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

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3
votes
2answers
598 views

Confidence interval of a random variable for an ordinary linear regression

I have a small problem. With my limited stats background I am not sure I am getting this one right. After fitting an ordinary linear regression model I get ...
0
votes
0answers
9 views

How to determine parameters of a normal distribution from a limited range of points?

In an experiment my data points are almost normally distributed with meanvalue != 0. My problem is I can only detect positive points (located on the right side of y ...
0
votes
1answer
16 views

Exchangeability and independence of random variables

I have a question on the relation between exchangeability and independence between random variables. Consider the random vectors $$u_1:= \begin{pmatrix} \epsilon_{1}\\ \epsilon_2\\ \epsilon_3 ...
-2
votes
0answers
9 views

What is the expectation of a product of a lognormal and a Poisson random variable [on hold]

How can I calculate the expectation of the product of a lognormal and a Poisson random variable? I cannot find a way to do this. Needless to say, the random variables here need not be independent.
1
vote
1answer
35 views

Questions on probability law

I'm trying to prove/disprove the following true or false statements, and I want to know if they are correct For every measurable function $g:\mathbb{R}\to \mathbb{R}$, $\mathbb{E}[g(X)]$ is ...
0
votes
0answers
13 views

How can I calculate definite integral of chi-squared pdf with one degree of freedom

enter image description here I need a calculating process of the above definite integral please help me.. (sorry for my poor English)
1
vote
1answer
12 views

Weibull distribution: from mean and variance to shape and scale factor

I need to sample values from a Weibull distribution whose mean and variance are provided (respectively 62 and 4275). I am running a Matlab code, therefore if I want to use wblrnd(shape,scale) I need ...
4
votes
0answers
56 views

Characterization of point process, given the number of points

For a point process with independent and identically distributed (i.i.d) inter-renewals, with distribution $p(x)$, we observed $N$ points on $[0,T]$. What is the joint probability distribution ...
0
votes
1answer
52 views

Is it Possible to Derive a State Transition Matrix from an Unscented Transformation?

I have an application where I am using an unscented Kalman filter to process data. While the unscented transformation eliminates the linearization assumption used with the typical state-transition ...
0
votes
0answers
7 views

deriving the profit function given probability distributions

I can't seem to get much further in deriving the profit function for part (c). I've attached the question and my attempt, but I'm not sure on what to do next, or if I've done something completely ...
-1
votes
1answer
28 views

probability functions

The total time, measured in units of $100$ hours, that a teenager runs her hair dryer over a period of one year is a continuous random variable $X$ that has the density function $$f(x)= \begin{cases} ...
0
votes
0answers
29 views

Odditiies in a StackExchange reputation distribution

As we know from for eg M.SE reputation distribution the reputation distribution is a neat power function, that's to be expected. However, on Travel StackExchange (possibly elsewhere, didn't research ...
-1
votes
1answer
32 views

If a student is selected at random and is found to be taller than $1.8~\text{m}$, what is the probability that the student is a girl? [on hold]

In a college $4\%$ of the boys and $1\%$ of the girls are taller than $1.8~\text{m}$. Furthermore, $60\%$ of the students are girls. If a student is selected at random and is found to be taller than ...
0
votes
0answers
7 views

what is the spatial distribution of waiting time?

Suppose rimu trees are spread in the territory of some area according to a time homogeneous Poisson process. Suppose a rimu tree is at point x, what is the distribution function of the distance to its ...
0
votes
0answers
17 views

Finding the survival and distribution function of a system.

We have a random variable $X\sim Gamma(3,c)$, so that means $f(x)=\frac{c^3}{\Gamma(3)}x^2e^{-cx} ; \ x>0$, with $c$ being appropriately selected scale parameter. We also have $P(U_2 \leq x)=x^2$ ...
0
votes
1answer
26 views

Consider a random variable X having the following PDF…

So I have to calculate the value of $c$ that makes this a legit PDF but I only know how to do it when it only has one function (set equal $1$ , integrate, solve for $c$). How do I calculate it for ...
0
votes
1answer
38 views

Simulate two centered normal random variables with given variances and given covariance [on hold]

How can I, by the central limit theorem, simulate two random variables $Z_{1}$ and $Z_{2}$ such that $$Z_{1}\sim N(0,\sigma^{2})\qquad Z_{2}\sim ...
0
votes
1answer
48 views

approximating a uniformly distributed random variable

Suppose that $U$ is a uniformly distributed (continuous) random variable on $[0,1]$. Let's say that I am interested in finding 3 discrete points $u_1,u_2,u_3$ which approximate $U$ in some sense. My ...
2
votes
2answers
33 views

Is covariance preserved under transformation?

