Random Definition and 1000 Threads

Hi guys, I was reading about random walks and i encountered one step of a proof which i don't know how to derive in a mathematically rigorous way. the problem is in the attached file and S is a random walk with X_i as increments, X_i = {-1,+1} I know that intuitively we can switch the...

I have 2x10^12 unique sequences of DNA, and I have an average of 47 copies of each sequence (so 94x10^12 DNA molecules total). How many molecules do I need to choose at random to be "confident" (defined as you please) that I have at least 10^10 unique molecules? 10^11? 10^12? I would really...

Homework Statement You have 7 apples whose weight (in gram) is independent of each other and normally distributed, N(\mu= 150, \sigma2 = 202). You also have a cabbage whose weight is independent of the apples and N(1000, 502) What is the probability that the seven apples will weigh more...

The Markov chain, as you know, is a sequence of random variables with the property that any two terms of the sequence X and Y are conditionally independent given any other random variable Z that is between them. This sequence (which is in fact a family, indexed by the naturals) can and has been...

Homework Statement Let X and Y be independent random variables. Prove that g(X) and h(Y) are also independent where g and h are functions. Homework Equations I did some research and somehow stumbled upon how E(XY) = E(X)E(Y) is important in the proof. f(x,y) = f(x)f(y) F(x,y) =...

Suppose you have a set of digits, for the sake of simplicity we'll make them binary, how would you determine the probability that the set is random? For example, given the following 3 strings of numbers... 1111111111111111111111111111111111111111111111111111111111111111...

I've been trying to find the solution to the following problem but it's evaded me thus far. Take the classic one dimension random walk scenario. I start at point 0 and can either step forward +1 step or step backwards -1 step (equal probability). I can countinue like this for N steps. If...

Young's doublt slit experiment - Systematic and random errors - URGENT! Homework Statement Can anybody think of any other systematic and/or random errors for young's double slit experiment when using it calculate the wavelength of a light source? I have an inclass prac write up tomorrow and...

What is the total random translational kinetic energy of 5.10 L of hydrogen gas (molar mass 2.016g/mo with pressure 1.04×10^5 Pa and temperature 301 K? K=3/2nRT Pv=nRT P=1.01*10^5 V=5.1*10^-3 Is it just PV/RT=n K=3/2(PV/RT)(RT) K=3/2 PV K=772.62 J Is that right because that seems too easy

Suppose I drop a rock off a cliff at height h. As it falls, I snap a million photographs at random intervals. On each picture I measure the distance the rock has fallen. Ignoring air resistance: x(t) = 1/2 g t^2 v(t) = g t total flight time = T = Sqrt[ 2 h/g] My confusion is that in...

Hi all, I am having trouble with the concept of joint pdf's. For example - a set Z1,Z2,...ZN are each gaussian rv. Let Z1~N(0,1), let X be +1 or -1 each with probability 0.5. Z2=Z1X1, so Z2 is ~N(0,1). (I assume this to be As Z2 is just Z1 multiplied by a simple factor, an instance...

Homework Statement There is a parking lot outside a building with n evenly spaced parking spaces that are numbered 1 through N with parking space #1 being closest to the entrance of the building, and parking space #n being the furthest away from the entrance to the building. A driver enters...

I'm reading a stat textbook and it says the following: Let a discrete-time random walk be defined by Xt = Xt-1 + et, where the et's are i.i.d. normal(0,σ2). Then for t≧1, (i) E(Xt) = 0 (ii) Var(Xt) = t σ2 However, the textbook doesn't have a lot of justifications for these results and...

If the set of real numbers is considered as a sample space with the Borel sigma algebra for its events, and also as an observation space with the same sigma algebra, is a pdf or pmf a kind of random variable? That is, are they measurable functions?

Ok after searching google through and through I have decided this is the closest related place I could ask about my situation that I could find. After this happening Friday morning I have been unable to stop thinking about it and would like to hear someone else's views. On this past Friday...

Hoel: An Introduction to Mathematical Statistics introduces the following formulas for expectation, where the density is zero outside of the interval [a,b]. E\left [ X \right ] = \int_{a}^{b} x f(x) \; dx E\left [ g(X) \right ] = \int_{a}^{b} g(x) f(x) \; dx He says, "Let the random...

