Random variables Definition and 351 Threads

Homework Statement Let X and Y be i.i.d normal random variables with mean 0 and variance (that is, N(0,1)). If Z=min(X,Y). Prove that the square of Z is a Gamma distribution and identify the parameters. My problem is that the cdf of a normal random variable has no exact form. I need the cdf...

Dear Friends, I want to find an upper and lower bound for the expected value of the minimum of independent binomial random variables. What paper/book do you suggest for this problem? In other words, I need to find bounds for E(min(X1,X2,...,Xn)), where Xi 's are independent random variables...

I am working on correcting an exam so that I may study for my probability final. Unfortunately, I don't have the correct answers, so I was hoping that someone here might be able to check my thought process. 1) Pick three numbers without replacement from the set {1,2,3,3,4,4,4}. Let T be the...

Homework Statement Problem 1 – Normal Random Variables B) Y ~ N(300, 100). Pr (300 < Y < 320) = 0.4772 D) H ~ N(4000, 25). R = f(H) = 0.5H – 60. E(R) = 1940; Var(R) = 156.25I have a problem solving these problems above...I missed the class when we covered this subject and now I am lost...

Homework Statement Given the joint density, f(x,y), derive the probability density function for Z = X + Y and V = Y - X. Homework Equations f(x,y) = 2 for 0 < x < y < 1 f(x,y) = 0 otherwise. The Attempt at a Solution For Z = X + Y, I can derive the fact that, f_Z(z) =...

Homework Statement Given 2 independent uniform random variables X,Y = U [0,1], consider the random variables Z = g (X,Y) for g = (x,y) = sqrt (-2ln(x) . cos(2piy). Since finding the distribution of g(X,Y) analytically is quite tough, I need to generate MATLAB program for 1 - 10,000...

Z = X + Y Where X and Y are continuous random variables defined on [0,1] with a continuous uniform distribution. I know we define the density of Z, fz as the convolution of fx and fy but I have no idea why to evaluate the convolution integral, we consider the intervals [0,z] and [1,z-1]...

Homework Statement X, Y, Z random variables, independent of each other, with uniform distribution in (0,1). C = XY and R = Z2. Without using the joint probability function, find P(C>R). The Attempt at a Solution So far: P(C > R) = P(C - R > 0) = P(XY - Z2 > 0) = P(g(X,Y,Z) > 0)...

Homework Statement [1] A random variable X is distributed as fX(x) = 1/9*(1+x)^2 1{-1<= x<= 2}. a) Find the density function of Y = -X^2 + X + 2. b) Find the cummulative distribution function of Y = X1{-1<=X<=1} + 1{X>=1} [2] Find the function that transforms a variable X with fX(x) =...

I'm sitting here with an interesting problem that I can't seem to figure out. I'm given two random variables X=a*exp(j*phi) Y=b where both a and b are known constants. phi is uniformly distributed on the interval [0,2pi) a third random variable Z=X+Y. My goal, is to find the...

Hi, I've been working on this problem but I feel like I'm over complicating it. If you have a random variable X= a*e(j*phi), where phi is uniform on the interval [0,2pi) and a is some constant, and another random variable Y= b where b is a constant. I'm looking to find the probability density...

I am a little fuzzy on the meaning of Independent Identically distributed random variables. I understand the independent part but still not 100% on the identically distributed part. I understand that identically distributed means they have the same pdf and cdf but does this mean that they have...

Homework Statement I have two random variables with two corresponding means and standard deviations. I need to calculate the upperbound that one of the random variables is greater than the other. Any ideas I'm stumped? Homework Equations I've used the Markov inequality to calculate the...

Hello, I am having difficulty approaching this problem: Assume that K, Z_1, Z_2, ... are independent. Let K be geometrically distributed with parameter success = p, failure = q. P(K = k) = q^(k-1) * p , k >= 1 Let Z_1, Z_2, ... be iid exponentially distributed random variables with...

