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

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5 views

Transition density of an AR(1) process?

If we have an AR(1) process, i.e: $X_{t+1} = \alpha X_t + e_{t+1}$ then what is its Markov Chain transition density? We know that for a Markov chain, the following holds: $P(X_{t+1}\leq x_{t+1} \mid ...
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1answer
26 views

Moment generating functions…which distributions to use?

Q: You hired a terrible programmer and the moment generating function for the distribution of software bugs is M(t) = (1 - $\theta$t)$^{-\alpha}$. Groups of bugs can be detected within $\mu$ = 47 ...
1
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0answers
21 views

Regression in Bivariate Normal

Suppose $(X_i \hspace{4pt} Y_i)'$ are $i.i.d$ $N_2 (\bf{\mu,\Sigma})$, $i=1(1)n$ where $E(X)=\mu_x$, $E(Y)=\mu_y$ and $\Sigma$ is given by \begin{bmatrix} \sigma^2_x & \rho\sigma_x\sigma_y\\ ...
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0answers
15 views

Find conditional probability of a mixture model

given is the following: A mixture model comprises a non-observable $\{ 0,1\}$-valued random variable $X$ such that $P(X=1)=1-P(X=0)=\pi$ and an observable variable $Y$ such that $Y\mid X=0$ is ...
1
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1answer
14 views

Linear regression relationships

Velocity $= X$, distance to stop $= Y$ $\beta_0= -17.5791$, $\hat{\operatorname{se}}(\beta_0)=6.7584$ $\beta_1 = 3.9324$, $\hat{\operatorname{se}}\beta_1 = 0.41.55$ degrees of freedom $=48$ (a) is ...
0
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1answer
13 views

What am I plugging in wrong to my normal distribution calculator?

I am trying to find the probability of the following question: Cans of regular Coke are labeled as containing 12 oz. Statistics students weighed the contents of 7 randomly chosen cans, and found the ...
1
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1answer
26 views

Distribution Technique Question of two independent Exponential Distributions

If $X_1$ and $X_2$ are two independent random variables having exponential densities then $f(x_1,x_2)$ is defined as $$f(x_1,x_2)=\exp(-(x_1+x_2))\,{\bf 1}_{(0,\infty)}(x_1){\bf ...
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0answers
24 views

Weighting the data by the history

I have a input stream 3D data that comes every time frame. Each point is defined by 3D vector of x,y,z. There is a evaluation function [say f(x)] that computes if the point at time t is valid or ...
1
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2answers
48 views

Runs of white balls in sampling without replacement

There are $m$ white balls and $n$ black balls in a box. Balls are randomly drawn from the box with no return. Denote $X_1$ : number of white balls that been drawn before the first black. For $2 \leq i ...
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1answer
29 views

Normal distribution calculations

We have a gaussian distribution $$ X \sim N(\mu,\sigma^2)$$ where $\mu = 4$ and $\sigma^2 =1.5$ . Probability is given by : $P(x<c)=0.35$ $c$ needs to be calculated. And we got ...
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2answers
30 views

Derivation of mean and variance of Hypergeometric Distribution

I need clarified and detailed derivation of mean and variance of a hyper-geometric distribution. If a box contains $N$ balls, $a$ of them are black and $N-a$ are white, and $n$ number of balls are ...
5
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1answer
51 views

Multivariate normal density function of function of random variable

Let $X_1,\dots,X_n$ be i.i.d random variables and $g$ be a symmetric function such that $$g(X_i,X_j)\sim N(\mu,\sigma^2)$$ for all $1\le i<j\le n$. I wish to know the density function of the joint ...
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1answer
15 views

Distribution of specific distributions

I have a normal distribution of independent variables, and there are a specific number of samples to this distribution: say 1 million samples. A function is set by the largest value of these million ...
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1answer
31 views

Seeking an example for Bayes estimator of two unknown parameters

I searched the web, taking advantage of several search approaches; however, due to redundancy of the existing information about Bayes estimator of one unknown parameter of random variables (either in ...
0
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1answer
18 views

If X + Y is truncated normal and X and Y are identitically (but not independently) distributed? What is the distribution of X and Y?

