Probability theory Definition and 129 Threads

I am stuck at obtaining the joint pmf of ##T_3## and ##T_1##. It is clear I think that ##T_1\in\text{Ge}(p)##, where the pmf of ##T_1## is given by ##p(1-p)^{k-1}##, ##k=1,2,\ldots##. Now, the negative binomial distribution counts the number of trials with ##n## successes and with success...

The exercise that appears in the text is: What I find confusing about this exercise is that the author has, up until now, not derived any results for order statistics when it comes to discrete distributions. I know the formula for the density of ##X_{(1)}## and the range when the underlying...

I'm not sure how to solve this. Intuitively, we'd want to know $$P(X_{(1)}>9.8,X_{(8)}<9.9),$$but is the above probability simply ##P((X_{(1)},X_{(8)})\in (9.8,10.0)\times(9.6,9.9))## and do I integrate the joint density over ##(9.8,10.0)\times(9.6,9.9)## then?

The first question is fairly straightforward. The density of ##X## (i.e. one of the iid r.v.s. ##X_1,\ldots,X_{100}##) is just ##f(x)=1## for ##0<x<1## and ##0## otherwise. The cdf ##F## is therefore ##F(x)=x## for ##0<x<1##, ##F(x)=0## for ##x<0## and ##F(x)=1## for ##x>1##. In the first...

Note, it's not assumed anywhere that ##X_1,\ldots,X_n## are independent. My solution to (a) is simply ##1/n!##, since we've got ##n!## possible orderings, and one of the orderings is the ordered one, so ##1/n!##. However, I am not sure this is correct, since I don't understand why the...

I'm studying branching processes, and I am bit confused about this exercise. The assumption of the processes I am studying (Galton-Watson processes) is that ##X(0)=1##, i.e. the population starts with one single ancestor. Then ##X(1)## denotes the number of children obtained by the ancestor. The...

I am paraphrasing from An Intermediate Course in Probability by Gut. Background: Let $$X(n)=\text{ number of individuals in generation } n.$$ We assume ##X(0)=1## and that all individuals give birth according to same probability law, independent of each other. Also, the number of offspring...

Here's my attempt. So, let ##N\in \text{Fs}(1/37)## be the number of bets on number ##13## (here ##\text{Fs}(1/37)## is the geometric distribution that models the first success), and let ##Y_1,Y_2,\ldots## be the losses in the bets on number ##36##. Thus $$Y_k=\begin{cases} 1,&\text{if number 36...

There's a theorem in An Intermediate Course in Probability by Gut that says if ##E|X|<\infty\implies EX=g_X'(1)##, where ##g_X## is the probability generating function. Now, consider the r.v. ##S_N##, which is the sum of a random number ##N## of terms of i.i.d. r.v.s. ##X_1,X_2,\ldots##...

I am asked to solve the challenging problem above (I don't see the purpose in this exercise actually, since transforms just make it harder I think). Here's my attempt; denote by ##\varphi_X## the characteristic function (cf) of ##X##, then...

I will omit the theorem and its proof here, since it would mean a lot of typing. But the relevant part of the proof of the theorem is that we are considering the set ##H## of functions consisting of ##\psi_\lambda(x)=e^{-\lambda x}## for ##x\geq 0## and ##\lambda\geq 0##. We extend the...

I quote from An Intermediate Course in Probability by Gut: First, I don't think it is clear that all moments exist. Integrating ##(1)## and making the substitution ##y=x^\alpha##, and rewriting the integral in terms of a gamma integral, I get that ##C=\Gamma(\beta/\alpha)##. The only condition...

I'm reading in my probability book about characterizations of the law of a random variable, that is, the probability measure ##\mathbb P_X(A)=\mathbb P(X\in A)##. I read the following passage (I'm paraphrasing slightly): This extract is basically saying that if $$\mathbb E[\varphi(X)]:=\int...

There are also two hints, which I will share with you now. The first hint says to start with the case ##n=2##. I've drawn a unit disc and a circle inside this unit disc, but I do not know how to reason further. The second hint says that the volume of an ##n##-dimensional ball of radius ##r## is...

What troubles me about this exercise is that I don't get the answer that the book gets regarding the expected quadratic prediction error. ##c## is determined by $$1=\int_0^1\int_0^{1-x} c\,dydx=c\int_0^1(1-x)\,dx=c\left[-\frac{(1-x)^2}{2}\right]_0^1=\frac{c}2,$$so ##c=2##. The marginal density...

Maybe this is a simple exercise, but I don't see how to prove the below theorem with the tools I've been given in the section (if it is possible at all). That's the theorem that I'm looking to prove. Now I'll just state some definitions and a theorem that has been given in the section prior to...