I have a random variable problem. I need to prove that my equation I came up with is a valid probability mass function. In the problem, I came up with this for my probability mass function: \Sigma 12/(k+4)(k+3)(k+2) Maple says that this does in fact converge to 1, so it's valid...

Homework Statement Let X and Y have JD f(x,y) = e^-y, 0<x<y Find: a) E(X|Y=y), E(Y|X=x) b) density function of R.V. E(X|Y), E(Y|X) The Attempt at a Solution a) I have found E(X|Y=y) = y/2 for y>= 0 E(Y|X=x) = x +1 for x>= 0 by finding fx(x) = ∫(x to infinity) e^-y dy = e^-x...

Could someone point me to a book that has a proof of the above statement? Thanks in advance!

Homework Statement Ok, I have 2 questions: 1. Nicotine levels in smokers can be modeled by a normal random variable with mean 315 and variance 1312. What is the probability, if 20 smokers are tested, that at most one has a nicotine level higher than 500? 2. fX,Y (x,y) = xe-x-y...

Homework Statement Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function of Y appears in the accompanying...

Homework Statement Let X1...XN be independent and identically distributed random variables, N is a non-negative integer valued random variable. Let Z = X1 + ... + XN (assume when N=0 Z=0). 1. Find E(Z) 2. Show var(Z) = var(N)E(X1)2 + E(N)var(X1) Homework Equations E(Z) = EX (E(X|Z))...

Homework Statement When reading a book, you detect each mistake with probability p, independent of other mistakes. Let M denote the amount of mistakes on a certain page and D be the number that you detect on that page. Write down P[D=k|M=m] and find for k>=0 P[D=k]. Homework Equations...

Homework Statement A text file contains 1000 characters. When the file is sent by email from one machine to another, each character (independent of other characters) has probability 0.001 of being corrupted. Use a poisson random variable to estimate the probability that the file is transferred...

please corret me if i am incorrect in my understanding of a RV,PMF or anything else but as i understand it a random variable simply maps a expirmental outcome to a real number. And a probability mass function simply gives the probabilty that a number will occur. Now my question is this: why...

I was reading some proofs about the convergence of random variables, and here are the little bits that I couldn't figure out... 1) Let Xn be a sequence of random variables, and let Xnk be a subsequence of it. If Xn conveges in probability to X, then Xnk also conveges in probability to X...

Let X1, X2, ...Xn be independent exponential variables having a common parameter gamma. Determine the distribution of min(X1,X2, ...Xn). The Attempt at a Solution I know how to do it with one X and one parameter but I am at a loss with these multiple ones... Thanks so much!

Homework Statement Let X be the number of 1's and Y be the number of 2's that occur in n rolls of a fair die. Find Cov(X, Y) Homework Equations Cov(X,Y) = E(XY) - E(X)E(Y) The Attempt at a Solution Both X and Y are binomial with parameters n and 1/6. Thus it is easy to find E(X)...

Homework Statement There are two urns, A and B. Let v be a random number of balls. Each of these balls is put to urn A with probably p and to urn B with probability q = 1 - p. Let v_a and v_b denote the numbers of balls in A and B, respectively. Show that random variables v_a and v_b are...

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 Suppose X and Y are independent random variables with X following a uniform distribution on (0,1) and Y exponentially distributed with parameter \lambda = 1. Find the density for Z = X + Y. Sketch the density and verify it integrates to 1. Homework Equations If Z =...

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 Let X be the height of a man and Y be the height of his daughter(both in inches). Suppose that the joint probability density function of X and Y is bivariatenormal with the following parameters: mean of X=71, mean of Y=60, std. deviation of X=3, Std. deviation of Y=2.7...

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...

Hi: e, z, mu are vectors of size N I need to show that E(e|z+mu) = E(e|mu) or at least E(e|z+mu) converges in probability to E(e|mu) as N goes to infinity, under the assumption that Z is not correlated with e. My guess is that to get this result I also need z to be orthogonal to mu...

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...