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

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3answers
121 views

A probability distribution and random variable

Assume we have $X_1,...,X_n$ independent poisson random variables. What is the cdf or pdf of $ \sum_{i=1}^{n} X_i$ ??
2
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1answer
375 views

Combining 1D normal distributions into a 2D distribution

First of all, apologies for my poor terminology - I have a particular problem which I understand in own terms, but I am having difficulty in applying the mathematics in the correct manner. My problem ...
2
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1answer
183 views

Why is likelihood not always 0 in continuous case?

Let's say I have some data, for example $d = 0.112$. And I have a known model $m$ which just produced uniformly distributed values over the interval $[0,.5]$. I am interested in computing the ...
2
votes
1answer
406 views

Joint distribution of non homogeneous Poisson event times?

I am trying to calculate the density of $(T_1,T_2)$ where $T_1$ is the time of the first event and $T_2$ is the time of the second event. I have been looking at the Wiki article on Poisson process and ...
2
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1answer
1k views

distribution of the normal cdf

I am wondering what is the probability density function for the normal cdf $\Phi (aX+b)$, where $\phi$ is the usual standard normal cumulative distribution function I want to calculate ...
2
votes
2answers
147 views

a question about Erlang/chi square distribution

Suppose you have $$ Y = X^2+Y^2 $$ where $X$ and $Y$ are both Gaussian with zero mean and variance $\sigma^2/2$ (you can think of $y$ as the square norm of $Z = X + jY$). The pdf should be ...
2
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2answers
2k views

PDF of product of variables?

could anyone please indicate a general strategy (if there is any) to get the PDF (or CDF) of the product of two random variables, each having known distributions and limits? After having scanned ...
2
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1answer
405 views

Integral with analytical solution with normal distribution

I received very good answers a couple of days ago in a simpler related problem, see Integral with Normal Distributions, but I am struggling with this new question: Let's define a function ...
2
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4answers
6k views

Convolution of two Gaussians is a Gaussian

I know that the product of two Gaussians is a Gaussian, and I know that the convolution of two Gaussians is also a Gaussian. I guess I was just wondering if there's a proof out there to show that the ...
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1answer
43 views

Check for independence of variables when the density (or distribution) is known.

This question is closely related to a previous one: Determine correlation and independence when only the joint density is given? Nonetheless, the setting is reproduced below. The joint pdf of $X = ...
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2answers
29 views

Finding CDF for PDF

¡bom dia! I need to find the CDF for the following: $$ f(x) = \begin{cases} 6(1-x^2), & -1<x<0, \\ 6/x^2, & 1<x<2, \\ 0, & \text{otherwise}. \end{cases} $$ This is more ...
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1answer
37 views

Probabability of Joint Distribution

Let the continuous random variables $X$ and $Y$ have the joint probability density function given by $f(x) = 3/2x$ for $0<x<2$, $0<y<1$, $x<2y$. Find Pr $(x<1.5|y>0.5)$. This was ...
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0answers
19 views

Does existence and uniqueness of a classical solution impose uniqueness of weak solutions to a pde?

I wonder if one knows that there exists a unique classical solution of a pde (for instance: Fokker-Planck equation), is one able to conclude that there isn't any weak solution of the pde, which ...
1
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1answer
61 views

How does a Nakagami Random Variable behave?

A Nakagami random variable has the following pdf $$f_{\Omega,m}= \frac{2m^m}{\Gamma(m)\Omega^m} x^{2m-1}e^{-\frac{m}{\Omega}x^2}$$ I have two questions regarding this random variable, 1- Is a sum of ...
1
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1answer
48 views

Finding $E(X \mid X > Y )$ when $X, Y \sim N(0,1)$ [duplicate]

This is the problem: The random variables $X$ and $Y$ are independent and $N(0,1)$-distributed. Determine $E(X \mid X > Y )$, $E(X + Y \mid X > Y )$. I go by the definition ...
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1answer
45 views

Central limit theorem kind of statement for records

I am trying to prove the following statement, but I do not know how to go on: Let $F(x)$ be an arbitrary continuous distribution function. Then there are constants $A_n, B_n > 0$ such that, as ...
1
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2answers
158 views

Can the expected value of a PMF be zero, as in E[X] = 0?

