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

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3
votes
1answer
57 views

What does it mean that log-normal distribution is positively skewed?

I'm writing an economic overview and I need to get an explanation in the context of log-normal distribution being derived from the idea of multiplicative influence of factors and in order to explain ...
2
votes
0answers
12 views

Additivity of cumulants of dependent random variables?

What sequences of real-valued random variables $X_1,X_2,X_3,\ldots$ exist for which for all $n$ and all $k$ $$ \operatorname{cum}_k (X_1+\cdots+X_n) = \operatorname{cum}_k(X_1)+\cdots + ...
3
votes
2answers
40 views

Why can we 'choose' continuity points?

Let $F$ and $F_n$ be distribution functions with $\lim_n F_n(x)=F(x)$ for all continuity points $x$ of $F$. In a proof there is the following part: Block quote [...] choose the finite points ...
1
vote
0answers
16 views

Confirming the triangular inequality for Lévy-metric $d_L(F,G)$ and $d_L(F,G)<\infty$

Let $F,G$ be cumulative distribution functions. The Lévy-metric is defined to be $$ d_L(F,G)=\inf\left\{h\geq 0: F(x-h)-h)\leq G(x)\leq F(x+h)+h~\forall x\in\mathbb{R}\right\} $$ I would like to ...
0
votes
0answers
16 views

Continuity points of cumulative distribution function

Let $F_n,n\geq 1$ and $F$ cumulative distribution functions and assume that for all $x$ and for arbitrary $\varepsilon >0$, if $n\geq N$, we have $$ F_n(x-2\varepsilon)-2\varepsilon\leq F(x)\leq ...
-1
votes
1answer
36 views

Probability Mass Function of having both loaded & fair coins [closed]

Suppose a box contains many coins that are either biased (loaded) or balanced. A loaded coin has probability of landing on its head as p ∈ (0.5, 1.0), and a balanced coin, of course has probability ...
2
votes
1answer
57 views

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 ...
0
votes
1answer
36 views

Arg min E(X-b)^2

Question is that Find argmin E(X-b)^2 Where X is a continuous random variable. I think the minimum of E(X-b)^2 is 0. Because (X-b)^2 is nonnegative. But how can i find argmin E(X-b)^2 ? And can i ...
1
vote
1answer
25 views

Maximum of standard brownian motion on an interval

I'm trying to find the probability that the maximum of standard Brownian motion on the interval $(t_1, t_2)$ exceeds a value $x$, i.e., $$P(max_{t_1 \le s \le t_2}B(s) \gt x)$$ I initially ...
1
vote
1answer
33 views

When is $\nabla^2 f (x, y, z)= $ probability measure?

When is $\nabla^2 f(x, y, z) $= probability density function ? That is $\nabla^2 f(x, y, z)= \mu (x, y, z)$ $\int \mu (x, y, z) dxdydy = 1 $ What conditions must $f(x,y,z) $ satisfy? It is known to ...
0
votes
1answer
28 views

derive the pdf for “difference of log-normal distributions”

Can someone please help me to derive pdf for $X$, $$ X = \frac{\ln(f_1) - \ln(f_2)}{b_2-b_1} $$ here $f_1$ and $f_2$ are normal distributions with different means and standard deviations, and $b_1$ ...
0
votes
1answer
56 views

Landon derivation of the Gaussian distribution

This is the follow up to the question Taylor series of a convolution. Continuing the derivation given at Probability Theory: The Logic Of Science By E. T. Jaynes, chapter 7 "The Central Gaussian, Or ...
0
votes
0answers
19 views

Negative Utility Function

I'm using a negative utility function to compute portfolio allocation, $u(x) = -p_1e^{-X/T} + -p_2e^{-Y/T} + -p_3e^{-Z/T}$ where $p_1 + p_2 + p_3 = 1$ Certainty equivalence of this I get through, ...
8
votes
2answers
11k views

Proof of $\frac{(n-1)S^2}{\sigma^2} \backsim \chi^2_{n-1}$

It's a standard result that given $X_1,\cdots ,X_n $ random sample from $N(\mu,\sigma^2)$, the random variable $$\frac{(n-1)S^2}{\sigma^2}$$ has a chi-square distribution with $(n-1)$ degrees of ...
2
votes
1answer
30 views