So I'd like to "integrate out" the ##x##-variable, like $$f_Y(y)=\int_0^1 \mathbf{1}_{(0,x)}(y)\frac1x\,dx.$$ I am a bit hesitant on how to proceed, since I feel like I will get an unbounded density. Something's fishy, but I don't know what.

Questions: 1. I am a bit unsure why ##g(x)=h(x)##. Clearly ##g(x)\geq h(x)##, but why is ##g(x)\leq h(x)##? Here's my explanation, which is kind of lengthy, but maybe you have a better one. If ##(a,b)\in\mathcal E_{\varphi}## is such that ##\varphi(x)>ax+b## for all ##x\in\mathbb R##, then...

I am reading a proof of Jensen's inequality. The proof goes like this. I do not know much about convex functions, but why does (1) hold? The definition of convex I'm using is that $$\varphi(tx+(1-t)y)\leq t\varphi(x)+(1-t)\varphi(y)$$ holds for all ##x,y\in\mathbb R## and all ##t\in[0,1]##.

I feel like I'm doing something wrong. I have computed $$E(Y\mid X=x)=\int_\mathbb{R}y f_{Y\mid X=x}\, dy,$$with pen and paper, and I get the same that WolframAlpha gets, namely ##0##. Can this be right? If this is indeed true, then is ##E(Y\mid X)=E(E(Y\mid X))=0## too? How do I go about...

I'm working through an exercise to prove various identities of the conditional expectation. One of the identities I need to show is the following $$E(f(X,Y)\mid Y=y)=E(f(X,y)\mid Y=y).$$ But I am a little concerned about this identity from things I've read elsewhere. I am paraphrasing from...

Let ##(\Omega,\mathcal{F},P)## be a probability space, and let us define the conditional expectation ##{\rm E}[X\mid\mathcal{G}]## for integrable random variables ##X:\Omega\to\mathbb{R}##, i.e. ##X\in L^1(P)##, and sub-sigma-algebras ##\mathcal{G}\subseteq\mathcal{F}##. If...

I don't really know how to approach this problem, but my plan is to find ##p_{X,Y}(x,y)=P(X=x,Y=y)##. The two conditions ##X=x## and ##Y=y## in terms of ##U_1## and ##U_2## read (I think) $$U_1=y,U_2 = x-y \text{ or }U_2 = y, U_1 = x-y,\qquad x\geq 2y.$$ So $$P(X = x, Y = y) = \begin{cases}...

This is the follow-up problem to my previous problem. "Integrating out" the ##y##-variable and ##x##-variable separately, we see that ##f_Y(y)=2## and ##f_X(x)=\min(1,x)-\max(0,x-1)##. From my previous post, we see that ##X## is the sum of two independent ##U(0,1)##-distributed r.v.s. What is...

Let's recall the densities of ##X## and ##Y##: \begin{align} f_X(x)=\mathbf{1}_{(0,1)}(x), \quad f_Y(y)=\frac{1}{\alpha}\mathbf{1}_{(0,\alpha)}(y) \end{align} Let ##z\in (0,1+\alpha)##. So we know that ##f_Z(z)## is given by: \begin{align} f_Z(z)=\int_\mathbb{R} f_X(t)f_Y(z-t)\,dt \end{align}...

I'm reading this article on transformation of random variables, i.e. functions of random variables. We have a probability space ##(\Omega, \mathcal F, P)## and measurable spaces ##(S, \mathcal S)## and ##(T, \mathcal T)##. We have a r.v. ##X:\Omega\to S## and a measurable map ##r:S\to T##. Then...

I'm tempted to write the joint density ##f_{XYZ}## as $$f_{XYZ}(x,y,z)=\begin{cases}\frac1{4\pi}&\text{if }x^2+y^2+z^2=1, \\ 0&\text{otherwise.}\end{cases}$$However, from other sources, I've read that a uniform distribution on the unit sphere does not have a density in three variables. If this...

Hi, in the Karhunen–Loève theorem's statement the random variables in the expansion are given by $$Z_k = \int_a^b X_te_k(t) \: dt$$ ##X_t## is a zero-mean square-integrable stochastic process defined over a probability space ##(\Omega, F, P)## and indexed over a closed and bounded interval ##[a...

Some time ago we had a discussion of the sleeping beauty problem https://www.physicsforums.com/threads/the-sleeping-beauty-problem-any-halfers-here.916459/ which is a well known problem in probability theory. In that thread, there was no consensus whether the probability of heads is 1/2 or 1/3...

A recent question about interpretations of probability nicely clarified the role of the Kolmogorov axioms: [... some excursions into QM, negative probabilities, and quasiprobability distributions ...] Conclusion: the Kolmogorov axioms formalize the concept of probability. They achieve this by...

