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

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19 views

Looking for a certain probability distribution

I would like to know if one can find a probability distribution with finite mean and the following property: $F(2^{i+1})-F(2^i)\le p$ for given parameter $0<p<1$ all $i$. That is, if we ...
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1answer
165 views

Expected value of a complicated function of more than one random variable.

Assume we have random variables with Probability Density Functions (pdf) as follows $$\omega_i \sim f_{1},\,\,\,\,\ i \in[1:n]$$ $$ \gamma= \{\gamma_1,\cdots,\gamma_n\} \sim f_2: \text{joint pdf of ...
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1answer
15 views

If $X$ is distributed exponentially with $\lambda=2$, how can I find $Y=\sqrt{X}$?

I have a random variable $X$, that is distributed exponentially with $\lambda=2$. Therefore it's probability density function is as follows: $$f(x) = \begin{cases} 0, & x \leq0\\ 2\cdot e^{-2x}, ...
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1answer
22 views

PDF of distance from the center of a random point in the unit disk

if find in a certain website ( also in an IEEE paper) that the probability function for the distance mentioned in the title is given by the following: P(d)=2d, but no one is giving the way to derive ...
1
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0answers
53 views

Distribution of an area

I've got the problem of trying to guess the probability distribution of the following area $S$: We draw two lines, $r_1$ and $r_2$, randomly in the plane passing through the origin. So we have two ...
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1answer
27 views

Dependence of RVs exponentially distributed

Let $X$ and $Y$ be random variables that are both exponentially distributed with different parameters. I dont get the idea how can they be dependent then... and we can only manipulate one parameter in ...
-1
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1answer
30 views

How to find the distribution of a function of multiple, not necessarily independent, random variables? [closed]

If $Y$ is a random variable defined as $Y=g(X_1,X_2)$, where $X_1$ and $X_2$ are two different random variables whose distributions are known (say with pdf's $f_{X_1}$ and $f_{X_2}$), how do we find ...
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1answer
19 views

Unfamiliar syntax in two-variable functions, solving for C.

$$ f (x, y) = c(x + y), x = 1, 2, 3, y = 1, . . . , x. $$ I'm sure this is a pretty basic question. I've done problems of this kind, solving for see before. But x and y have always been clearly ...
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1answer
26 views

Does $P(X\leq a) = P(X^2\leq a^2)$ if $X$ is a positive random variable and $a>0$?

The answer looks positive to me, since $$P(\omega:X(\omega) \leq a) = P(\omega:X(\omega)^2\leq a^2)$$ Am I right?
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0answers
27 views

Is a limit of a sequence of distribution functions is necessarily a distribution function

I got a question: "is a limit of a sequence of distribution functions is necessarily a distribution function". My answer is NO. I have a counterexample: $F_n(x)=0$ if $x<n$ and $1$ otherwise. It ...
0
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0answers
30 views

How to get the pdf( or cdf ) of a random variable Y, such that Y=1/X, X>=a, a<0?

Given a random variable Y, such that Y = 1/X, X>=a, a<0, suppose the pdf of X is g(x), then what is the pdf of Y, which is f(y)? or what is the cdf of Y, which is F(y)? Recently I came across this ...
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0answers
32 views

What is the interval for an exponential random variable?

Suppose that I have generated a random number $x$ using an exponential distribution with rate parameter $\lambda$. How can I find an interval $[a,b]$ such that $x$ is in this interval with probability ...
0
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0answers
42 views

How to find the variance of a function of a normally distributed random variable. The function is cummulative function of normal distribution

I am wondering how to find the variance of a special function of a normally distributed random variable. More specifically, I am confused by the following question: assume $Y\sim N(\mu,\sigma^2)$, how ...
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1answer
19 views

Determining which test statistic is better under two different scenarios

Problem: I have 9 independent normal observations from a normal dist. with $\sigma=10$. And we want to test $H_0:\mu=150$vs $H_A:\mu\neq 150$. Also $\alpha=0.05$ which uses the test statistic ...
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2answers
27 views

Finding probabilities - Elementary probability.

Suppose $\mathbb P(A) = 0.3$, $\mathbb P(B) = 0.5$, and $\mathbb P(B |A) = 0.6$. a. Find $\mathbb P(A \text{ and } B)$. b. Find $\mathbb P(A \text{ or } B)$. c. Find $\mathbb P(A|B)$. ANSWER: a. ...
1
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1answer
9 views

Expectation of distance from centre of a circular scattering

A random point $(X,Y)$ has a normal distribution on a plane with circular scattering with $E[X]=E[Y]=0$ and var$[X]$=var$[Y]$=$\sigma^2$. The distance of the point $(X,Y)$ from the centre of ...
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0answers
12 views

central limit theorem and sampling dist.

If you takes samples from a distribution, and you can see that they have different variances, can the central limit theorem still be applied. The computer vision teqnique i am referring to is this ...
2
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1answer
29 views

How to determine a conditional distribution

Consider: $$X\stackrel{d}{=}N(\mu, \sigma^2)\qquad Y\mid X \stackrel{d}{=}N(\alpha+\beta X, \tau^2) \qquad U\mid Y,X \stackrel{d}{=} N(0,\nu^2)$$ Determine the distribution of $W=X+U$ given ...
1
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1answer
33 views

How do you represent a random choice of random variables mathematically? What is its mean, variance, etc.?