Let $(aX + bY)$ be a truncated normal and assume $X,Y$ are both identically distributed (but necessarily NOT independent) what is the distribution of $X$ and $Y$? More importantly can the pdf of $X$ ...
2
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2answers
55 views

Probability distribution of number of waiting customers in front of a counter

The number of customers arriving at a bank counter is in accordance with a Poisson distribution with mean rate of 5 customers in 3 minutes. Service time at the counter follows exponential distribution ...
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0answers
9 views

Lower bound for (function of) density of well-behaved random variable

Suppose we have a non-negative random variable $\tilde{\theta}$ such that $\mathbb{E}\tilde{\theta} = a > 0$, with finite variance $\sigma^2$. Let its CDF be given by $F(\theta) := ...
4
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1answer
32 views

Flat product distribution

When $X,Y$ are iid random variables, uniformly distributed on $[0,1]$, then $Z=X*Y$ has the density $-\log(z)$, as it is shown here: product distribution of two uniform distribution, what about 3 or ...
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1answer
54 views

A related problem regarding Normal Distribution (Continuous Probability) [on hold]

A circus performer who gets shot from a cannon is supposed to land in a safety net positioned at the other end of the arena. The distance he travels is normally distributed with a mean of 140 feet and ...
-2
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1answer
24 views

The probability that two randomly selected $2$ year old male feral cats will live to be $ 3$ years old is? [on hold]

The probability that a randomly selected $2$ year old male feral will live to be $3$ years old is $0.82666$. (a) what is the probability that two randomly selected $2$ year old male feral cats will ...
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1answer
25 views

Suppose that $E$ and $F$ are two events? [on hold]

Suppose that $E$ and $F$ are two events and that $P(E\cap F)= 0.4$ and $P(E)= 0.8$. What is $P(F\mid E)$ ?
2
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2answers
29 views

Probability Distribution

I'm thinking about a set of n users on Facebook. Between each of the $\binom{n}{2}$ pairs of distinct friends, lets say an edge (indicating that the two people are friends) is independently present ...
2
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0answers
59 views

Of strings and substrings: A problem of probability

Problem Let $\Sigma=\{a, b\}$. Let $\Sigma^*$ denote the Kleene star of $\Sigma$: \begin{equation*} \Sigma^* = \{\varepsilon, a, b, aa, ab, ba, bb, aaa, aab, \ldots\} \end{equation*} where ...
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0answers
13 views

Reconstruct multivariate binary distribution from marginals

I'm have a random vector $\bf a$ with binary entries, $a_i \in \{0,1\}$. The probability distribution $P({\bf a})$ is not fully specified, but I have the marginals $p_i$, which are the probabilities ...
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2answers
24 views

how to estimate parameters of a triangular distribution? [on hold]

I have a set of observations, and they come from a triangular distribution. Now I want to estimate its parameters, but how?
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0answers
27 views

Distribution Problem based on unknown function [on hold]

I got struck at this problems as Function is not given. Any help will be appreciated
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1answer
16 views

Geometric distribution with given probability value.

The probability of a man hitting a target is $2/3$. If he doesn't stop shooting until he hits the target for the first time, a) What is the probability of taking 5 shots to hit the target? b) Which is ...
1
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1answer
19 views

What is the “Cumulative Distribution of the magnitude of the N-dimensional standard gaussian”

I am confused by this line from a paper: "Let $F_1(x)$ be the cumulative distribution of the magnitude of an $n$−dimensional standard Gaussian random variable and $F_2(x)$ be the cumulative ...
2
votes
1answer
18 views

Properties of unimodal functions

A probability density function $f$ is said to be unimodal if the value at which the maximum value of the function is attained is unique. I am reading some papers on econometrics that appear to use ...
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2answers
34 views

Question about probability distributions

I've recently came across this question: ...
0
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1answer
35 views

Infrequent fail of the popular parameter estimators, having several beta-distributed random variables to be estimated

I have a project in which there exist $N$ Beta-distributed Random variables each of which should be estimated, having a sample for each of them. The sample domain is $\{0.1,0.3,0.5,0.7,0.9\}$ and the ...
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0answers
31 views

How do I calculate conditional PDF?

Obtain $$P(2 < Y < 3 | X = 1)$$ where the joint pdf of X and Y is $$f_{X,Y}(x,y) = (6-x-y)/8$$ where $$0 < x < 2$$ and $$2 < y < 4$$? so first, I did $$f_Y|X=1(y) = ...
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0answers
29 views

Interpretation of integral as ratio of joint and conditional densities?