A random variable is defined as a function from one set, called a sample space, to another, called an observation space, both of which must be underlying sets of probability spaces. But often when people talk about a random variable - as in definitions of a particular, named distribution, such...

I'm wondering how conditional probability relates to concepts of sample space, observation space, random variable, etc. Using the notation introduced in the OP here, how would one define the standard notation for conditional probability "P(B|A)" where A and B are both subsets of some sample...

Hello all, I have a question concerning the best way to access random elements in a large file. I have a single data file of numbers, 12G in size with dimensions 1024x768x16384. My goal is to, as quickly as possible, load, say 16384 of those numbers into memory. Assume these numbers are not...

In a box with 80 tickets, 10 people buy 10 tickets each. If there are 4 winning tickets drawn at random find the probability that 1) one person gets all 3 winning tickets 2) there are 3 different winners 3) some person gets two winners and someone else gets just one for 1)...

Hi, Why there is a half factor in the definition of the correlation of complex random variables, like: \phi_{zz}(\tau)=\frac{1}{2}\mathbf{E}\left[z^*(t+\tau)z(t)\right]? Thanks in advance

hello , i wanted to get a random number generator for colored as well as white noise. all numbers in the range should be equally likely to be produced.. ie, say within 1-100, 50 shouldn't hav an extra high probability. on the net i found this fortran program. http://fortran.com/gauss_random...

So I ran an ODE solver with an additional, random (using randn) injected input. Then I took the same ODE solver and the same random line, and multiplied it by 5, so now I have (5*randn). The runtime is already ten times as long (and still not finished). Why does it take so much longer to...

I'm studying for the probability actuarial exam and I came across a problem involving transformations of random variable and use of the Jacobian determinant to find the density of transformed random variable, and I was confused about the general method of finding these new densities. I know the...

Definition: Let X1,X2,... be a sequence of random variables defined on a sample space S. We say that Xn converges to a random variable X in probability if for each ε>0, P(|Xn-X|≥ε)->0 as n->∞. ==================================== Now I don't really understand the meaning of |Xn-X| used in...

Does quantum mechanics say "Everything's random at the quantum scale, but on a macroscopic level, it's pretty much deterministic. It's still random, but deterministic in the sense that the variance attributable to quantum events is vanishingly small"?

Hello everyone, I have a (noob!) question about Markov Random Field potential function. So, I am looking at some literature where the markov random field potential function used is the so called Huber function, which is defined as: V(t, a) = t^2 if (a < some_value) V(t, a) = 2*a*|t| -...

Homework Statement I'm wondering how I go about calculating the expectation of a random variable? Is it a different process for a discrete and a continuous? Can you show me an example? Say Poisson and expoential? Also, in the formula Var(X) = E[X^2] - (E[X])^2 how does one...

Homework Statement part iv confuse me,especially the limits for y please look to my answer for this part and comment Homework Equations The Attempt at a Solution i) I got c = 1/3 ii) P(X^2 >=1)=P(X>=1) + P(X<= -1) = 7/9 iii) P(X-1>=-1/4) = P(X>= -1/4+1)=37/576 iv) we find...

What is the PDF of the exponential of a Gaussian random variable? i.e. suppose W is a random variable drawn from a Gaussian distribution, then what is the random distribution of exp(W)? Thank you!

Homework Statement Lets say I roll 2 fair dice and take the sum of the square of each dice. What formula will be the variance? Homework Equations var(x)=e(x^2)-e(x)^2The Attempt at a Solution For dice A; E(A)=3.5 E(A^2)=91/6 ^ same for dice B. VAR(A^2+B^2)=E(A^4)-E(A^2)^2+E(B^4)-E(B^2)^2 ?

In a recent federal appeals court case, a special 11-judge panel sat to decide on a certain particular legal issue under certain particular facts. Of the 11 judges, 3 were appointed by political party A, and 8 were appointed by political party B. Of the party-A judges, 2 of 3 sided with the...

Hi Experts, I'm working in industry and have an application requiring some expert knowledge on statistics/probability. I have a probability distribution function (PDF) for a Gaussian random variable. I know the standard deviation of the PDF. I also know total number of experiments conducted...