Suppose that I have six random variables $X_1, X_2,\ldots,X_6$ (say, e.g., six coins with different biases). We should be able to get a new random variable $Y$ by rolling a die to get a number $n\in ...
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0answers
14 views

Partial derivative of a random vector

If $x$ indicates a $1\times n$ random vector of any distribution, then is the partial derivative of $x$ w.r.t $x$ equal to the derivative of the individual elements in the matrix, or are they just the ...
0
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1answer
24 views

Given common probability density function $f(x, y) = e^{-(x+y)}$ where $0 \le x ,y < \infty$, calculate $P(X < Y)$.

First time I'm getting to such question: Given common probability density function $f(x, y) = e^{-(x+y)}$ where $0 \le x ,y < \infty$, calculate $P(X < Y)$. What is the way to approach ...
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0answers
63 views

Duration of a Gambler's Ruin game against an opponent with infinite credit

A gambler enters the casino with $x\$$ in his pocket and sits on some table. At each iteration he bets $1\$$ and either wins $1\$$ with probability $p\leq\frac{1}{2}$ or loses $1\$$. Assuming that ...
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1answer
36 views

Find the distribution of sum and product of standard normal random variables

Let $X,Y$ and $Z$ be three independent real valued random variables. All with finite second moment and all with mean $0$ and variance $1$. Define $$ W= \frac{X+YZ}{\sqrt{1+Z^2}} $$ Find the ...
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2answers
51 views

Derive the distribution of a lower censored s.v.

I could use some pointers solving this problem: Given a certain s.v. $X$ with cdf $F_x(x)$ and pdf $f_X(x)$. Let s.v. $Y$ be the lower censored of $X$ at $x=b$. Meaning: $$Y = \begin{cases}0 ...
4
votes
2answers
69 views

Limit $ \lim\limits_{n\to\infty}\Bigl[\frac{1}{2^{n/2}\Gamma(n/2)} \int_{n-\sqrt{2n}}^{\infty} t^{\frac{n}{2}-1}e^{\frac{-t}{2}}\,dt\Bigr]$

Find $ \lim\limits_{n\to\infty}\left[\dfrac{1}{2^{n/2}\Gamma(n/2)} \displaystyle \int_{n-\sqrt{2n}}^{\infty} t^{\frac{n}{2}-1}e^{\frac{-t}{2}}\,dt\right]$ This looks like the p.d.f. of a chi-square ...
0
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0answers
16 views

$\Sigma$ meaning when describing gaussian distributions?

What does it mean when the $\Sigma$ is used instead of $\sigma$ when describing a distribution? Does it have distinct features, or is related to something else?
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1answer
32 views

Prove $\mathbb{P}( k < l/2 ) \geq \frac{l}{2} \times \mathbb{P}( k = l/4 ) $ for binomial variable $k$

Suppose we have a Binomial variable: $$ k \sim Bin(l,\alpha) $$ Is it possible to prove/disprove that: $$ \mathbb{P}( k < l/2 ) \geq \frac{l}{2} \times \mathbb{P}( k = l/4 ) $$ EDIT: it's been ...
0
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1answer
41 views

Multivariate Hypergeometric Cumulative Distribution Function

I think my problem is unique in that it hasn't been posed here. Starting with a simple case to which I think I have an answer: I have 11 cards, 3 of which are bad. These cards are used in a game ...
2
votes
1answer
22 views

Rule of thumb for the number of unique values from a distribution drawing

My main problem is to count the number of unique values (number of symbols) drawn from a specific finite distribution $(p_i)$ of $m$ symbols, after $n$ drawings. The difficulty is to estimate it ...
0
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0answers
21 views

Does sequence of quantiles $x_{p_n}$ converge to endpoint $x_F$ of cdf F?

Let $X$ be a RV with continuous cdf $F$. Let $p_n$ be a sequence in $(0,1)$ with $\lim_{n \to \infty}p_n = 0$ and $x_{p_n}=\inf \{y \in \mathbb{R}:F(y) \ge 1-p_n\}$ the $p_n$-quantile of F. Define by ...
0
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0answers
31 views

Marginal density from joint distribution of 3 random variables

Suppose random variables X, Y, Z have joint probability density function given as \begin{equation} f_{XYZ}(x,y,z) = \frac{2x}{(A_2-A_1)(R_2^{2} - R_1^{2})y^{2}z} e^{-z^{2}/2} \end{equation} where ...
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0answers
9 views

Different continuous distributions applicable to crowd scenarios

I working on different distributions of crowd exiting from a subway train. Right now I have studied normal distribution, that is centred around the exit. What could some other such distributions that ...
0
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1answer
13 views

PDF of a uniform RV with a RV upper bound

I am trying to find the PDF of a uniform RV that has a uniform RV upper bound -- in particular, the variable is uniformly distributed between 0 and Q, where Q is a random variable. The RV Q has PDF ...
0
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0answers
17 views

