Random variables Definition and 351 Threads

Dear friends, I have divided the time into slots of fixed size. And i toss a coin of probability of heads 1/2 in the first slot. In the next slot, i toss a coin of probability of head 1/4, and in the i^th slot i toss a coin of prob of head 1/2^i. I do this until i get a head. What is the...

Homework Statement Compute P(X=k l X+Y=p)Homework Equations The Attempt at a Solution No idea. Kind of understand page #1. Although it seems like there's a lot of unnecessary stuff. Could have gone straight from the top to the bottom. And I don't know why/if you even have to substitute the...

Homework Statement X1 and X2 are two independent discrete random variables with P(X1 = k) = c3-k P(X2 = k) = d4-k for k in natural numbers and where X1, X2 in natural numbers is almost always valid. 0 is not include in N. Find constants c and d. Homework Equations The Attempt...

Ok, so I have this written in my notes and while going over it I have a few questions: Suppose cubical boxes are made so that the length, X (in cm) of an edge is distributed as f(x)=\frac{1}{2} for 9≤X≤11 0 otherwise What sort of distribution will the volume, Y, of the boxes have...

Ive been working with random variables for a while and only today have I come up with a basic question that undermines what I thought I knew... If I have two random variables X and Y, when am I allowed to multiply them? i.e. Z=XY Let S_1 and S_1 be sigma algebras such that S_1 is contained in...

Homework Statement This is an example in a textbook; it already has the solution. I don't understand how E[X|Y] was obtained though. So my question is how do I calculate E[X|Y] from the information given? http://img689.imageshack.us/img689/484/20120413134552578.jpg...

Hi Guys, Long time reader first time poster... This simple question has stumped me all day and I think I've finally cracked it! I'm hoping someone can confirm that, or tell me how wrong I am - either is fine :) One in 1000 cows have a rare genetic disease. The disease is not contagious...

Here is the homework question. I only have an issue with part c but have shown all my work up until then. Any help is appreciated! Mr and Mrs Brown decide to continue having children until they either have their first boy or until they have five children. Assume that each child is equally...

Homework Statement A submarine has three navigational devices but can remain at sea if at least two are working. Suppose that the failure times are exponential with means 1 year, 1.5 years, and 3 years. What is the average length of time the boat can remain at sea?Homework Equations Density...

Homework Statement So, I know the pdf for independent random variables is found by using the convolution; the pdf is f[sub:X+Y](a) = ∫ f[sub:X](a-y)f[sub:Y](y)dy, but can I just use the density function for a function of a random variable instead; that is: f[sub:X+Y](x[u,v], y[u,v])(Jacobian...

Homework Statement Two variables, X and Y have a joint density f(x,y) which is constant (1/∏) in the circular region x2+y2 <= 1 and is zero outside that region The question is: Are X and Y independent? Homework Equations Well, I know that for two random variable to be independent...

Hey everyone. I haven't taken statistics yet, but as a matter of interest I was contemplating the fact that uniform random variables added together seem to generate "bell curve" like distributions. My question is if I add up an infinite number of equally distributed random variables will the...

I have two multivariate normally i.i.d random variables, x and y, that are size n vectors. Let us assume for simplicity that their variances are 1. From these random variables, I form two vectors that contain their means, and denote these mx and my. I know that if mx = my, then W = (mx -...

[b]1. X_1,X_2\cdots X_n\:\text{are IID Random Variables with CDF}\,F(x)\:\text{and PDF}\,f(x)\\ \text{then What is the CDF of Random variable }\,Max(X_1,X_2\cdots X_n) Homework Equations [b]3. \text{Since Y will be one among}\,X_1,X_2\cdots X_n,\text{why cannot its CDF be }\,F(x)\\\text{I need...

I have some questions I could not find answer I hope here to get the correct answer my questions here in this picture ( attached ) 2 questions

Homework Statement During a typical Pennsylvania winter, I80 averages 1.6 potholes per 10 miles. A certain county is responsible for repairing potholes in a 30 mile stretch of the interstate. Let X denote the number of potholes the county will have to repair at the end of next winter. 1...

Homework Statement 1. A test consists of 10 true-false questions. (a) In how many ways can it be completed? (HINT: The task of completing the test consists of 10 stages. Use the Product Rule.) (b) A student answers the questions by flipping a coin. Let X denote the number of correctly...

