Random variables Definition and 318 Threads

I am looking for a Hoeffding-type result that bounds the tail of a sum of independent, but not identically distributed random variables. Let X_1,..,X_n be independent exponential random variables with rates k_1,...,k_n. (Note: X_i's are unbounded unlike the case considered by Hoeffding) Let S=...

If x and y are independent and identically distributed exponential random variables, and z = x+y w = x-y are z and w also independent? Do I have to actually find the joint pdf of z and w, then find the marginals and then see if they multiply to equal the joint pdf? Or is there a way to just...

I am studying calculus and statistics currently, and a possible relationship between them just occurred to me. I was thinking about two things: (i) is a differentiable function from R to R a manifold, and (ii) in what way is a random event really unpredictable? So I don't know much about either...

Homework Statement Let X1,X2,X3,Y1,Y2,Y3 be random variables. If X1 and Y1 have the same distribution, X2 and Y2 have the same distribution, X3 and Y3 have the same distribution, then is it true that X1+X2+X3 and Y1+Y2+Y3 will have the same distribution? Why or why not? 2. Homework...

Homework Statement Vicki owns two separtment stores. Delinquent charge accounts at store #1 show a normal distribution, with mean $90 and std. deviation $30, whereas at store #2, they show a normal distribution with mean $100 and std. deviation $50. If 10 delinquent accounts are selected...

Homework Statement The distribution of the IQ of a randomly selected student from a certain college is N(110,16). What is the probability that the average of the IQ's of 10 randomly selected students from this college is at least 112? Homework Equations I think we need P(Sample Mean...

Hello, What is the difference between independent and uncorrelated random variables? Practical examples of both? Regards

I've been trying to solve the following question: Let X be a random variable s.t. Pr[|X|<+\infty]=1. Then for every epsilon>0 there exists a bounded random variable Y such that P[X\neq Y]<epsilon. The ideia here would be to find a set of epsilon measure so Y would be different than X in that...

1 Let X be a normal variable with mean 0 and variance 1. Let Y = ZX where Z and X are independent and Pr(Z = +1) = Pr(Z = -1) =1/2. a Show that Y and Z are independent. b Show that Y is also normal with mean 0 and variance 1. c Show that X and Y are uncorrelated but dependent. d Can you...

If X_1 and X_2 are independent random variables in \mathbb{R}^n, and \rho_{X_1} and \rho_{X_2} are their probability densities, then let \rho_{X_1+X_2} be the probability density of the random variable X_1+X_2. Is it true that \hat{\rho}_{X_1+X_2}(\xi) =...

Good Evening: I'm given this problem: A device that continuously measures and records seismic activity is placed in a remote region. The time, T, to failure of this device is exponentially distributed with mean 3 years. Since the device will not be monitored during its first two years of...

Consider a character string randomly generated from an alphabet {T,H} of length L, where T and H each have a probability of 0.5. For an arbitrary finite L the probability of a given string is p=(0.5)^L. A probability is the sole determinant of Shannon entropy (S). Therefore I'm claiming that...

A student is getting ready to take an important oral examination and is concerned about the possibility of having an "on" day or an "off" day. He figures that if he has an on day, then each of his examiners will pass him independently of each other, with probability .8, whereas if he has an off...

Homework Statement Can anyone help me with question 5d on this paper, I just don't get it. I have done 5a,5b and 5c. How do I find the values for x1 and x2 ? http://www.mathspapers.co.uk/Papers/edex/S1Jan03Q.pdf Thanks.

Hi everyone. I have this problem. Given three random variables X, Y, Z with joint pdf (probability density function) f(x,y,z)=\exp(-(x+y+z)) if x>0, y>0, z>0; 0 elsewhere find the pdf of U (f_U), where U is the random variable given by U=(X+Y+Z)/3. Now I know how to find the joint pdf...

Hi, All, Let x1 x2... Xn be correlated random events (or variables). Say P(X1), P(X2)..., P(Xn) can be computed, in addition to that, covariance and correlated between all X can be computed. My question is, what is P(X1) * P(X2) *... * P(Xn)?

Homework Statement Find E(XY), Cov(X,Y) and correlation(X,Y) for the random variables X, Y whose joint distribution is given by the following table. X 1 2 3 Y -1| 0 .1 .1 0| 0 .5 .6 1| .2 0 0The Attempt at a...