Sampling from general multivariate Gaussians

Suppose you have access to a sampler such as randn in Matlab/Octave that returns samples from a simple one-dimensional Gaussian distribution (a normal distribution) ...
0
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1answer
35 views

Joint PDF of maximum of $n$ and maximum of $n+1$ random variables

Let $X_1, X_2, . . .$ be i.i.d. r.v.s with CDF $F$ , and let $M_n$ = $max(X_1 , X_2, . . . , X_n)$. Find the joint distribution of $M_n$ and $M_{n+1}$ , for each $n \geq 1$. \begin{align} ...
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1answer
15 views

Exponential Distribution - serving time of customer

A post office has 2 clerks. Ashima enters the post office while 2 other customers are being served by the clerks. She is next in line. Assume that the time a clerk spends serving a customer is ...
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2answers
27 views

Value of constant k which makes the function $f(x)=\frac{k|x|}{(1+|x|)^4}$ a p.d.f.

Let $f(x)=\dfrac{k|x|}{(1+|x|)^4}$, $-\infty<x<\infty$. Then, what is the value for which f(x) is a probability density function ? f(x) will be a p.d.f. if $\displaystyle ...
0
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1answer
24 views

scatter versus covariance

I am a bit confused about the difference between a scatter matrix (I believe it is sometimes called a scatter matrix) and the covariance. Some distributions, such as elliptical distributions, may not ...
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1answer
21 views

PDF of the ratio of two independent Gamma random variables

Let $X \sim \operatorname{Gamma}(a,\lambda)$ and $Y \sim \operatorname{Gamma}(b,\lambda)$ being independent. Find the PDF of the ratio $W=X/Y$. I found $$ f_W(w) = \frac{\Gamma(a+b)}{\Gamma(a) + ...
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0answers
22 views

How to identify the future distribution of a stochastic variable from its SDE

I would like to know some common practice to identify the future distribution of a random variable modelled by an arbitrary SDE. Would you study it empirically (like generating Monte-Carlo ...
1
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1answer
52 views

A question on a certain transformation of a normal random variable

I have to solve the following exercise: Suppose $X\sim N(\mu,1)$ and consider $Y=\dfrac{1-\Phi(X)}{\phi(X)}$, where $\phi,\Phi$ are the pdf and cdf of the standard normal distribution with ...
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2answers
45 views

How to add two random variables?

Given that $$\begin{array}{cccc} \text{X} & -1 & 1 & 3 \\ \text{p} & 0.2 & ? & 0.3 \\ \end{array}$$ and $$\begin{array}{cccc} \text{Y} & 1 & 2 & 3 \\ \text{p} ...
0
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0answers
37 views

PDF of product of 3 random variables

I am trying to find the PDF of S = XY/Z^2, where X,Y,Z are independent with X is uniform [-A1, A2], Z is uniform [B1, B2] and Y positive random variable with any distribution (for example Rayleigh). ...
3
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0answers
40 views

Estimating the support of a probability density function

The inverse moment problem deals with the reconstruction of a probability density function (PDF) of a random variable (RV) by means of its statistical moments. In the special case of the Hausdorff ...
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1answer
12 views

Gamma-Poisson Conjugacy

Busses arrive at a certain bus stop according to a poisson process with rate $\lambda$ buses per hour, where $\lambda$ is unknown. The uncertainty about $\lambda$ is quantified using the prior ...
0
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1answer
18 views

Let $X,Y \sim \operatorname{Expo}(\lambda)$ i.i.d, and $T = X + Y$, $W = X/Y$. Find joint and marginal PDF of $T$ and $W$

Let $X$ and $Y$ be i.i.d. $\operatorname{Expo}(\lambda)$, and transform them to $T = X + Y$, $W = X/Y$ . (a) Find the joint PDF of $T$ and $W$ . Are they independent? (b) Find the ...
8
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1answer
99 views

Heavy-tailed distributions

I have encountered the following two definitions of heavy-tailedness (right tail) for a $[0,\infty)$-valued random variable $X$ satisfying $\mathbb{E}[X]<\infty$: (i) ...
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1answer
18 views

Deriving the marginal posterior

Context of the question: You can take everything below as given. $E_2$ is a $k$ by $1$ matrix and $V_{22}$ is a $k$ by $k$ matrix. Let $X$ denote the data. I have derived so far the joint posterior ...
0
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1answer
26 views

Distribution of the power of an exponential random variable

Three students are working independently on their probability homework. They start at the same time. The times that they take to finish it are i.i.d. random variables $T_1$, $T_2$ , $T_3$ with ...
2
votes
1answer
32 views

Intuition for proof of Slepians Inequality

If z is a centered gaussian random variable and $ x_1 ,x_2 ,..,x_n ,y_1,y_2,..,y_n $ are points in $ \mathbb{R}^{2n} $ satisfying $ |x_i-x_j |_2 \leq |y_i -y_j |_2 \ \ \ \forall i,j \in [n] $ then $ E ...