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

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1answer
24 views

Poisson distribution of false alarms

I have this problem: The daily amount $X$ of burglar alarms has the Poisson distribution with parametr $\lambda>0$: $$P[X=k]=\dfrac{\lambda^k}{k!}e^{-\lambda}, k=0,1,...$$ It is known that ...
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0answers
20 views

Joint distribution of two normal marginal distributions

My question is related to the possibility of stating joint convergence in distribution from marginal weak convergence. Consider two sequences of random vectors $X_n$ and $Y_n$ defined on the ...
2
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2answers
29 views

Convergence rate of mean and standard deviation.

I have a random variable simulator with Normal distribution $(\mu,\sigma^2)$. I repeatedly conduction simulation. Each time, the simulation gives $N$ numbers $x_1,x_2,\ldots,x_N$. I use the $N$ ...
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0answers
11 views

The exact usage of Sequential Monte Carlo for distributions over time?

I have wondered the usage of Sequential Monte Carlos and it is used as an alternative to Kalman filter for example. However I wonder if this can be also used for simulating a distribution over time? ...
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0answers
30 views

Find a Markov chain transition kernel

Let $f_{X}$ be a density we would like to sample from. For some reasons, $f_{X}$ may be analytically intractable or expensive to evaluate. A solution consists in considering a density $(x,y) \in X ...
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0answers
20 views

Alternative way to compute gamma integral of Chi-squared distribution

I am developing a software with Java, which is not comfortable at all wih integral computations. So I must compute the following integral (I need it to compute Chi-Squared CDF): $$ ...
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2answers
48 views

How to find the joint PDF?

Ley X,Y two random variables with density $f(x,y)=8xy \ \text{if} \ 0<x<y<1$ Find the joint distribution F(x,y). I find that $F(x,y)=2x^2y^2-x^4\ \text{ if} \ 0<x<y<1$ but I ...
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0answers
20 views

Sum of two random variables - uniform distributions [duplicate]

I have two continuous uniform random variables I need to add. I read that to get the sum of two pdfs you convolve them. I'm getting a bit confused on the limits of integration though. If both RVs are ...
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1answer
34 views

Please help me to understand how to read statistical tables

Sorry I never learnt from a professor or class how and now when I look at them I don't know what to do. Here is an example. The Chi Squared table, ...
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0answers
21 views

Given difference of two RV draws, what is the new expectation of each individual draw?

We have two draws, $d1$ and $d2$, from a known distribution $F()$. Provided that we know $d1-d2$, can we update our expectations/distribution for an individual draw? That is, what are: $ ...
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1answer
41 views

Is the following a distribution function [closed]

Is the following a distribution function? $$F(x) =\begin{cases} e^{-1/x} &\text{ if } x>0\\ 0 &\text{ otherwise} \end{cases}$$ If so, give the corresponding density function. If not, ...
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0answers
25 views

Beta distribution decomposition

I'm doing an exercise and seem to be stuck. Let $(X_k)_{k=0}^\infty$ be a sequence of iid $\text{Beta}(\alpha, \beta)$-distributed random variables and let $X \in \text{Beta}(\alpha, \alpha + ...
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0answers
50 views

Prove that the limit in probability of normally distributed random variables is normally distributed, too [duplicate]

Let $X_n\sim\mathcal N_{\mu_n,\sigma_n^2}$ for some $(\mu_n,\sigma_n^2)\in\mathbb R\times(0,\infty)$ and $X$ be a real-valued random variable with $$X_n\stackrel{\text{in probability}}\to X\;.\tag 1$$ ...
1
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1answer
35 views

Joint characteristic function of $x$ and $y=x^2$ if $x$ is the standard normal variable

How to find the joint characteristic function of $x$ and $y=x^2$ if $x$ is standard normal variable with mean $0$ and variance $1$?
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0answers
20 views

Probability of intersecting events continuous in the probability of the events

Consider two (dependent or independent) events A and B. It seems natural $P(A \cap B)$ is continuous in the sense that $P(A \cap B)$ does not "jump" if the probabilities of A and B are slightly ...
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2answers
25 views

The discrete probability density of $Z=XY$

$X$ and $Y$ are independent random variables. $X$ : Bernoulli with $\frac{1}{4}$ success parameter $Y$ : Bernoulli with $\frac{1}{2}$ success parameter Calculate the discrete probability density ...
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0answers
19 views

Joint distribution probability problem. 2 [on hold]

I need to find $P [X+Y\lt z]$. The joint CDF is given by $0.5xy(x+y)$ for $0\lt x\lt 1$ and $0\lt y\lt 1$. $0.5x(x+1)$ for $0\lt \lt 1$ and $y\gt1$ $0.5y(y+1)$ for $0\lt 1$ and $x\gt1$ Now I ...
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3answers
305 views

Maximum of a sum of random variables

Let $X_1, \dots, X_n$ be independent and identically distributed random variables with $E(X_i) = 0$ and $$S_k = \sum_{i \leq k} X_i$$ What is the probability distribution of $M_2 = \max \{ X_1, ...
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1answer
23 views

Cumulative distribution functions and random variable problem

So the question is: If X is a random variable with a cdf $FX (t)$, and Y is the random variable given by Y = aX + b. Express the cdf $FY (t)$ of Y in terms of $FX (t)$. (Consider separately the ...
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1answer
64 views

Limits involving integration.