Homework Statement Let X and Y be two independent random variables with distribution functions F and G, respectively. Find the distribution functions of max(X,Y) and min(X,Y). Homework Equations The Attempt at a Solution Can someone give me a jumping off point for this problem...

Homework Statement Let Z be a standard normal random variable and \alpha be a given constant. Find the real number x that maximizes P(x < Z < x + \alpha)/ Homework Equations The Attempt at a Solution Looking at the standard normal tables, it seems obvious to me that x=0 gives the...

Homework Statement Let X be a standard normal random variable. Calculate E(XcosX), E(sinX), and E\left(\frac{X}{1+X^{2}}\right) Homework Equations The Attempt at a Solution I have no idea where to start with this. I am not seeing any connection between it and the chapter...

Homework Statement Let \psi(x) = 2\phi(x) - 1. The function \psi is called the positive normal distribution. Prove that if Z is standard normal, then |Z| is positive normal. Homework Equations The Attempt at a Solution I am not really sure where to begin with this. Can anyone...

Hi. The question is: Given X, Y and Z are all continuous, independant random variables uniformly distributed on (0,1), prove that (XY)^Z is also uniformly distributed on (0,1). I worked out the pdf of XY=W. I think it's -ln(w). I have no idea at all how to show that W^Z is U(0,1). What...

Homework Statement A communications channel transmits the digits 0 and 1. However, due to static, the digit transmitted is incorrectly received with probability 0.2. Suppose that we want to transmit an important message consisting of one binary digit. To reduce the chance of error, we...

Homework Statement \Omega is a set of points \omega ; C_{i} i = 1, 2, ... 7 are subsets of \Omega; and ( \Omega, F, P) = (B_{i}, i/10, i = 1, 2, 3, 4 ) is a probability modal with B_{1} = C_{1} \cup C_{7}, B_{2} = C_{2} \cup C_{6}, B_{3} = C_{3} \cup C_{5} and B_{4} = C_{4}. State...

"Factorizing" random variables Suppose we have a (discrete) random variable X. Consider a random variable Y "equivalent" to X if there are functions f, g such that X = f(Y) and Y = g(X). Among other things, this implies H(Y) = H(X). Y = Y1 x Y2 x ... x Yn, where x is the cartesian product...

Hello, Can somebody pls explain to me what is the difference between generating random numbers and random variables. The confusion is mainly because most of the time texts write that for n (iid) random variables in the limiting sense reaches the expectation of the first random variable. I...

Homework Statement Suppose X is a discrete random variable with probability mass function pX(x)=1/5, if x=-2,-1,0,1,2 pX(x)=0, otherwise Let Y=X2. Are X and Y independent? Prove using definitions and theorems. Homework Equations The Attempt at a Solution The random variables X and Y...

I am studying for the FRM and there is a question concerning the captioned. I try to start off by following the standard Expectation calculation and breakdown the pdf into Bayesian Conditional Probability function. Then i got stuck there. Anyone can help me to find a proof on it? Many thanks.

Let X and Y be two random variables. Say, for example, they have the following joint probability mass function x -1 0 1 -1 0 1/4 0 y 0 1/4 0 1/4 1 0 1/4 0 What is the proper way of computing E(XY[/color])? Can I let Z=XY and find...

If Z1,Z2...Zn are standard normal random variable that are identically and independently distrubuted, then how can one prove that squaring and summing them will produce a Chi- squared random variable with n degrees of freedom. Any help on this will be greatly appreciated. I am new to this...

Homework Statement An ambulance travels back and forth, at a constant speed, along a road of length L. At a certain moment of time an accident occurs at a point uniformly distributed on the road. (That is, its distance from one of the fixed ends of the road is uniformly distributed over...

Suppose I have a sample X_1, ..., X_n of independently, identically distributed exponential random variables. One result I deducted was that the ratio of any two of them (eg. X_1 / X_2) is independent of the sample average 1/n * \sum_{i=1}^{n} X_i. (Aside: that ratio, as a random variable...

Man I hate probability...anyhow could some help me with this Q as I am not understanding how to set it up... Suppose that the force acting on a column which helps to support a building is normally distributed with mean 15.0 kips and standard deviation 1.25 kips: What is the probability...

Q: If X_1 and X_2 are independent exponential random variables with respective parameters \lambda_1 and \lambda_2, find the distribution of Z = X_1 / X_2. Discussion The best method to attack this problem apparent to me is coming up with a cumulative distributive function for Z and then...

