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

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1answer
26 views

Finding the probability of $R_1$

Consider a binary communication system that consists of a transmitter, a receiver and a Chanel that transfers bits from the transmitter to the receiver. The nature of the channel is such that it ...
1
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1answer
91 views

Relation between two distributions expressed in terms of their CDFs

Not great at stats, and having trouble wrapping my mind around this. Would love an explanation, not overly detailed, in plain english of what these transformations mean. The bias correction ...
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0answers
29 views

Limiting distribution $\displaystyle\frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n Y_i}$

let $X_1,X_2,\ldots,X_n$ be random sample of bernoulli distribution with parameter of $\displaystyle\frac{\theta_1}{\theta_1+\theta_2}$ and let $Y_1,Y_2,\ldots,Y_n$ be random sample of geometric ...
3
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1answer
60 views

How to find the CDF of the sum of independent uniformly distributed random variables?

$X,Y$ are independent random variables with uniform distribution on $[0,1]$, and let the random variable $Z=X+Y$. The density of $Z$ is: $$f_{X+Y}(z)=\int_0^z f_X(x)f_Y(z-x)dx$$ What is the formula ...
0
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1answer
39 views

Determine the target weight so that no more than 5% of boxes with normal weight distribution contain less than 500 g [closed]

Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation 12 g. Suppose a law states that no more than 5% ...
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3answers
336 views

An increasing probability density function?

Could anyone come up with a probability density function which is: supported on [1,∞) (or [0,∞)) increasing discrete
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0answers
34 views

Conditional distribution for sum of random variables

Let $X$ be some discrete random variable, i.e. $\mathbb P(X=x_j)=p_j$ for $x_j\in\mathbb R$ and $\sum_{j=1}^J p_j=1$. Furthermore, let $L_1,\ldots, L_n$ be a list of be random variables of which we ...
1
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1answer
260 views

Generalized chi distribution

Let $v\in\mathbb{R}^n$ follow a multivariate Gaussian$(0,I)$ distribution, and $M\in\mathbb{R}^{n\times n}$ a matrix. Has the distribution of the Euclidean norm $\|Mv\|$ been studied? I know that its ...
1
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0answers
58 views

Approximation for the convolution of normal and lognormal distributions

$$X \sim \ln\mathcal{N}(\mu_X,\,\sigma_X)$$ $$Y \sim \mathcal{N}(0,\,1)$$ $$Z = X + Y$$ I want to find the probability density functions and cumulative distribution functions of $Z$. As the below is ...
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0answers
29 views

Poisson process distribution of arrival and interarrival times

Suppose buses arrive at a stop as a Poisson process with a mean rate of 20 per hour. What is the distribution of time between the first and second afternoon buses? I was thinking that it is the ...
1
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1answer
46 views

Limits of the joint pdf $(2/3)(x + 2y)$

I'm given equation that the joint pdf is $(2/3)(x + 2y)$ when $0 < x, y < 1$ and we want to find the probability that $X < 1/3 + Y$. I understand how to do the actual math part, and that I ...
1
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1answer
38 views

cdf is $F_X(x) = 1-(1-x)^k$, is that a “famous” distribution?

The question is the following. I found that the cdf of X is $F_X(x) = 1-(1-x)^k$, where $k$ is a parameter. I was wondering if that is some famous distribution (like the one that has a name, for ex., ...
2
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2answers
31 views

Product of binomial and uniformly distributed variables

I've got a question that I really think should be quite simple to answer, but I just can't see what I'm missing. We have the random variables $X \sim R(0,1)$ and $Z\sim b(1,1/2)$. I want to determine ...
1
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1answer
17 views

${\bf E}[Y]$ of a joint distribution

So, I have that a joint probability density function is given by the formula: $$ 5e^{-5x} / x, \quad 0 < y < x < \infty $$ and I have to find the $\operatorname{Cov}(X,Y)$. I know that ...
3
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1answer
40 views

Expectation of first and second order statistics in a random distribution

Let $E(f_{i}^{n})$ and $E(s_{i}^{n})$ denote the expected first and second order statistics for $n$ draws from the distribution $V_i$ .i.e set $X_{i}^{n}=\{x^1,.....,x^n | x^j \sim V_i \}$ and let ...
1
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0answers
41 views

how to calculate remaining waiting time in exponential distribution?

ABC corp conducted a study of service times at the drive-up window of fast-food restaurants. The average time between placing an order and receiving the order at restaurant is 2.45 minutes. Assume ...
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0answers
19 views

Relationship between two beta distributions

We have experiment results, which are ratios of poisson variables, of the form X/X+Y, where X and Y are poisson, and X+Y is not fixed. I read that under high lambda, one can approximate these with ...
0
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1answer
18 views

How to find Reliability of a rectangular distribution function?