Let $X_1,X_2$ be normally distributed random variables with $\rho = 0.5$, mean equal to $0$ and variance equal to $1$. Let $U_i = \Phi(X_i)$ where $\Phi$ is the marginal distribution of $X_1,X_2$. We ...
0
votes
0answers
11 views
+100

Which probability distribution(s) $f(x)$ allow for a closed form solution to $\int\left(x-a\right)^{-\gamma}f\left(x\right)dx$?

I'm trying to find if there is a specific probability distribution $f\left(x\right)$ (or many) such that the following integral $$\int\left(x-a\right)^{-\gamma}f\left(x\right)dx$$ has a closed form ...
2
votes
0answers
21 views

Hypothesis Testing on Renewal Processes

We have time $[0,T]$ to observe a renewal point process, where the inter-renewal timings are i.i.d, and then decide whether the observation is according to a renewal process in which the pdf of ...
0
votes
1answer
19 views

Density of a distribution given by a Gaussian copula and a set of marginals

Suppose the distribution of an $n$-dimensional random vector $X$ is characterized by a Gaussian copula $C_R$ with correlation matrix $R$ and a set of marginal $\{(F_{X_i}, f_{X_i})\}_{i=1}^n$ (pairs ...
1
vote
0answers
311 views

Hidden Markov Model, transition probabilities which are modeled with an exponential distribution

I'm looking at implementing an algorithm described in a paper, but I'm having trouble understanding how the transition probabilities for a Hidden Markov Model are defined. In the first sections, I ...
0
votes
0answers
72 views

Transition matrix in left-right hidden semi-Markov model

I'm developing a hidden semi-Markov model left-right . In a left-right model a sequence of $M$ states starts in state $1$ and ends in state $M$, with no repetition of states. Since the model is ...
2
votes
2answers
22 views

Simple Probability - Enumeration and Geometric Distributions

I am not sure as to why this particular practice problem does not use a geometric distribution. A prize is randomly placed in one of ten boxes, numbered from 1 to 10. You search for the prize asking ...
2
votes
1answer
82 views

simplify expectation definition Hidden Markov Model

I am reading Rabiner's paper entitled "A tutorial on hidden markov models and selected applications in speech recognition". There is a very simple example where he simplifies the calculation of an ...
0
votes
0answers
20 views

Binary Hidden Markov Model

Consider a binary HMM with 2 observed variables $O_n \in \{0,1\} \; \forall n \in \mathbb{N}$. Suppose that the hidden Markov process $X_n$ is characterised by a known transition probability matrix ...
2
votes
2answers
38 views

uniqueness of joint probability mass function given the marginals and the covariance

Let X and Y be two nonnegative, integer-valued random variables. Is there a way to find the joint probability mass function, i.e. $$ \mathbb{P}(X= k, Y= h) $$ for some $k,h\geq 0$, given the ...
-1
votes
0answers
33 views

Question about order statistics [on hold]

$X_i, i = 1,2,3,\ldots,n$ be IID continuous RV's, with common CDF $F$. For $y \in \mathbb{R}$ define: $N(y) =$ number of $X_i$'s less than or equal to $y$ ($N(y)$ takes values $0,1,2,\ldots$ Show ...
2
votes
1answer
27 views

($N_t$) is Poisson process with $\lambda = 1$. Calculate $E(N_2\mid N_1)$ and $E(N_1\mid N_2)$

($N_t$) is a Poisson Process with constant rate $\lambda = 1$. $1)$ Calculate $E(N_2\mid N_1)$: So this is how far I've gotten: Let $N_2 = N_1 + (N_2 - N_1)$ $E(N_2\mid N_1) = E(N_1\mid N_1) ...
0
votes
3answers
38 views

100 lottery tickets are distributed, only 2 of them have a prize.

100 lottery tickets are distributed, only 2 of them have a prize. Rupert Murdoch buys n of these tickets (n of course is some number between 0 and 100). What is the probability that Murdoch wins ...
1
vote
1answer
19 views

slot machine can display the numbers 1, 2 or 3.

A slot machine can display the numbers 1, 2 or 3. 1 has probability 0.1, 2 has probability 0.3, and 3 has probability 0.6. The machine is run 10 times, and the numbers that show are added. At ...
4
votes
1answer
31 views

Question on proving tight sequences.

I was just wondering how you would go about showing that a sequence of random variables is a tight sequence. For example suppose $X_{n}$ is distributed Exponentially($\lambda_n$) how would I show that ...
1
vote
0answers
30 views

Limit of monotone decreasing function on generalised inverse.