The whole question is: Let X be a discrete random variable and let Y = 0.5 X + 3. (i) Assume that the PMF of X is given by where k is some suitable constant. Determine the value of k. (ii) Find E ...
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1answer
91 views

Determine the accuracy of Poisson approximation to birthday problem

I'm currently doing an exploration of the Birthday Problem, and noticed that the formula given to calculate the probability for $m$ people in a room is: $$1-\frac{365!}{365^m (365-m)!}$$ And this ...
1
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1answer
71 views

Exponential of Squared Brownian Motion

Long time lurker, first time posting! Have a problem, that looks familiar but I can't put my finger on it. Need to calculate $\mathbb{E} [\exp(aW_T^2)|F_t]$ where $W_t$ is an $F_t$ adapted standard ...
1
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0answers
54 views

Poisson distribution given Gamma Distribution

I'm struggling with this one: If $\theta $ is a Gamma$(p,\lambda)$ random variable with $p>1$ and $\lambda>0$. We give the density of the gamma distribution: $ f(x) = \frac { { \lambda }^{ p ...
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0answers
30 views

Probability: NEED HELP to Understand with the follow [duplicate]

I need help to understand the probability derviation of a paper. Please help me. For the following, please only treat $|h_{R,B}|^2$ and $|h_{A,R}|^2$ as random variables (other parameters can be ...
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1answer
64 views

Question about transformations and sums on uniformly distributed random variables.

I'm looking into a few problems as a hobby of mine, and found myself with the following problem: let $X$ be a random variable uniformly distributed on $[0,1]$. What is the probability that after $N$ ...
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1answer
102 views

Generate random numbers with a modified PERT distribution

I want to generate random numbers based on the modified PERT distribution. The modified PERT distribution is a special case of the beta distribution and is defined as: $$f_X(x) = ...
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2answers
48 views

Poisson approximation to binomial distribution: $f(x)\geq g(x)$ or $f(x) \leq g(x)$

We have a random variable $X$ which has a Binomial Distribution Bin(n,p) and a random variable $Y$ which has a Poisson Distribution Poisson(np). We are interested in $$f(x):=Pr[X \geq x].$$ For ...
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2answers
68 views

Approximate distribution for the sample mean?

A random variable $X$ is said to follow a discrete uniform distribution if its probability function is given by $$p_X(x) = \left\{ \begin{array}{ll}\frac{1}{\theta}, & x = 1, 2, \ldots, ...
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1answer
75 views

Finding the pdf of an estimator

We have a set of unidimensional data, $X_1, \ldots , X_n$ drawn from the positive reals. We define a model for its distribution: The data are drawn from a uniform distribution on the interval $[0, ...
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1answer
27 views

Hypergeometric Distribution Confusion

I'm having trouble understand the part of the pmf for the Hypergeometric Distribution highlighted in green: $$\Pr[X = k] = \frac{\dbinom{m}{k}\!\!\color{green}{\dbinom{N-m}{n-k}}}{\dbinom{N}{n}}$$ ...
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1answer
60 views

Finding joint density of dependent variables

I've got two random variables, $x$ and $y$, where $x=u(y)$ and $y=v(x)$. How do I find the joint density of the two variables? The eventual purpose is to find the integral $\int^a\int^b f(x,y) ...
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2answers
51 views

Proving this random variable problem

$X_1,X_2,X_3,\ldots$ are IID random variable taking values in $(-1,\infty)$. Also $t\in(0,1)$. Define random variables $Y_1,Y_2,Y_3,\ldots$ recursively like $$Y_1 = (1+tX_1)$$ $$Y_n = ...
1
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1answer
51 views

Conditional expectation of conditional sum(Not fully complete)

I have a question: Determine the conditional expectation $\mathrm{E}(A|B)$ for: The number $B(\geq 0)$ of bats that leave a cave at the time of a nuclear explosion has a geometric distribution ...
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2answers
156 views

Show that $S = \sqrt{S^2}$ is a biased estimator of $\sigma$ given a random sample from a normal distribution …

Suppose $Y_1, \ldots, Y_n$ is a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. Let $S^2$ be the sample variance, which is unbiased for $\sigma^2$. GOAL: Show that ...
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1answer
36 views

relation between multivariate probability generating function and univariate ones

Suppose I have two independent integer random variables $X_1$, $X_2$ (with constraint that $X_1+X_2\le N,0\le X_1\le N,0\le X_2\le N$), with probability generating functions $g_1(z)$, $g_2(z)$. Now I ...
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3answers
76 views

Given $X$ and $Y$ are independent N(0,1) random variables and $Z = \sqrt{X^2+Y^2}$ from the marginal pdf of $Z$

Let $X$ and $Y$ be independent $N(0; 1)$ random variables. Let $Z = \sqrt{X^2+Y^2}$. (a) Derive the marginal pdf of $Z$ and then using the marginal pdf to compute ${\rm E}[Z^2]$ (b) Can you propose ...
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1answer
124 views

Is the distribution of a product of M iid uniform random variables really Log Normal?