Hi, Guys, I'm new to this forum, and don't have strong background in probability theory, so please bare with me if the question is too naive. Here's the question, In a problem I'm trying to model, I have a random variable (say, R), which is a sum of random number (say, N) of random variables...

Pdf (or mgf) of maximum of dependent exponential random variables ? max of Z1, Z2, Z3, Z4 where Z1 = |X1+X2+X3|^2 + |Y1+Y2+Y3|^2 Z2 = |X1-X2+X3|^2 + |Y1-Y2+Y3|^2 Z3 = |X1+X2-X3|^2 + |Y1+Y2-Y3|^2 Z4 = |X1-X2-X3|^2 + |Y1-Y2-Y3|^2 Xi, Yi are independent zero mean normal with...

Hi all, I want to find maximum of two random variables which are correlated and are non gaussian too. Baiscally I need an analytical orr approximate solution to their bivaraite distribution with means and varaince of resulting distribution. There is some work by Clark 'The greatest of finite...

Homework Statement Let X_1, \ldots, X_6 be a sequence of independent and identically distributed continuous random variables. Find (a) P\{X_6 > X_1 \, | \, X_1= \max(X_1, \ldots, X_5)\} (b) P\{X_6 > X_2 \, | \, X_1 = \max(X_1, \ldots, X_5)\} The attempt at a solution (a) is the...

a discrete random variable has range space {1, 2, ..., n} and satisfies P(X=j) = j/c for some number c. Find c, and then find E(X), E(X^2), E(1/X) and Var(X). thanks

For two independent and identically distributed random variables having the exponential distribution, do they have the same lambda value, or are the lambda values different?

Hi I have a question regarding i.i.d. random variables. Suppose X_1,X_2,\ldots is sequence of independent and identically distributed random variables with probability density function f_{X}(x), mean = \mu and variance = \sigma^2 < \infty. Define Y_{n} = \frac{1}{n}\sum_{i=1}^{n}X_{i}...

Hi all, assume we have a sample space with 2^n points. (it size is 2^n for some natural n) I need to prove that the maximal number of independent binary (indicator) random variables (which are not trivial, i.e. constant) is n... Thnks, Pitter

I have two independent standard normal random variables X1,X2. Now I want to construct two new normal random variables Y1,Y2 with mean\mu1, \mu2 and variance (\sigma1)^2, (\sigma2)^2 and correlation \rho. How do I approach this problem?

Can anyone tell me how to find the joint PDF of two random variables? I can't seem to find an explanation anywhere. I'm trying to solve a problem but I'm not sure where to go with it: Y is an exponential random variable with parameter \lambda=4. X is also an exponential random variable and...

Homework Statement A random variable has distribution function F(z) = P(y<= z) given by (this is a piecewise function) f(z) = 0 if z < -1 1/2 if -1 <= z < 1 1/2 + 1/4(z-1 if 1 <= z < 2 1 if 2 <= z What is P(Y = 2)? Find all the numbers t with the property that both P(Y <= t) >=...

Homework Statement A random variable has a distribution function F(z) given by F(z) = 0 if z< -1 F(z) = 1/2 if -1 <= z < 2 F(z) = (1-z^{-3}) is 2 <= z Find the associated mean and variance. The Attempt at a Solution I drew the distribution function. I started with the associated...

Homework Statement If X is represented by the Gaussian distribution, that is, f_{X}(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp{(-\frac{x^2}{2\sigma^2})} find an expression for the pdf fZ(z) of Z = arctan(x). The Attempt at a Solution If Z =g(X), then g(X) is multivalued unless the range of...

Dear all, I wonder wheather there exsits a probability inequality for the sum of independent normal random variables (X_i are i.i.d. normal random varianble with mean \mu and variance \sigma^2): P\left(\frac{1}{n}\sum_{i=1}^n X_i - \mu> \epsilon\right)\leq f(\epsilon, \sigma^2,n) \right). We...

Hi fellow members, I would appreciate if you could help with the following problem, it has had me stumped! Prove the statistical distance between random variables X & Y Thank You, and have a great day!

I have two random variables X and Y. Now the distribution of X and Y, is a bit complicated. Basically they follow Gamma distributions, X=\Gamma(k1,\theta) and Y=\Gamma(k2,\theta), but k1 and k2 are Poisson distributed. But I do have a closed form expression for the distribution of X and Y, and...