Homework Statement There is a population of 30 elk. 6 elk are captured, tagged and then released into the wild. Then later 5 elk are captured, what is the probability that k elk are tagged? Homework Equations p=6/30 = 1/5P = \stackrel{n}{k} * pk * (1-p)k \stackrel{n}{k} is n...

I hope I wrote that correctly but I'm trying to find the joint. I heard it was impossible from someone. X = A/R A~BIN(n1, p1) R~BIN(n2, p2) I know I shouldn't be using the Jacobian method for Discrete distributions but I have to do it anyway. Anyone know?

Let X_1, X_2, ... be a sequence of random variables and define Y_i = X_i/E[X_i]. Does the sequence Y_1, Y_2, ... always convergence to a random variable with mean 1?

Let be $X_1, X_2, ..., X_n, ... $ independent identically distributed random variables with mutual distribution $ \mathbb{P}\{X_i=0\}=1-\mathbb{P}\{X_i=1\}=p $. Let be $ Y:= \sum_{n=1}^{\infty}2^{-n}X_n$. a) Prove that if $p=\frac{1}{2}$ then Y is uniformly distributed on interval [0,1]. b) Show...

Given the fact that X and Y are independent Cauchy random variables, I want to show that Z = X+Y is also a Cauchy random variable. I am given that X and Y are independent and identically distributed (both Cauchy), with density function f(x) = 1/(∏(1+x2)) . I also use the fact the...

I have: $Z=X_1+\ldots+X_N$, where: $X_i\sim_{iid}\,\text{Exponential}(\lambda)$ $N\sim\,\text{Geometric}_1(p)$ For all $i,\,N$ and $X_i$ are independent. I need to find the probability distribution of $Z$: $G_N(t)=\frac{(1-p)t}{1-pt}$ $M_X(t)=\frac{\lambda}{\lambda-t}$...

I think I understand the concept of random variable (for example, the number of heads when three coins are tossed together or the temperature of a place at 6.00am every morning). I am, however, confused as I have seen some material which refers even the values taken by a random variable (or...

Homework Statement If R1 and R2 are two uniformly distributed random variables on the interval [0,1]. What is the density function Z=R1*R2? Homework Equations I'm not sure actually The Attempt at a Solution I have tried to manipulate with moment generating function (which i...

Homework Statement The n random variables X_{1}, X_{2},..., X_{n} are mutually independent and distributed with the probability density f(x)=\frac{1}{\pi(1+x^{2})} i) Find the probability density of the average Y=\frac{1}{n}\Sigma^{i=1}_{n}X_{i} ii) Explain why it does not converge...

Hi All, By definition, the sum of iid non-central chi-square RVs is non-central chi-square. what is the sum of ono-identical non-central chi-square RV. I have a set of non zero mean complex Gaussian random variables H_i with a mean m_i and variance σ_i . i=1...N. H the result of their...

Hi all, I have a random variable (RV): X=\text{max}X_i+X_j where Xi and Xj are two different RVs from a set of i.i.d N RVs. I need to find the distribution of X. What is the most efficient way? Thanks in advance

Hello! I am trying to understand an example from my book that deals with two independent Poisson random variables X1 and X2 with parameters λ1 and λ2. The problem is to find the probability distribution of Y = X1 + X2. I am aware this can be done with the moment-generating function technique...

Hi. I would like to find out the probability distributions function of the sum of 5 independant random variables. They are a sum of errors: 1%, 1%, 0.1%, 0.1%, 1%. I think this is the convolution of all these. So the limits are +/- 3.2% I know the convolution of 2 square pulses becomes a...

Homework Statement Suppose we have a function, f(x,y) = e^-x * e^-y , 0<=x< ∞, 0<=y<∞, where X and Y are exponential random variables with mean = 1. (For those who may not know, all this means is ∫(x*e^(-x) dx) from 0 to ∞ = 1, and the same for y) Suppose we want to transform f(x,y) into...

Hi. Why in all literature bus arrivals are referred as independent random variables (Poisson as well)? Is there any reference where there is some math explanation except intuitive approach which of course tell that there is no correlation between 2 bus arrivals? Best regards

Hi all, I am currently doing my Final Year Project on the topic of Optimal Placement of Suicide Bomber Detectors. Given 2 dependent bomb detectors, I am trying to prove that the probability of detection in the intersected area will be larger than the individually covered areas, by working...