A common exercise in Bayesian statistics is specifying a prior $p(\theta)$ on some parameter $\theta$. We then observe a collection of data $D=(X_1,\dots,X_N)$, the distribution of which is ...
1
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2answers
24 views

Find the required Chi-square score for an arbitrarily low p-value (2 degrees of freedom)

I'm trying to use the Chi-Square test to find the significance of data that suffers from the multiple testing problem. Because I have this multiple testing problem, the required p-value to view a test ...
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1answer
21 views

Uniform distribution and real values [closed]

If the random variable $k$ is uniformly distributed in $(0,5)$, What is the probability that the roots of the equation $4x^2+4xk + k + 2 = 0$ are real?
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1answer
27 views

Slow convergence simulating log-normal sample from the normal

I am trying to simulate a log-normal random variable $Y$ with mean $m = \mathbb{E}[Y] = 0.001$ and standard deviation $s = 0.094$ by simulating a normal sample instead, and then exponentiating it. ...
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votes
1answer
22 views

Probability of a user references in a network [closed]

I am trying to figure out no of possible referrals of a user in a network. Where the size of a network is not fixed but we can set an assumption of 1000 persons. Edit: A user knows few users in a ...
3
votes
1answer
18 views

CDF of the difference of two Gaussian mixtures

I have two Gaussian mixtures, $X_D$ and $X_{\overline{D}}$: $$ f(X_D) = \sum_{c=1}^m f(X_D\mid C=c)P(C=c) = \sum_{c=1}^m \phi(x-\mu-g(c))P(C=c), $$ $$ f(X_\overline{D}) = \sum_{c=1}^m ...
3
votes
1answer
65 views

Distribution of $\sin(X) *\cos(Y)$ where $X,Y$ are iid r.v., uniformly distributed on $[0, 2 \pi]$

What is the probability density of $R = \sin(X) * \cos(Y)$ where $X,Y$ are independent random variables, uniformly distributed on $[0, 2 \pi]$? I am stuck with complicated integrals, not sure if ...
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votes
1answer
34 views

Does these inequalities hold in General for probability distribution? [closed]

Let $Q(y)$ be a probability density of $y \in [-1,1]$. Then for $t> 0$, the inequalities are $\displaystyle \int_{0 \leq y <t} y^2 Q(y) \, dy \leq t^2 \int_{0 \leq y <t} Q(y) \, dy $. ...
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1answer
17 views

computing weight from distance metric

I have a distance between two points in meters. I want to convert this distance into weight such that as distance increases the weight decreases. What are some good weighting function that can ...
2
votes
1answer
22 views

Probability in knockout games.

Suppose in a knockout tournament 32 players p1 , p2 .....p32 participate. In each round players are divided into pairs at random and winner goes to the next round. If p5 reaches semifinal what is ...
0
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0answers
12 views

characteristic function problem 4

which of the following is not a characteristic function? a) 1 b) $e^{it} $ , $t \in R$ c) $\frac{1}{1-it} $, $t\in R$ d)$e^{|-t|}$, $t \in R$
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0answers
78 views
+50

Find a probability density

I am going through a paper trying to understand all the single steps, but I got stuck. I need to calculate $$p(x+\delta t) \mid x(t), t)= \int p(x(t+\delta t) \mid \mu , x(t), t)p(\mu\mid x(t), t) ...
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1answer
17 views

How to calculate a posterior probability with a given Gaussian Mixture Model?

I'm building a GMM-based classifier in speech processing and I'm using GMM as a probabilistic scoring mechanism (therefore I don't intrinsically care about the underlying mixture components). For ...
3
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2answers
89 views

Given a variable $X$ with a PDF, what is the PDF of $\sqrt{X}$

I feel this is simple and I'm overlooking something really basic. Let's say a have a variable $x$ which obeys the exponential distribution. So if collect 100000 occurrences of $x$ and plot its ...
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1answer
28 views

How to find median from a probability distribution?

Having trouble on something that should be really, really easy. I need to find the median of the following probability distribution...but according to the website I linked below...I'm doing it ...
0
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1answer
31 views

Lottery probability with payout system

Assume we have a lottery which has following payouts 1,2,5,6,9,10,16. The organizer expects 4% profit from the lottery. I wrote ...
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0answers
17 views

What are interventions / intervention distribution?

I don't understand what interventions are: Definition 2.2.1 [Intervention Distribution] Consider a distribution $\mathbb{P}^\mathbf{X}$ that has been generated from an SEM $\mathcal{S} := ...
0
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0answers
34 views

Binomial-like distribution

Starting with $1$, for $n$ trials multiply by either $1+p$ or $1-p$, with $0 \le p< \le 1$. Does this distribution have a name? What are its properties, such as density (PDF)? It is like a skewed ...