Distributions Definition and 309 Threads

Homework Statement Suppose we have a discrete random variable whose values $X = x$ can include the value $0$. Some examples are: ##X\sim \text{Binomial}(n,p)## with ##x = 0,1,2,\ldots,n## and ##X\sim \text{Poisson}(\lambda)## with ##x = 0,1,2,3,\ldots## Sometimes we can only observe these...

For random variables (X,Y) = (R*cos(V),R*sin(V)) I have R_PDF(r) = 2*r/K^2 and V_PDF(v) = 1/(2*pi) where (0 < r < K) and (0 < v < 2*pi) Is XY_PDF(x,y) the joint density of X and Y that I get by using the PDF method with Jacobians from the distribution R_PDF(r)*V_PDF(v)? So without having...

Hello, For this type of question: There are 5 green and 45 red marbles in the urn. Standing next to the urn, you close your eyes and draw 10 marbles without replacement. What is the probability that exactly 4 of the 10 are green? I understand that I can use Hypergeometric distribution, which...

I need some help to make sure my reasoning is correct. Bear with me please. I have a time distribution for a process and I want to construct a distribution for the time it takes to perform two processes. So I would define ##\tau = t + t## This would create a new distribution with is a...

Find the standard score for a the data value 9 from a normal distribution which has a mean of 16.8 and a standard deviation of 2.3

Homework Statement Let ##\{p_x\}## and ##\{q_x\}## be two probability distributions over the same index set ##\{x\}={1,2,...,N}##. Then, the trace distance between them is given by ##D(p_x,q_x):=\frac{1}{2} \sum_x |p_x-q_x|##. Prove that ##D(p_x,q,_x)=max_S |p(S)-q(S)|=max_S | \sum_{x \in S}...

One hundred pennies are being distributed independently and at random into 30 boxes, labeled 1, 2, ..., 30. What is the probability that there are exactly 3 pennies in box number 1? I tried using a Poisson distribution f(x) = (e^-λ)*(λ^x)/x! , with λ = 100/30 = 10/3 and x = 3. I got 0.22021 (5...

so I was reviewing my textbook on calculating electric field when we can assume a continuous charge distribution and they said three useful tools are (1) making use of symmetry (2) expressing the charge dq in terms of charge density lambda (3) and checking the answer at the limit of large r...

Homework Statement I have a few data-analysis problems due to Thursday, 1. Assume that a sandbag is dropped at different heights and the observations are (z i;ti) pairs. Physical model for a free fall is z=½gt2. Assume that the height measurement z has an additive random error v. The...

Homework Statement A manufacturer has designed a process to produce pipes that are 10 feet long. The distribution of the pipe length, however, is actually Uniform on the interval 10 feet to 10.57 feet. Assume that the lengths of individual pipes produced by the process are independent. Let X...

I looked in forum and i couldn't find a useful thread (useful for me). I am currently using C# and PHP and I am going to learn C and java for university soon For general uses i mostly use windows (Photoshop , ...) . I currently have ubuntu on my laptop (wanted to install kali too). I want to...

Homework Statement I need to solve a bit of a simple "reverse" problem that I'm unable to find treated in any detail, probably because it's actually reasonably straightforward. What I need to do is to "compute the charge and current distribution that give rise to [the following] field."...

Hi all, I study binding of analytes in a platform where I have 10.000 sensors. Theres is one binding event per sensor and I identify it as a sudden positive change in the signal. I do first a control experiment without analytes. I measure the maximum change in the signal for each sensor and I...

Homework Statement 50 students live in a dormitory. The parking lot has the capacity for 30 cars. If each student has a car with probability 12 (independently from other students), what is the probability that there won't be enough parking spaces for all the cars? Homework Equations P(A) =...

Homework Statement How does P(-1<Z<1) equal to 1-2P(Z>1)? (So you can find the values on the Normal Distribution Table) Homework EquationsThe Attempt at a Solution I tried P(-1+1<Z+1<1+1) but ended up with P(1<Z+1<2).

Homework Statement A disk with a radius of 0.6 m is given a uniform charge density of -7.2*10-9 C/m2. The disk is in the xz plane, and centered at the origin. A. What is the size and direction of the electric field at the point A, whose coordinates are (0m, 1.5m, 0m)? B. You now add an...

