Random variables Definition and 351 Threads

Homework Statement D = (L + E) / S Where L, E, and S are mutually independent random variables that are each normally distributed. I need to find (symbolically), the conditional PDF f(d|s). Homework Equations The Attempt at a Solution Not sure what to do with so many...

I am stuck on the following problem: Five items are to be sampled from a large lot of samples. The inspector doesn't know that three of the five sampled items are defective. They will be tested in randomly selected order until a defective item is found, at which point the entire lot is...

Hi, I have the following question : How do we estimate the joint probability Pr(X_1, ... X_n) when n \rightarrow \infty ? Thanks a lot.

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...

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) =...

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?

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...

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...

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

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...

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, 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...

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...

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)...

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...

Homework Statement Consider independent random variables X1, X2, X3, and X4 having pdf: fx(x) = 2x over the interval (0,1) Give the pdf of the sample maximum V = max{X1,X2,X3,X4}. The Attempt at a Solution I can't find ANYTHING about how to solve this in the book, please help!

Two "Sum of Random Variables" Problems Homework Statement Problem A: Consider two independent uniform random variables on [0,1]. Compute the probability density function for Y = X1 + 2X2. Problem B: Edit: never mind, solved this one Homework Equations fY(y) = F'Y(y) FY(y) = double integral...

If X is some RV, and Y is a sum of n independent Xis (i.e. n independent identically distributed random variables with distribution X), is the mean of Y just the sum of the means of the n Xs? That is, if Y=X1+X2+...+Xn, is E[Y]=nE[X]? I know that for one-to-one order-preserving functions, if...

Homework Statement The random variables X1 and X2 are independent and identically distributed with common density fX(x) = e-x for x>0. Determine the distribution function for the random variable Y given by Y = X1 + X2. Homework Equations Not sure. Question is from Ch4 of the book, and...

Homework Statement Consider a random variable X having cdf: 1, x ≥ 4, 3/4, 1 ≤ x < 4, FX(x) = 1/2, 0 ≤ x < 1, 1/4, −1 ≤ x < 0...

HELP!Sums of Random Variables problem: Statistics Homework Statement 3. Assume that Y = 3 X1+5 X2+4 X3+6 X4 and X1, X2, X3 and X4 are random variables that represent the dice rolls of a 6 sided, 8 sided, 10 sided and 12 sided dice, respectively. a. If all four dice rolls yield a 3, what...

Suppose X and Y are Uniform(-1, 1) such that X and Y are independent and identically distributed. What is the density of Z = X + Y? Here is what I have done so far (I am new to this forum, so, my formatting is very bad). I know that fX(x) = fY(x) = 1/2 if -1<x<1 and 0 otherwise The...

Hi, I have been trying to solve the problem of finding the random variable that results from the difference between two other random variables. Let me use the following notation: y=r^2 and x=2 r d cos\gamma, where y is Gamma distributed and therefore r is Nakagami. I would like to find...

Hello, How do we interpret the fact that a random variable can have no mean? For example the Cauchy distribution, which arises from the ratio of two standard normal distributions. I seek intuitive explanations or visualisations to understand math "facts" better.

Suppose X and Y are independent Poisson random variables, each with mean 1, obtain i) P(X+Y)=4 ii)E[(X+Y)^2] I m trying to solve this problem but have difficulty starting ... If some one could give me a some pointers

Question : Let xi, where i = 1,2,3,..,100 be indepenedent random variables, each with a uniformly distributed over (0,1) . Using the central Limit theorem , obtain the probability P( <summation> xi > 50)

Let Xi, i=1,...,10, be independent random variables, each uniformly distributed over (0, 1). Calculate an approximation to P(\sumXi > 6) Solution E(x) = 1/2 and Var(X) = 1/12 [How should is calulate the approxmiate ]

Hi everyone, here's a probability problem that seems really counter-intuitive to me: Find four random variables taking values in {-1, 1} such that any three are independent but all four are not. Hint: consider products of independent random variables. My thoughts: From a set perspective...

Homework Statement I am trying to work out how to find the distribution function F_{Y} of Y, a random variable given the distribution function F_{X} of X and the way that Y is defined given X (see below). Any pointers to get me started would be brilliant. I have done a similar question to...