Is it true that when X and Y are independent, E ({X+Y}3) = E (X3)+E(Y3)?

Mathematics uses Lebesgue measure for probability theory. However it is well known that it comes with a flaw that is not all sets are measurable. Is there a reason why the choice is also preferred in physics?

For concretness I'll use atoms and photons but this problem is actually just about probabilities. There's an atom A whose probability to emit a photon between times t and t+dt is given by a gaussian distribution probability P_A centered around time T_A with variance V_A. There's a similar atom...

I am interested in stomach acid and heat expansion, for instance the stomach will become heated due to an athelete competing. The heat causes atheletes to live shorter than people who don't have their body heated so often. I do a lot of differential equations and number theory, but I was...

I submitted this solution, and it was marked incorrect. Could I get some feedback on where I went wrong? Let S represent the event that Party A wins the senate and H represent the event that Party A wins the house. There are 4 cases: winning the senate and house (##S \cap H##), winning just...

I got (a) and (b) but I'm still working on (c). The solutions can be found here for your reference: https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/unit-ii/lecture-9/MIT6_041SCF13_assn05_sol.pdf. But...

I tried to derive the right hand side of the Radon-Nikodym derivative above but I got different result, here is my attempt: \begin{equation} \label{eq1} \begin{split} \frac{\mathrm d\mu_{\Theta\mid X}}{\mathrm d\mu_\Theta}(\theta \mid x) &= f_{\Theta\mid X}(\theta\mid x) \mathrm \space...

Because I do have a background in the latter it was originally very difficult for me to understand some aspects of QP (quantum physics) when I initially learned it. More specifically whenever probabilities were involved I couldn’t really make full sense of it while I never had any problems...

Greg Bernhardt submitted a new blog post The Sum of Geometric Series from Probability Theory Continue reading the Original Blog Post.

Homework Statement Let ##U_1, U_2, U_3## be independent uniform on ##[0,1]##. a) Find the joint density function of ##U_{(1)}, U_{(2)}, U_{(3)}##. b) The locations of three gas stations are independently and randomly placed along a mile of highway. What is the probability that no two gas...

Homework Statement Hi all, I have some difficulty understanding the following problem, help is greatly appreciated! Let ##U_1, U_2, U_3## be independent random variables uniform on ##[0,1]##. Find the probability that the roots of the quadratic ##U_1 x^2 + U_2 x + U3## are real. Homework...

Homework Statement [/B] Hi all, I have an issue understanding the concepts pertaining to the following problem, assistance is greatly appreciated. I understand the "flow" of the problem; first find the probability of obtaining balls of the same colour, then use the geometric distribution...

Homework Statement Discrete random variables ##X,Y,Z## are mutually independent if for all ##x_i, y_j, z_k##, $$P(X=x_i \wedge Y=y_j \wedge Z=z_k ) = P(X=x_i)P(Y=y_j)P(Z=z_k )$$ I am trying to show (or trying to understand how someone has shown) that ##X,Y## are also independent as a result...

Homework Statement Hi all, could someone give my working a quick skim to see if it checks out? Many thanks in advance. Suppose that 5 cards are dealt from a 52-card deck. What is the probability of drawing at least two kings given that there is at least one king?Homework Equations The Attempt...

<Moderator's note: Moved from homework.> Hi all, I have an issue understanding a statement I read in my text. It first states the following Proposition (Let's call it Proposition A): The number of unordered samples of ##r## objects selected from ##n## objects without replacement is ##n...

Suppose I am trying to approximate a function which I do not know, but I can measure. Each measurement takes a lot of effort. Say the function I am approximating is ##y=f(x)## and ##x \in [0,100]## Supose I know the expectation and variance of ##f(x)##. Is there a way to compute the confidence...

Hi everyone. As a graduate student in statistics, I had taken a graduate course in measure-theoretic probability theory. In a conversation with the professor, he had remarked that if I wanted to pursue further research on some of the topics covered, it may be wise to do background reading or...

Homework Statement Homework Equations All needed are in the picture above (i hope so) The Attempt at a Solution to me it is extremely difficult because it is so complicated with many notations. Also, I actually don't know how to read the question properly to answer it Is E(beta) is the...

Homework Statement The time (minute) that it takes for a terrain runner to get around a runway is a random variable X with the tightness function fX = (125-x)/450 , 95≤x≤125 How big is the probability of eight different runners, whose times are independent after 100 minutes: a) Everyone has...

Homework Statement In a vessel is a 5 cent coin and two 1-cent coins. If someone takes up two randomly chosen of these coins, and we let X be the total value of the coins taken, what is the probability function for X? Homework Equations I know that X has a value {2,6} The Attempt at a...