I have a question inspired by a recent thread that I did not want to hijack (https://www.physicsforums.com/threads/distributions-on-non-test-functions.831144/) I realize that weaker requirements on the space of test functions results in a more restricted set of distributions. For example, if...

The definitions of distributions that I have seen (for instance https://en.wikipedia.org/wiki/Distribution_(mathematics)#Distributions ) define a distribution as a map whose domain is a set of test functions. A defining quality of test functions is that they have compact support, which for most...

I have made an interesting observation that I can't explain to myself. Think about a prior probability P and a posterior probability Q. They are defined on an event space W with only three elements: w1, w2, and w3 (the number of elements won't matter as long as it's finite). The Kullback-Leibler...

Homework Statement I have to write equations of motion for a field, namely ## A ##. The full action has actually three terms, but my problem is with a part of the action reading: $$ S =\int d^{10}x \sqrt{-g} [ f(x_1, ... , x_{10}) + \delta (y) A ]^2 $$ In the 10 x's there is of course the...

Hi folks. I've a question. Let k be a parameter which must be estimated. It lies within the interval (a;b), a and b being finite real numbers. Let us further assume we dispose of a series of measurements X of known standard deviations. X is a complex function of k. What are Jeffreys...

As fields ##\phi ## are ill defined at precise time and position i read that fields have to be smeared. So we have test functions f in bounded regions in space time. We have a Hilbert space and ##\phi (f) ## is an operator which acts on H. Maybe we can retrieved the usual wave function when it...

I'm trying to get a better visualization of joint/marginal distributions. It's my weakest conceptual area as I pursue the actuary exams, and I want to fully understand this on a more statistical level. I've taken linear/diff-eq/multivariate, so I'm completely comfortable with the integration...

Hello community, this is my first post and i start with a question about the famous dirac delta function. I have some question of the use and application of the dirac delta function. My first question is: Using Dirac delta functions in the appropriate coordinates, express the following charge...

haruspex submitted a new PF Insights post Frequently Made Errors in Probability - Continuous and Discrete Distributions Continue reading the Original PF Insights Post.

The below is motivated by a problem I'm observing in my experimental data I have m boxes, where each box is supposed to contain k molecules of mRNA. The measurement process includes labeling all the molecules with a box-specific tag, mixing them, amplifying them to detectable levels and...

Homework Statement Problem: In an acid-base titration, a base or acid is gradually added to the other until they have completely neutralized each other. Let X and Y denote the milliliters of acid and base needed for equivalence, respectively. Assume X and Y have a bivariate normal...

Let $X_1, ..., X_{35}$ be independent Poisson random variables having mean and variance 2. Let $Y_1, ..., Y_{15}$ be independent Normal random variables having mean 1 and variance 2. (a.) Specify the (approximate) distributions of $\bar{X}$. (b.) Find the probability $P(1.8 \leq \bar{X}...

Hi All, In relation to the Boltzmann distribution vs the FD/BE distributions in different applications, I have 2 basic questions: 1. The Boltzmann distribution comes most easily from the Canonical Ensemble (constant N, V,T) while the FD/BE come from the Grand Canonical ensemble (constant .mu...

Homework Statement You purchase a chainsaw, and can buy one of two types of batteries to power it, namely Duxcell and Infinitycell. Batteries of each type have lifetimes before recharge that can be assumed independent and Normally distributed. The mean and standard deviation of the lifetimes...

I am a little confused about how variables are related to distributions as one moves from descriptive statistics to inferential statistics. I know that a variable in descriptive statistics is some measurable characteristic of some phenomenon, and its distribution is some description (table or...

Homework Statement Have 2 iid random variables following the distribution f(x) = \frac{\lambda}{2}e^{-\lambda |x|}, x \in\mathbb{R} I'm asked to solve for E[X_1 + X_2 | X_1 < X_2] Homework EquationsThe Attempt at a Solution So what I'm trying to do is create a new random variable Z = X_1 +...

Hi guys, If the value of v for which f(v) has its maximum value is known for a sample of a gas, is there a way of determining the corresponding maximum of f(E)/g(E)?