What causes the "random" decay of atoms in different interpretations? I'd like proponents of each interpretation to explain what their interpretation says about this issue? What happens/causes radio active decay? I know that atleast dBB and MWI needs to have a deterministic answer to this...

Which properties of the photon/quantum are random? Spin - Yes Position (within the "range/orbital") - Yes Momentum - Yes Phase - No Polarization - No Coherence (derivative of phase) - No Is the above correct? Please add the properties I missed

Hi, I am a bit confused. Basically if I have a pdf, fX(x) and i want to work out the distribution of Y=X^2 for example, then this involves me letting Y=X^2, rearranging to get X in terms of Y, substituting these into all values of x in my original pdf fX, and then multipying it by whatever dx...

I'm having a trouble doing this kind of problems :S Lets try this for example: The joint p.d.f of the continuous random variable X and Y is: f(x,y)= (2y+x)/8 for 0<x<2 ; 1<y<2 now we're asked to find a probability, say P(X+Y<2) I know i have to double integrate but how do I choose my...

The probability density function of the time customers arrive at a terminal (in minutes after 8:00 A.M) is f(x)= (e^(-x/10))/10 for 0 < x c) Determine the probability that: two or more customers arrive before 8:40 A.M among five that arrive at the terminal. Assume arrivals are...

Hi everyone, I'm confused about Durrett's formal definition of a random variable, as well his formal notions of probability spaces in general. I always try to make abstract definitions concrete through simple examples, but I can't wrap my head around this one: Durrett defines: X is a random...

Hi, Suppose X_1, X_2,\cdots be an independent and identically distributed sequence of exponentially distributed random variables with parameter 1. Now Let N_n:=\#\{1\leq k\leq n:X_k\geq \log(n)\} I was told that N_n\xrightarrow{\mathcal{D}}Y where Y\sim Poisson(1). Could anyone give...

I'm trying to understand the movement of particles due to quantum mechanics. I made this image. It is a magical receptor box. Every time an H atom strikes a receptor, the letter associated with that receptor shows up as a read out on a computer. For example if a H atom hits the top receptor, it...

Hi all, I have a question relating to the sum of continuous random variable probabilities that I hope you can help to answer. In any probability density function (pdf), dealing with discrete or continuous random variables, the sum of the probabilities of all possible events must equal 1...

For an exponential random variable X with rate u What is E{X|X>a} where a is a scale value from searching in internet I found that E{X|X>a}=a+E{x} but I can not prove it Help please

Homework Statement For an exponential random variable X with rate u What is E{X|X>a} where a is a scale value Homework Equations The Attempt at a Solution

Homework Statement Let X ~UNIF(0,1), and Y=1-e-x. Find the PDF of Y Homework Equations The Attempt at a Solution So i have Fy=Pr(Y<y) =Pr(1-e-x<y) =Pr(-e-x<y-1) =Pr(e-x>1-y) =Pr(-x>ln(1-y)...

Homework Statement 10 face cards are face down in a row on a table. Exactly one of them is an ace. You turn the cards over one at a time, moving from left to right. Let X be the random variable for the number of cards turned before the ace is turned over. What is the probability function...

Can anyone help me with the below question? for each of the following pairs of random variables X,Y, indicate a. whether X and Y are dependent or independent b. whether X and Y are positively correlated, negatively correlate or uncorrelated i. X and Y are uniformly distributed on the disk...

Homework Statement Given: The joint probability distribution function of X and Y: f(x,y) = 2xe^(-y), x > 0, y > x^2 0, otherwise Obtain the pdf of V = (X^2)/Y The Attempt at a Solution The interval of V is (0,1) because Y is always...

Homework Statement Suppose that X is uniformly distributed on (0,2), Y is uniformly distributed on (0,3), and X and Y are independent. Determine the distribution functions for the following random variables: a)X-Y b)XY c)X/Y The Attempt at a Solution ok so we know the density fx=1/2...

Playing around with Wolfram Alpha I discovered an elegant looking little equation. Judging by the decimal approximation of both sides, there seems to be an extremely high probability that it is true. A picture of the equation is attached but ill try to type it too...