $X$ which takes the values $+1$ and $-1$ with equal probability $\frac{1}{2}$. Can I approximate this with a normal distribution ?
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1answer
148 views

Independent distributions

I have a confusion regarding the notion of independence of distributions. what is meant by saying two distributions are independent..? suppose I have two normal distributions with means 1,-1 and ...
0
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1answer
68 views

Which distributions should be used to model the winning & 2nd bids in second price auctions?

With second price auction which distributions should I use to model the winning bids and 2nd bids (separately)? I'm thinking of using Gaussian. However for the winning bids r.v, it has to satisfy: $$ ...
1
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1answer
147 views

Large Deviations Result for non iid variables/ Conditional Large Deviations?

Let $X_n, n\in\mathbb{Z}$ be a sequence of independent random variables (finite, let's say the size of the alphabet is $2$ to simplify things) with mean zero and variance less than $1$. Is there a ...
2
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1answer
94 views

Law of Large Numbers for “Approximately” iid samples

Let $Z$ be a random variable taking values from the finite set $\mathcal{Z}$ with pmf $p_Z$. Let $Z^n$ be a random iid vector drawn according to $p_Z^n$, then LLN states that the sample mean of $Z^n$ ...
1
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1answer
317 views

The distribution when combining two samples together?

Suppose $X\sim N(0,{\sigma}^2)$ and $Y\sim N(0,{2\sigma}^2)$ . $X_1, ..., X_m$ are the samples from $X$ and $Y_1, ..., Y_n$ are the samples from $Y$. And then combine two samples as a new sample ...
1
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1answer
2k views

Product of two independent poisson variables

Suppose we have two independent poisson variables $X_1$ and $X_2$ such that $X_1 ∼ \operatorname{Poisson}(\lambda_1)$ and $X_2 ∼ \operatorname{Poisson}(\lambda_2)$. What will be the probability ...
3
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0answers
72 views

Find a density function for the endpoint of this stochastic process

$(X_t, Y_t, Z_t)$ is a three-dimensional stochastic process described as follows: $X_t$ is a Brownian Motion. $Y_t = \int_0^t X_s ds$ $Z_t = \inf_{s \in [0, t]} X_s$ I would like to find a density ...
2
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2answers
353 views

Relating two proofs of binomial distribution mean

There are two ways of calculating the mean of the binomial distribution. One is to observe that the distribution measures the number of successes in a sample size $n$ drawn from space of size $N$. ...
3
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1answer
172 views

Find the transition function of this stochastic process

Let $(X_t, Y_t)$ be a two-dimensional Markov stochastic process that runs on time interval $[t_0, t_f]$. Its infintesimal generator is described by the functions $\mu_X, \mu_Y, \sigma_X, \sigma_Y$. I ...
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3answers
648 views

Compute probability of a particular ordering of normal random variables

There are $m$ normally distributed, independent random variables $N_1, \ldots, N_m$ with distinct means $\mu_1, \ldots \mu_m$ and standard deviations $\sigma_1, \ldots, \sigma_m$. Then, we get a ...
3
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1answer
649 views

What's the difference between expected values in binomial distributions and hypergeometric distributions?

The formula for the expected value in a binomial distribution is: $$E(X) = nP(s)$$ where $n$ is the number of trials and $P(s)$ is the probability of success. The formula for the expected value in a ...
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1answer
36 views

When to use other transforms?

maple code int(g*f, x=-infinity..infinity) when $g$ is $\large exp^{i*t*x}$, Fourier transform between density function and characteristic function If $g$ are $x^t$, $|x^{t}|$, $t^{x}$, what do they ...
0
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2answers
691 views

binomial distribution with two random variables

my teacher said that when you have two random variables X and Y, both are binomially distributed, then X-Y can never be binomail distribued? Why? I recall he mentioned something because it only takes ...
6
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1answer
151 views

Characterizing a distribution by its projections

Consider the density $f(x,y)=\large\frac{1}{2\pi}\frac{1}{\sqrt{1-x^2-y^2}}$ on the unit disk centered at the origin. There is a particular characterization of this distribution: it is the unique ...
3
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0answers
66 views

Is this a valid method for time-integrating a stochastic process?

I have a stochastic process $X_t$, and I have a function $a(x | t)$ that reflects my beliefs about the value of $X_t$ ($a$ is a density function in its first parameter). I am studying the properties ...
4
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1answer
130 views

When is a function of two Normal variables Normal?

I know that for two independent Gaussian variables, the sum and the product is Gaussian as well. Is there a general form for this, ie a class of functions $f:\mathbb R^2 \rightarrow \mathbb R $ so ...
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1answer
56 views

Can we find the parameter(s) for gamma given the mean?

If were told that something is gamma distributed with mean $m$, are we able to use that to find one of the parameters? Like if were told that something is exponentially distributed with mean $m$, then ...
2
votes
0answers
67 views

Gaussian Bayesian filtering with bound observation ($b_1<x<b_2$)

Suppose we have a Normal r.v $$ x \sim \mathcal{N}(\mu, \sigma^2) $$ and a Normal prior of $\mu$ $$ \mu \sim \mathcal{N}(\theta, \delta^2) $$ I know how to do the Bayesian update with a ...
1
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1answer
114 views

Let $F(x,y)=1$ for $x+y\ge 0$ and be zero otherwise. Show that $F$ cannot possibly be the joint distribution function of a pair of random variables.

Let $F(x,y)=1$ for $x+y\geq 0$ and be zero otherwise. Show that $F$ cannot possibly be the joint distribution function of a pair of random variables. Ok so basically I need to show that there can't ...
0
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1answer
135 views

Probability density of vector sum

Consider two unit $\mathbb R^2$ vectors $v$ and $w$. Then $v+w$ lies within a (closed) circle with radius 2, that is, in the region $x^2+y^2\leq4$. Intuitively, the probability of $v+w$ lying close ...
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1answer
237 views

Don't understand the *derivation* of geometrically distributed random variables

I don't understand the derivation of geometrically distributed random variables as done here (only the first $10$ lines - everything until exercise $2$ - are relevant for me). Please bare with me, ...
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1answer
918 views

What does it mean that the probability density function is proportional to a function?

I'm studying for SOA/CAS Exam P and I have a problem that says that $X$ is a continuous and positive random variable whose probability density function is proportional to: $$\frac{1}{(1+x)^5}$$ Where ...