Distribution Function is absolutely continuous or singular?

$$F(x) = \begin{cases} 0,& x < -1\\ \frac{x}{3} + \frac13,& -1 \leq x \leq 0\\ \frac{x}{3} + \frac23,& 0 < x \leq 1\\ 1,& 1 \leq x \end{cases}$$ This $F(x)$ is a distribution ...
2
votes
0answers
20 views

Deriving $(n-1)\dfrac{S^2}{\sigma^2} \sim \chi^2(n-1)$ [duplicate]

I can accept the fact that $Z^2 = \dfrac{\left(X-\mu\right)^2}{\sigma^2} \sim \chi^2(1)$ without knowing too much about this mysterious $\chi$-function, but I'm wondering how I can show that ...
0
votes
0answers
14 views

Lagrangian method with objective function and constraints in expected value form.

Im reading a paper and over last two weeks I have been involved with a mathematical calculation. It is about maximizing the principal utility under uncertainty; max $\int G(x-s(x))f(x,a)dx $ , where ...
0
votes
1answer
15 views

Question about the support of a joint distribution

Let X and Y be continuous random variables having the joint pdf $$f(x,y) = 8xy , 0\leq{y}\leq{x}\leq{1}$$ Find $g(x|y=\frac{1}{2})$ the conditional pdf of $X$ given $Y = \frac{1}{2}$ I found that ...
0
votes
1answer
24 views

Meaning of symbol “$y\nearrow x$” in CDF Limit

Could somebody explain the meaning of "$y\nearrow x$"? $F_X$ is right continuous, that is, for any $x$, $\lim_{y \nearrow x} F_X (y) = F_X(x)$.
-2
votes
1answer
34 views

How do can i solve the integral, finding cdf [closed]

Let $X$ be an exponential random variable with mean 1 and Y a uniform random variable between $0$ and $1$. Assume X and Y are independent and let $Z =e^{X/2}$ Find the joint cumulative ...
0
votes
0answers
28 views

Probability with ordered statistics and exponential distribution involved

Assume that $X_1,X_2$ are independent random variables with exponential distribution with the same mean 100. Let $X_{(1)}=\min\{X_1,X_2\}$ and $X_{(2)}=\max\{X_1,X_2\}$. Calculate ...
0
votes
0answers
19 views

Distribution function of Sum of IID Exponentiation Variables of Variable amount

So I'm trying to determine the distribution function of a random variable, S, give: $N \sim Geo(\frac{1}{1+\lambda}) $ $S_i \sim Exp(\mu), \forall i\in [0,N]$ $S = \Sigma^{N}_{i=0}S_i$ $S = ...
0
votes
0answers
28 views

joint pdf for two independent uniform distribution

Suppose that $𝑋_1$ and $𝑋_2$ are independent and follow a uniform distribution over $[0, 1]$. Let $𝑌_1 = 𝑋_1 + 𝑋_2$, and $𝑌_2 = 𝑋_2 − 𝑋_1$. a) Find the joint pdf $𝑓_{𝑌_1,𝑌_2} (𝑦_1, 𝑦_2)$ ...
0
votes
0answers
33 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
vote
1answer
28 views

Find the conditional pmf of $Y$ given $X = 0$

Let $X$ and $Y$ have the joint pmf defined by $f(0, 0) = f(1, 2) = 0.3$, $f(0, 1) = f(1, 1) =0.2$ $(a)$ Tabulate the conditional pmf of $Y$ given $X=0$ $(b)$ Tabulate the conditional pmf of $X$ ...
1
vote
1answer
27 views

Finding the Probability of a Normal Distribution

The mean IQ scores of 30 primary school students is 108.56 and the Standard deviation is 12.33. Assume that IQ scores for primary school students that have been kept for 50 years illustrate a normal ...
1
vote
1answer
1k views

Proof for Standard Deviation Formula for a Binomial Distribution

I understand the concept of standard deviation as the square root of the square of the mean of each sample value - the mean of the sample values. Here is the mathematical representation (I've solved ...
1
vote
2answers
31 views

Method for separating 'randomness' and 'non-randomness'

