Random variables Definition and 318 Threads

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...

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...

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}...

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 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...

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,

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)] =...

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...

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 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 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...