Consider a right-continuous, monotone decreasing, non-negative function $\bar F(x)$ (its the tail of a probability distribution, but that doesn't matter). Now let \begin{equation} I_{n}=\{x : \bar ...
0
votes
0answers
20 views

Estimating distribution from two distributions

I have been doing a survey on Family Incomes in India. The income of male and females are denoted by x and y. x and y are strictly positive. Per chance, individual values of y were deleted. I only ...
-1
votes
0answers
25 views

Is the CDF of a mixture distribution uniformly distributed?

It is well-known that if $Y = F(X)$, such that $F$ is a continuous and a strictly increasing cumulative distribution function with a well-defined quantile function $F^{-1}$, then $Y \sim U(0,1)$. Now, ...
0
votes
0answers
12 views
0
votes
2answers
28 views

Finding the distribution of a function of random variables using the definition (without the convolution theorem)

I'm trying to find $f_Z(z)$ with $Z=2X-Y$, for $X$ and $Y$ with joint density function $f_{XY}(x,y)$: $$ \begin{cases} x/8 & 1 \le x \le 3 \land -1 \le y \le 1 \\ 0 & \text{elsewhere} ...
1
vote
1answer
80 views

Conditional distribution of random variable X given itself

I'm stuck with something that might seem trivial but gives me headache. What is the distribution of $X|X$, i.e. the conditional distribution of $X$ given $X$? I'm pretty confident that: $$\mathbb ...
0
votes
2answers
19 views

What is the difference between infinitely divisible and stable law?

I read somewhere that stable law is the special case of infinitely divisible. In other word, stable distribution is a special case of infinitely divisible distribution. But I am not quite sure what ...
1
vote
1answer
63 views

Distribution of Square of Rician Random Variable?

We know that the square of a Rayleigh random variable has exponential distribution, i.e., Let the random variable $X$ have Rayleigh distribution with PDF ...
-1
votes
0answers
15 views

Probability- random variables [on hold]

n biased coins are tossed, when every coin has the the probability of p to get "H" independently of the other coins. Then, all the coins that got "H" are tossed again. What I have to do is to find the ...
-1
votes
1answer
12 views

Does this distribution relation have a name: $\mathbb E(X^n)=b^n \mathbb E(Z^n)$ [on hold]

More precisely, if $X$ and $Z$ are distributions so that $X=bZ$, then $\mathbb E(X^n)=b^n \mathbb E(Z^n)$. I found it on this site and would like to know "where it comes from".
1
vote
1answer
22 views

Convergence of sequence of random variables 2

If I know $\lim\limits_{n \to \infty} \mathbb{P}(X_n<c-\gamma)=0$ for all $\gamma>0$, how can I prove supremum of all reals $\alpha$ for which $\lim\limits_{n \to \infty} \mathbb{P}(X_n\leq ...
3
votes
0answers
23 views

The distribution of the condition number for complex Wishart matrices.

I am trying to derive the distribution of the condition number for centered uncorrelated complex Wishart matrices $n\times n$ with $m$ degrees of freedom. The problem is with the solution I got (it's ...
0
votes
1answer
28 views

Even moments of distribution given probability density function

Given the probability density function $f(x)$, and the $𝔼[X] = \frac{2}{\sqrt{\pi\lambda}} $, how best should I go about deducing the even moments of this distribution? $f(x) = ...
-1
votes
0answers
92 views

Bayesian Network: Probability distribution of random variable itself a random variable [on hold]

I'm doing a thing with a Bayesian Network. There is a tool to analyze such networks and there is a "doubt" setting in [0, 1]. If the certainity of a prediction is less than that value, then it is ...
0
votes
0answers
188 views

How to model mutual independence in Bayesian Networks?

It's well known that 3 random variables may be pairwise statistically independent but not mutually independent, for an illustration see: example pairwise vs. mutual relations. Can mutual ...
0
votes
0answers
81 views

Conditional independence in Bayesian network with qualitative influences

I have some troubles solving an exercise from the book Probabilistic Graphical Models (pgm.stanford.edu). We are given the bayesian network with binary-valued variables. We do not know the CPDs, ...
-1
votes
1answer
39 views

How to find the Probability of $X \gt 1$ for $p(X)=2e^{(-2X)}$ [on hold]

A PDF is given by formula $p(X)=2e^{(-2X)}$, $x>0$. Determine P($X \gt 1$)? we input 2 and got $=.03663$ but we do know as $X$ goes to infinity $p(X)$ will be $2$.