Let's assume I have a random two signals: Sin(t) R(t) Sin(t) is of course the trignometric function, but R(t) is a random process. So let's now assume I ...
3
votes
1answer
41 views

Finding a joint probability mass function

I have to find the joint probability mass function (pmf) of (X,Y) for the following problem: Roll a die repeatedly until a five or six appears, and let X be the number of rolls before a five or six ...
0
votes
1answer
25 views

Cumulative distribution function (CDF) strictly less than

Suppose a distribution function for the random variable $X$ is given by $$F(x)=\left\{ \begin{array}{11} \hfill 0 \hfill & x \lt 0\\ \hfill \dfrac{x}{2} \hfill & 0 \leq x \lt 1\\ \hfill ...
3
votes
1answer
26 views

Is a subsequence of an exchangeable sequence exchangeable?

Consider a finite sequence of random variables $X_1,...,X_n$ (1) SUFF COND: Suppose $X_1,...,X_n$ are exchangeable, meaning that the joint probability distribution of $X_1,...,X_n$ is equivalent to ...
1
vote
1answer
30 views

Find $g(x|y=\frac{1}{2})$, the conditional pdf of $X$ given $Y = \frac{1}{2}$ (Need confirmation)

Let X and Y be continuous random variables having the joint pdf $$f(x,y) = 8xy , 0\leq{y}\leq{x}\leq{1}$$ I found that the marginal pdf of Y is $f_2(y) = 4y - 4y^3$. Does $g(x|y=\frac{1}{2}) = ...
1
vote
0answers
19 views

Generate Correlated Normals

I want to generate normals $X,Y,Z$ with the correlation matrix $R$ but with means $0, 1, 2$ and variances $4, 16, 25$ respectively. How can I do this? Is it possible to apply Cholesky?
1
vote
0answers
14 views

Generating 'bursty' traffic using probability distributions beyond Poisson

I'm trying to develop a more realistic vehicle generation model for populating a traffic microsimulator. I'm trying to model a real-world intersection from which I have historical flow data. Currently ...
7
votes
1answer
246 views

How well can the maximum of a Gaussian process be approximated by a finite-dimensional Gaussian variable?

Consider a compact set $K$ in $\mathbb{R}^p$, and let $W$ be a mean zero continuous Gaussian process on $K$, meaning that $W$ takes its values in the space of continuous functions from $K$ to ...
0
votes
1answer
1k views

Range of Uniform Distribution

I'm trying to compute the density for the range $R_n$ for samples of a random variable $X$ that are uniformly distributed on the interval $(a,b)$. We define the range as $$ R_n = X_{(n)} - X_{(1)}, ...
1
vote
1answer
32 views

Probability. Find the CDF of $Y = X^2 $

Let $X$ have the uniform distribution $U(−1, 3)$. Find the CDF of $Y = X^2$. I thought this would be simply $$G(y)= \int_{-\sqrt{(y)}}^{\sqrt{(y)}} \frac{1}{4} dx$$ where $0\leq{y}<9$. Which is ...
-2
votes
1answer
44 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} ...
1
vote
0answers
7 views

Variance of truncated multivariate Gaussian

Let $X \in R^n$ be distributed as the standard multivariate Gaussian i.e. $\mathcal{N}(0,I)$. I want to understand the covariance of the distribution conditioned on certain sets. Let $P_S$ be the ...
2
votes
0answers
35 views

How to evaluate the quality of the probability distribution output of a classifier?

In a classification problem, I have trained a neural network which outputs class probabilities for a given input. For a new input, I now want to evaluate the "quality" of the neural network's ...
1
vote
1answer
50 views

Expected value of Bernoulli with probability of success Gaussian distributed

I have a circle with centre $(0,0)$. I am generating Matlab code to include $N$ neurons in a neural network. The probability of including individual neurons in a network decays exponentially with ...
1
vote
1answer
14 views

Invariant distributions: Applications in the real World

I'm studying about invariant distributions for Markov processes; say in the context of dynamics of Random Neural Networks (biological Networks). I can't fully understand what does an invariant ...
3
votes
2answers
601 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
16 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 ...
1
vote
1answer
56 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
25 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
20 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
58 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
54 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
19 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 ...