Solve the limit $$\lim_{n\to \infty}\left[\frac{1}{2^{\frac{n}{2}}\Gamma({\frac{n}{2}})}\int_{n+\sqrt{2n}}^{\infty} e^{-\frac{t}{2}}t^{\frac{n}{2}-1} dt\right] $$ equals After looking at it for ...
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2answers
45 views

With/out replacement ball drawing distribution

So the question asks if an urn contains n balls labeled 1,2,..., n. We draw m balls from the urn, one at a time. Let Y be the largest label of a ball in the draw. Find the distribution of Y (including ...
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1answer
38 views

Poisson distribution question

The question says when an $8$-bit word is being transmitted. There is a chance of $0.1$ for an error in each bit independently. Every bit is transmitted $3$ times. When decoding, choose the bit that ...
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0answers
40 views

Distribution of Product of Matrices holding Multivariate Random Normal Variable Observations

We have matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ that hold observations from multivariate normal distributions. Each matrix represents $\boldsymbol{n}$ observations of $\boldsymbol{m}$ variables ...
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1answer
19 views

Finding value of constant c from joint pdf

$f(x,y) = c(x^2 + y^2)$, for $x = -1, 0, 1, 3$ and $y = -1, 2, 3$. I missed a couple classes so I've only seen joint pdf represented in a table ... not an equation. If someone could just walk me ...
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2answers
63 views

Is the composition of CDFs also a CDF?

Two functions $F(x)$ and $G(x)$ are known as cdf functions. Is $$F(G (x))$$ necessarily a cdf function? My answer is no, and here is my counterexample: Because a cdf function qualifies: ...
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0answers
28 views

Distribution of “range” of a process

Let $X_t$ be a stochastic process, for example a brownian motion (i.e. $X_{t+h} - X_t \sim \mathcal{N}(0,\sqrt{h}^2)$). The difference between now's value $X_t$ and a past value $X_{t-100}$ is $$M_t ...
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0answers
46 views

Log concavity of $\int_{x \in \mathcal{C}_1} t^n \, e^{-\sum_i x_i t} \; dx$

Consider a real function $f(x) > 0$ with $n$ strictly positive arguments $x_i>0$ with the following properties: i) $f(x)$ is homogeneous of degree one in $x$ ii) Strictly increasing in all ...
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1answer
25 views

Find the expected value of “Y”, exponential family with lots of questions here

I have a problem I don't know how to approach. It is A generalization of the 1-parameter exponential family, to allow 2-parameter distribution, is the family given by $$f(y;\theta, ...
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1answer
32 views

Alternative proof of $\prod_{n=1}^{+\infty}\frac{e^{it/2^n}+1}{2}=\frac{e^{it}-1}{it}$

Let $t\in \mathbb{R}$. I want an alternative proof of the following identity $$\prod_{n=1}^{+\infty}\frac{e^{it/2^n}+1}{2}=\frac{e^{it}-1}{it} \quad(\star)$$ I've came up with this identity observing ...
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1answer
21 views

propertis of Truncated Normal 2 [closed]

If x has a Truncated Normal Distribution (μ,σ) and $$E(x)=\mu+\sigma \frac{\phi(l)}{1-\Phi(l)}$$ what is the E(|x|)?
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1answer
66 views

Simple CDF Computation for Products of Random Variable

Let $X(k)$ be i.i.d random variable governed by uniform distribution $[-1,1]$ for $k=0,1,2,...N$. I would like to compute the following CDF $$ P\left( {\prod\limits_{k = 0}^{N - 1} {(1 + X(} k)) ...
2
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1answer
39 views

Cumulative distribution functions (cdfs) question

If I have two cumulative distribution functions $F(x)$ and $G(x)$, are $$F(x)G(x)\quad\text{ and }\quad [F(x)G(x)]^{0.5}$$ necessarily cdf? I feel a little weird, cause I am thinking to qualify to be ...
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3answers
51 views

Let $X\sim\text{unif} [0,1]$. Find the probability density function of $Y=(1/X)-X$.