I'm struggling with question 10. I'm not sure how to account for the different time? I'm probably just overthinking it. I have attached a picture of the answer as well. Again, trying to figure out question #10

Homework Statement The number of claims that an insurance company receives per week is a random variable with the Poisson distribution with parameter λ. The probability that a claim will be accepted as genuine is p, and is independent of other claims. a) What is the probability that no claim...

"We present a new distance determination to the Galactic globular cluster 47 Tucanae by fitting the spectral energy distributions of its white dwarfs (WDs) to pure hydrogen atmosphere WD models. Our photometric data set is obtained from a 121-orbit Hubble Space Telescope program using the Wide...

Do charges exist as a point or a distribution? Or does it depend on the situation? Or does the concept of image mean that it's very difficult to tell, and if so why is the point charge model being pushed so hard, what phenomena does it explain that distributions cant?

Hello Mathematicians! I'm doing some work on obtaining true measures of ability for students, and am trying to find a simple mathematical example that would show that a student's true ability is obtained by having a few equally weighted tests rather than one big test. The example I'm thinking...

I have an experiment in which I want to extract the distribution function of a process. I expect it to be Gaussian. Each data point measured is an entire distribution, f(x), but I am forced to average over many points such that the result of the experiment is the sum of many measurements of...

Homework Statement A plastic rod of finite length carries an uniform linear charge Q = -5 μC along the x-axis with the left edge of the rod at the origin (0,0) and its right edge at (8,0) m. All distances are measured in meters. Determine the magnitude and direction of the net electric...

Homework Statement An electric circuit has 5 components. It is known that one of the componenets is faulty. To detremine which one is faulty, all 5 componenets are tested one by one until the faulty component is found, The random variable X represents the number of test required to determine...

Hi all, I was going over the poll https://www.physicsforums.com/showthread.php?t=766275, and I was wondering how one would go about testing whether the distribution of PF's member nationalities is "the same" (up to some confidence level) than the distribution of the world's population...

I'm reading about the exponential family of distributions. In my book, I have the expression f(x|\theta) = h(x)c(\theta)exp(\sum_{i=1}^{k} w_i(\theta) t_i(x)) where h(x) \geq 0 , t_1(x), t_2(x),...,t_k(x) are real valued functions of the observation x, c(\theta) \geq 0 , and...

Homework Statement Here's the question: Given X is a discrete random variable. E(X-2) = 1/3 , VAR(X-2) = 20/9 . Detrmine value of E(X^2) the ans is 23/3 . but i ended up getting 3 . why i am wrong? Homework Equations The Attempt at a Solution

Homework Statement A charge lies on a string that is stretched along an x-axis from x = 0 to x = 3.00 m; the charge density on the string is a uniform 9.00 nC/m. Determine the magnitude of the electric field at x = 8.00 m on the x axis. Homework Equations \int_0^3 kλ/(8-x)^2\,dx...

Suppose, for a suitable class of real-valued test functions T(\mathbb{R}^n), that \{G_x\} is a one-parameter family of distributions. That is, \forall x \in \mathbb{R}^n, G_x: T(\mathbb{R}^n) \to \mathbb{R}. Now, suppose L is a linear differential operator. That is, \forall g \in...

I know that, for a random vector (X,Y,Z) jointly normally distributed, the conditional expectation E[X|Y=y,Z=z] is an additive function of y and z For what other distributions is this true?

I'm going through the Degroot book on probability and statistics for the Nth time and I always have trouble 'getting it'. I guess I would feel much better if I understood how the various distribution arose to begin with rather than being presented with them in all there dryness without context...

Homework Statement The probabilities of blood types O, A, B and AB are 0.46, 0.39, 0.12, 0.03 respectively. If a clinic is seeking either type O or B from six random individuals, what is the probability that at least 2 people have the desired blood type? Homework Equations The...

I am a bit puzzled about the distinction between integrable, and completely integrable distributions. Before I pose my question, let me first define the terms: A distribution ##D## of dimension ##k## on ##M## is a smooth collection of ##k##-dimensional subspaces ##D_p \subset T_pM## with ##D =...