Assume that the failing of a device is equally probable within an interval [a,b] such that the fault density is: f(x) = {1/b-a if a<= t <= b ...
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2answers
39 views

What does “Choose N ~ Poisson(ξ), Choose θ ~ Dir ( α )” mean in the context of Latent Dirichlet Allocation

I'm reading http://machinelearning.wustl.edu/mlpapers/paper_files/BleiNJ03.pdf and trying to understand the notation and concepts behind LDA, in order to implement it myself. I've followed some ...
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0answers
38 views

Joint distribution of random variables

Let $S$ and $T$ two independent random variables. Suppose that S is a standard Gaussian random variable with density $f(s)=(2\pi)^{-1/2}e^{-s^2/2}$ and T with, $2f(t)\mathbf 1_{\{\mathbb ...
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0answers
28 views

Exact probability distribution for hitting time of simple random walk

Consider simple random walk on the line starting from the site $y \in \mathbb{N}$. With probability $p$ the walker moves to the right and with probability $1-p$ to the left. Call $\tau$ the first time ...
1
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1answer
270 views

Unknown number of colours Bernoulli Urn

Okay, so, in the traditional Bernoulli Urn problem, we have an urn with a number N, possibly infinite, of coloured balls, and there are k possible colours. That one I grok. However, what if I don't ...
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2answers
54 views

Conditional Expectation Discrete and Continuous

Find $E[X]$ and $Var[X]$ So for the expectation so far I got that: $$E[X] = E[X|N=n]P(N=n) = \large\frac{n+1}{\lambda} \frac{\lambda^{n}}{n!}e^{-\lambda}$$ but for conditioning on both a discrete ...
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0answers
12 views

Evacuation schedule length for a linear wireless network?

There is a linear wireless network consisting of node 0, 1, 2, ..., n. Node i can only communicate with node (i - 1) and (i + 1). Each node (i > 0) generates a packet and transfers it to node 0, via ...
0
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1answer
401 views

Find unknown value in probability density function

"Suppose that a random variable $Y$ has a p.d.f. given by $f (y) = ky^3*e^{-y/2}$ when $y > 0$, and otherwise 0. Find the value of $k$ that makes $f (y)$ a density function." I found that $k=1.$ ...
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0answers
15 views

Modeling Counts with Small Number of Observations

I have a large data set that contains $5$ different fields. The fields are ...
0
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1answer
26 views

limiting joint distribution

Let $X_n\xrightarrow[d]{}N(0,\sigma^2_x)$ and $Y_n\xrightarrow[d]{}N(0,\sigma^2_y)$. $X_n, Y_n$ are not independent. Can I say that $\left( \begin{array} {} X_n \\ Y_n \end{array} ...
2
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1answer
14 views

Convergence , conditional distribution

here my short question. I saw the following conditional distribution, which converges, in a book: $\lim_{s \to \infty}P\left(\frac{X-f(s)}{g(s)}\leq x\mid X>s\right)=G(x)\ \forall x$ in the set ...
4
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2answers
480 views

How do you calculate the average number of lands in your hand in a game of MTG?

(This is actually a question and a half; Please tell me if this should be done otherwise) I have a magic deck of 60 cards. Some of them (say 25) are lands. The first card I draw has a 25/60 chance of ...
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0answers
22 views

Implementing the Delta method to assess the confidence and prediction intervals

I want to calculate the table of confidence and prediction intervals for a custom Cumulative Distribution Function or CDF, and I am following the forums and articles aid. My major cuestions that I ...
1
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1answer
26 views

Relation between the distribution functions of random variables $Y$ and $-Y$

I'm having trouble understanding a certain property of CDFs for negative random variables. Let $Y$ be an exponential random variable and let $f_y, F_Y$ denote the PDF and CDF respectively. My book ...
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2answers
343 views

joint distribution of two dependent uniform variables

Let $X$, $Y$ be two random variables uniformly distributed in $[0,1]$ satisfying the constraint $ X \leq Y$. I want to calculate the joint density $f_{X,Y}(x,y)$ of $(X,Y)$ but i don't know how to ...
0
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1answer
33 views

Hypothesis Testing, P-value, T-test Statistic, Confidence Interval

I am writing a report for my class project. I am taking statistics and I am REALLY panicking with the results I have in my report. I do not think my calculations for t-test statistic or confidence ...
1
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0answers
26 views