Let $X\sim\text{Uniform} [0,1]$ and let $$Y=\frac{1}{X}-X.$$ Find the pdf of $Y$. $f(x)=1$ if $0≤x≤1$, $0$ otherwise $Y$ is a decreasing function. Thus, the pdf should be: ...
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1answer
40 views

Question about weak convergence of random variables

When you start to learn probability theory, for instance the central limit theorem, you learn about convergence in distribution $X_n\to X$ (where, say, both $X_n$ and $X$ are $\mathbb R$-valued random ...
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1answer
41 views

Computing the expectation of a Tricky R.V. (Brought form Neuroscience).

I need to compute $\Bbb{E}(\tau^{X} \ \Bbb{1}_{\{\tau^{X}<+\infty\}})$ where: $1) $ $\tau^y$ is a r.v. representing the time spent by a particle until it "jumps", ( $ y \in R_{\geq 0} $ is the ...
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1answer
17 views

Chance of something occuring multiple times

How do I calculate the chance that some event, that has $p\%$ chance of occurring per observation, will occur at least $k$ times over $n$ observations?
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1answer
27 views

CDF for Laplace distribution

According to the Wiki article on the Laplace distribution, $$F(x)=\int\limits_{-\infty}^x f(u)du=\begin{cases} \frac{1}{2}\exp(\frac{x-\mu}{b}) && \text{if }x< \mu \\ ...
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1answer
26 views

Is difference of two independent Gaussian R.Vs and their sum independent?

I have been trying to answer a question I have been carry on from probability course. I'd appreciate if anyone can help me. Suppose we have two independent Gaussian distributions, both zero-mean with ...
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0answers
19 views

Random Variable with p.d.f. as product of two $1d$ Gaussians?

$X_1 \sim N(\mu_1,\sigma^{2}_1)$ $X_2 \sim N(\mu_2,\sigma^{2}_2)$ What random variable as function of $X_1$ and $X_2$ has p.d.f. which is equal to product of two Gaussians?
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21 views

Which distribution is fully decided by the first four moments?

For example, Gaussian distribution is fully decided by the first two moments, i.e., mean and variance. Another example is the generalized Gaussian distribution which has one additional parameter, ...
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0answers
26 views

Finding a contour of constant density in a joint distribution

I would appreciate any help to find the solution for the following problem. In short the problem is explained below: What I need: I need to find a contour of constant density in joint distribution of ...
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0answers
14 views

How do I find the marginal distribution with this summation/series?

Let $X$ be a Poisson(2) and $Y$ be Binomial(10,3/4) random variables, If $X$ and $Y$ are independent, then $P(XY=0)$ is I thought of using transformation to find the distribution of $XY$ so I let ...
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1answer
27 views

Finding the conditional distribution of 2 dependent normal random variables

Here's the situation $X \sim N(\mu, \sigma^2)$ and given $X=x$, $Y \sim N(x, \tau^2)$ I need to find the distribution of $X$ given $Y=y$ From what's given, I know the pdf's of $X$ as well as ...
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0answers
39 views

How to derive the distribution of a variable linearly related to two others?

Say i have $$x= \beta + \alpha \ln(y) + \varepsilon, $$ where $E(\ln(y)) = 0$, so $E(m) = \beta$, and $\varepsilon$~ $N(0, \sigma^2)$. Assume $$f(y) = \tau f^A(\ln(y)) + (1 - ...
4
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1answer
74 views

From marginal distribution to joint distribution

Consider two sequences of real-valued random variables, $\{X_n\}_n$ and $\{T_n\}_n$. Let $\rightarrow_d$ denote convergence in distribution. Assume (1) $X_n\rightarrow_d L$ as $n\rightarrow \infty$, ...
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6 views

Signing *change* of probability that one random variable is lower than another

Let $\tilde{z}_L \in [0,1]$ and $\tilde{z}_H \in [0,1]$ denote two random variables, with $F_L(z|\theta) := \Pr\{\tilde{z}_L \leq z|\theta\}$ and $F_H(z|\theta) := \Pr\{\tilde{z}_H \leq z|\theta\}$. ...
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1answer
42 views

zero covariance but not independent - normally distributed random variable $X$ and $X^2$

This is one of my homework question, which the answer sheet has already been given out. However, I still don't understand it. Exercise 1.1. It is well known that for two normal random variables, zero ...
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0answers
23 views

Where does the name of the hypergeometric distribution come from?

I understand what it does and how to get there, but why is it called hypergeometric? All the other distributions I know of have rather self-explanatory names like "binomial" or "exponential", or are ...
2
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0answers
56 views

Sampling from a given pdf

I have the following pdf: $$ f(x) = C x^d I_0\left(b \sqrt{- \log\left(\frac{x}{A}\right)}\right)$$ for $0 < x \leq A$, $C$ is a normalizing constant, $b$, $d$ are constants, and $I_0$ is the ...
2
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2answers
25 views

cumulative distribution of intersection of events

Let $X_1,\dotsc,X_n$ be independent identically distributed random variables having common distribution function $F_X(\cdot)$. Express the event 'the smallest of the $X$s exceeds $k$' as an ...