Conventional wisdom says yes or mostly. But consider the following simple derivation: Let $y = \prod_{i=1}^M x_i$ where $x_i\sim U(0,1)$. Then from independence, $E[y] = 2^{-M}$. Now, if we let ...
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0answers
91 views

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$, find $V(S'^2)$.

If $S'^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n}$ and $S^2 = \frac{\sum_{i=1}^n (Y_i - \bar{Y})^2}{n-1}$ then $S'^2$ is a biased estimator of $σ^2$, but $S^2$ is an unbiased estimator of the same ...
1
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1answer
50 views

Determine a distribution with no parameters?

I'm confused by this question and I was hoping for some guidance some one to point me in the right direction Let $X_1.........X_n$ be a random sample from a population with mean $\mu$, that is ...
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2answers
36 views

Randomized Algorithm

I asked this question earlier but I wanted to change the problem. A band has tour sites A, B, and C. They get paid every time they play at each tour site, specifically: ...
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2answers
67 views

Two probability questions.

I have two questions. (1) solution(1): Sample size $=|S|=12^{20}$ $11^{20}\rightarrow$ guarantee that one box is empty. $10^{20}\rightarrow$ guarantee that two boxes are empty. ...
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1answer
103 views

Hypergeometric Distribution : probability that more than half is good

To simplify the context, let's say that 34 % of people are ugly. haha... lets take a sample of 15 people. (n = 15) a) What is the probability that 3 or less out of the 15 are ugly ? I went on and ...
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1answer
44 views

what is the expected return of this game when played with best strategy?

In a two people's game, you start with one dollar and you are betting 1 dollar coin at the very start of the game, then if you win you have double amount of money and you can choose of choose not to ...
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2answers
194 views

Integrating a special skew normal — the CDF of a convolution of a normal with a truncated normal

I am having a little trouble trying to compute an integral. In short, I wish to solve the following: $$F(x) = \int_{-\infty}^x \phi(au-b)\,\Phi(au+b)\,du $$ My intuition is that this might be ...
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2answers
91 views

independent chi squares mean independent non central chi square?

Let $Y$ be a multivariate normal random vector with covariance $\Sigma$. Let $A_0,A_1$ be matrices such that $$A_0\Sigma A_1=0.$$ It is known that in this case $Y'A_0Y$ and $Y'A_1Y$ are independent ...
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2answers
135 views

Expected values in a sequence

We draw 2 numbers from a normal (gauss) distribution with mean $\mu$ and variance $\sigma$ and we add them to find the first value $a_1$ of a sequence. The second value $a_2$ of this sequence is the ...
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1answer
43 views

Got Stuck with these probability problems

I tried my best to solve 'em , but after waiting a few sheets of paper , I got nothing on me . A litle help from you guys might do the trick , Thanks ! ...
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1answer
157 views

Mean and variance of geometric function using binomial distribution

Can anyone help solving this question please? I tried but not sure of the steps to reach the conclusion.
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2answers
152 views

poisson random variable distribution using probability

Let $X$ be the number of emails that a company receives in a day. Assume that $X$ is a Poisson random variable with parameter $\lambda$. The company classifies each email as spam or not spam. The ...
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1answer
78 views

Covariance combined with normal distribution

We have $N_1$ and $N_2$, normal distributed random variables with averages $µ_i=E[N_i]$ and variances $σ_i^2=Var[N_i]$ and $c = Cov(N_1, N_2)$. We want to compute $E[e^{N_1} I(N_2>0)]$, where I is ...
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1answer
49 views

Proof this form

I am trying to proof this form: Let $g(.)$ be a function, for $y_n$ is a a nonnegative random variable, $\varepsilon>0$, $g(x)>0$ is increasing function for $x>0$, and $E[g(x)]>0$, then ...
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1answer
164 views

Conditions for positive dependence

Consider two random variables $X$ and $Y$ with joint distribution $F_{X,Y}$ and strictly positive density function $f_{X,Y}$. Additionally, let $x^*$ be the value of $x$ that solves: $$ \Pr[Y\leq ...
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1answer
3k views

Continuous uniform distribution over a circle with radius R

I started to do this problem with the standard integration techniques, but I cant help but think that there has got to be something I am not seeing. Since it is a uniform distribution, even though x ...