Homework Statement The Langevin equation for the Ornstein-Uhlenbeck process is \dot{x} = -\kappa x(t) + \eta (t) where the noise \eta has azero mean and variance <\eta (t)\eta (t')> = 2D(t-t')\delta with D \equiv kT/M\gamma. Assume the process was started at t0 = - \infty. Using...

we know that, if, for example, the variable X has a probability distribution f and that the variable Y has a probability distribution g, and both are independent then the variable Z=X+Y has a distribution f*g, where " * " stands for convolution. if Z=XY then the probability distribution of Z is...

Hi, everybody. My problem is about Probability and Random Process. i can't understand the probability density function of sum of two random variables and function of product of two random variables. Here is my question with a part of a solution: how can i find these problems solutions and...

Homework Statement The expectation value of the sum of two random variables is given as: \langle x + y \rangle = \langle x \rangle + \langel y \rangle My textbook provides the following derivation of this relationship. Suppose that we have two random variables, x and y. Let p_{ij}...

Homework Statement Show that the function defined by f(x,y,z,u) = 24*(1+x+y+z+u)^(-5) for x,y,z,u>0 and f=0 elsewhere is a joint density function. Find P(X>Y>Z>U) and P(X+Y+Z+U>=1). Homework Equations distribution function = quadruple integral from 0 to x (or y or z or u) here of the...

Let x_1, x_2, ..., x_n be identically distributed independent random variables, taking values in (1, 2). If y = x_1/(x_1 + ... + x_n), then what is the expectation of y?

Homework Statement X1, X2, X3 are three random variable with uniform distribution at [0 1]. Solve the PDF of Z=X1+X2+X3. Homework Equations The Attempt at a Solution PDF of Z, f_z=\int\intf_x1(z-x2-x3)*f_x2*f_x3 dx2 dx3 I saw the answer at http://eom.springer.de/U/u095240.htm, but I cannot...

Homework Statement Suppose X has an exponential with parameter L and Y=X^(1/a). Find the density function of Y. This is the Weibull distribution Homework Equations The Attempt at a Solution X~exponential (L) => fx(s)= Le^(-Ls) Fx(s)=P(X<s) = 1-e^(-Ls)...

Homework Statement Let X have mean u and variance s^2. Find the mean and the variance of Y=[(X-u)/s]Homework Equations The Mean is linearThe Attempt at a Solution I thought to just plug in the mean of X anywhere i saw it in Y so mean of Y would be 0 and then for the variance I was kind of...

Homework Statement H = X + Y where X and Y are two continuous, dependent random variables. The Joint PDF f(x,y) is continuous. All the literature that I have looked at concerning this matter have dealt with the convolution of two independent random variables. Homework Equations All I know...

Homework Statement 1. Suppose u flip a coin Z = 1 if the coin is heads Z = 3 if the coin is tails W = Z^2 + Z a) what is the probability function of Z? b) what is the probability function of W? 2. Let Z ~ Geometric (theta). Compute P(5<=Z<=9). Homework Equations The Attempt at a Solution...

Well, I thought I understood the difference between (weak) convergence in probability, and almost sure convergence. My prof stated that when dealing with discrete probability spaces, both forms of convergence are the same. That is, not only does A.S. convergence imply weak convergence, as...

Homework Statement Suppose X1,X2... are iid mean 1 exponential random variables. Use large deviation methodology to give a lower bound for the rate function R(a) for a>1 Homework Equations R(a) \leq \frac{-logP[Sn >n*a]}{n} The Attempt at a Solution I know that a sum of exponential random...

What is the motivation behind random variables in probability theory? The definition is easy to understand. Given a probability space (Ω, μ), a random variable on that space is an integrable function X:Ω→R. So essentially, it allows you to work in the concrete representation R instead of the...

If X is uniformly distributed over [0,a), and Y is independent, then X + Y (mod a) is uniformly distributed over [0,a), independent of the distribution of Y. Can anyone point me to a statistics text that shows this? Thanks,

Hi friends/colleagues, Let X1, X2, ..., Xn be a sequence of independent, but NOT identically distributed random variables, with E(Xi)=0, and variance of each Xi being UNEQUAL but finite. Let S be the vector of partial sum of Xs: Si=X1+X2+...+Xi. Question: What is the limiting...

Homework Statement Say, it is known that E_X[f(X)] = E_X[g(X)] = a where f(X) and g(X) are two functions of the same random variable X. What is the relationship between f(X) and g(X)? Homework Equations The Attempt at a Solution My answer is f(X) = g(X) + h(X) where E_X[h(X)] =...