Infinite fourth moment and maximum entropy

Alright, I expect this is a silly question, but I don't actually know, so. Suppose there is some random variable that's distributed on the reals, and all I know about the distribution is its mean ...
1
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1answer
241 views

Affine transform of multivariate gaussian

If $X_1, \ldots, X_n$ are iid $N(0,1)$ or in other words $\mathbf{X}=(X_1, \ldots, X_n)$ is distributed $N(\mathbf{0}, \mathbf{I})$, then $A\mathbf{X}+\mu$ is distributed $N(\mu, AA^t)$. Showing that ...
0
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1answer
29 views

Proving Continuity and equivalence

I have posted ths on the Quant Finance page as it is part of a QF problem but realised I may get a swifter response here! Iam working on a problem where I have successfully reduced a version of Black ...
0
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0answers
54 views

Compound Gaussian distribution

Let $\mathbf{a},\mathbf{b}\sim \mathcal{N}(\mathbf{0},\sigma^2\mathbb{I})$ and let $A$ be the circulant matrix defined to have $\mathbf{a}$ as its first column. I'm trying to study the behaviour of ...
1
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1answer
32 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 ...
0
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1answer
20 views

Combining biased coin flips

Suppose one has a biased coin $C_1$ with probability $p$ of landing heads and $(1-p)$ prob. of landing tails. If one wants to sample a coin $C_2$ with $p^2$ probability, one can flip the coin $C_1$ ...
1
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1answer
17 views

Convergence in distribution for changing domains.

I am trying to consider whether this is possible and/or reasonable: Let $X_n:\Delta_n \to \mathbb{R}$ be a sequence of random variables, defined over a unique space $\Delta_n \subseteq \Omega$ for ...
0
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1answer
28 views

Distributions of local times of a single excursion of 1D random walk

Consider Simple Random Walk in one dimensions, starting from $x \in \mathbb{Z}^+$. The walker jumps to the right with probability $p$ and to the left with probability $1-p$. Assume $p \leq ...
3
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0answers
35 views

Probability and sums of prime factors

Of the first N natural numbers, we select two different numbers at random. We'll call the greater one A and the lesser one B. What is the probability (P) that the sum of A's prime factors is LESS than ...
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1answer
41 views

Why do different probability distributions have different restrictions on their parameters?

Is it correct that the parameters of the following distributions must be taken from the intervals given below? Bernoulli. $p$ from $[0, 1]$ Binomial. $n$ positive integer, $p$ from $[0, 1]$ ...
2
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1answer
39 views

$X$ ~ $\Gamma(s,\lambda)$. Using $M_X(t)$ find the following…

a) $E(X) =$ ? b) $E(X^2) = $ ? c) $Var(X) = $ ? My thoughts: I know that moment-generating function for $\Gamma(k,\theta) = ( 1 - t\theta)^{-k}$ for $t < \frac{1}{\theta}$. I also know that ...
3
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1answer
63 views

Prove Number of Arrivals $N(s)$ up to time $s$ follows $\mathrm{Poisson}(\lambda s)$ Distribution

This comes from my self-study of Durrett's "Essentials of Stochastic Processes" book, page 97. Definition Let $\tau_1,\tau_2,\ldots$ be independent $\mathrm{exponential}(\lambda)$ random variables. ...
0
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1answer
41 views

Mixing continuous and discrete distributions

I'm wondering how, if it is at all possible, to write the p.d.f. for the following random variable. Given RVs $X_1$ and $X_2$ distributed according to some joint distribution having known density ...
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0answers
32 views

Prove Joint distribution of estimators

Let $X_1,...,X_n$ iid r.v. with distribution F, with mean $\mu$ and median $\theta$.Assume that $Var(X_i)=\sigma^2$ and $F'(\theta)>0$. If $\hat{\mu}_n$ is the sample mean, and $\hat{\theta}_n$ the ...
2
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1answer
40 views

Deriving a lower bound for a probability involving a random variable $X$ with the Gamma distribution.

Question Let $X$ have the $Gamma(\alpha, \beta)$ density. I.e. $$f_X(x) = \frac{1}{\gamma(\alpha)\beta^\alpha}x^{\alpha-1}e^{-\frac{x}{\beta}}$$ when $x >0$ and $0$ elsewhere. The moment ...
2
votes
1answer
302 views

Maximum likelihood estimators, hypergeometric and binomial

I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
0
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2answers
30 views

Probability density use for biased outcome

I am not a mathematics pro so do not mind if this is dumb let us suppose I have a method for generating random real values between 0 and 1 . All the values between 0 and 1 are equally likely to be ...