Suppose the random varaible Y has non-zero probability at 0,1,2,3,... (i.e. the support of Y is the set of non-negative integers). Define a random variable W: W=0 ,if Y=0,1,2,or 3 --=Y-3 ,if Y=4,5,... Define a random variable Z: Z=max{0,Y-3}=0 ,if Y≦3 --------------=Y-3 ,if Y>3 And...

Hi, I’m looking for a way to combine two discrete random variables (which I have as probability distributions). The combination should be the product (or other operation) of the two variables. This would be easy if they were independent, but they’re not. There is a known correlation between...

If you were to pick two random numbers on the interval [0,1], what is the probability that the sum of their squares is less than 1? That is, if you let Y_1 ~ U(0,1) and Y_2 ~ U(0,1), find P(Y_1^2 + Y^2_2 \leq 1). There is also a hint: the substitution u = 1 - y_1 may be helpful - look for a beta...

Hi all, I was having some troubles with a practise question and thought I'd ask here. Given an r.v. X has a pdf of f(x) = k(1-x2), where -1<x<1, I found k to be 3/4. And I found the c.d.f F(x) = 3/4 * (x - x3/3 + 2/3) Now I have to find a value a such that P(-a <= X <= a) = 0.95. I thought...

Homework Statement Given f(x,y) = x + y 0≤x≤1, 0≤y≤1 zero , otherwise Show that X+Y has density f(z) = z^2, 0 < z < 1, and z(2-z), 1 < z < 2. Homework Equations Also, how to find the density f(z) for X*Y? The Attempt at a Solution Even if f(z) is not given, some...

Homework Statement If X and Y are independent uniformly distributed random variables between 0 and 1, what is the probability that X^2+Y^2 is less than or equal to one. Homework Equations P(Z<1) = P(X^2+Y^2<1) For z between 0 and 1, P(X^2<z) = P(X < √z) = √z The Attempt at a Solution...

Homework Statement The joint probability density function of the random variable (X, Y) is given by: f(x,y) = \frac{2x}{y^2} \text{where} \; 0 \leq x\leq 1 \; \text{and} \; y\geq 1 and 0 elsewhere. Find the probability density function of the folowing random variable: U=X+Y...

Hi, I am stuck with the problem of solving this problem for my research. I have 3 random variables say X, Y, and Z and say Pr[X > Y] = p_xy, Pr[X > Z] = p_xz, and Pr[Y > Z] = 0.5. Note that p_yx = 1 - p_xy. Similarly, p_zx = 1 - p_xz, p_yz = p_zy = 0.5 I need to find out the Pr[X >...

if X1 and X2 are two uniformly distributed random variables and if Y = X1 + X2 why is that the probability density function of Y is convolution of probability density functions of X1 and X2 ? I tried many ways, I'm not able to get at this conclusion

For a sum of two independent uniform discrete random variables, Z = X + Y, what is the probability mass function of Z? X and Y both take on values between 1 and L I know that for the sum of independent rv's the PMF is a convolution so... Ʃ(1/k)(1/n-k) from k = 1 to L but I'm wondering...

Homework Statement Two toys are started at the same time each with a different battery. The first battery has a lifetime that is exponentially distributed with mean 100 min; the second battery has a lifetime that is Rayleigh-distributed with a mean 100 minutes. a) Find the CDF to the time...

Homework Statement X, Y, (X_n)_{n>0} \text{ and } (Y_n)_{n>0} are random variables. Show that if X_n \xrightarrow{\text{P}} X and Y_n \xrightarrow{\text{P}} Y then X_n + Y_n \xrightarrow{\text{P}} X + Y Homework Equations If X_n \xrightarrow{\text{P}} X then...

Homework Statement Before I get started here I have one really quick basic question: Lets say I want the probability that an survives two hours, and that the probability an engine will fail in any given hour is .02. Then I can get 1 - .02 - .98(.02) = .9604. This is found by a geometric...

Distribution of difference of two second degree non central chi squared random variables. This problem can be cast as an indefinite quadratic form for which there are a number of general numerical techniques to determine the CDF. Alternatively, it may be written as a linear combination of...

Homework Statement 1) Let X have the p.d.f f(x) = 3(1-x)2, 0≤x<1. Compute: a) P(0.1 < X < 0.5) etc... 2) Find the mean and variance, and determine the 90th percentile , of each of the distributions given by the following densities: a) f(x) 2x, 0≤0<0 etc.. 3) Find the 50th...

attached in a file. I will be grateful for some help here. Thanks :smile:

For the density function for random variable Y: f(y) = cy^2 for 0<= y <= 2; 0 elsewhere We are asked to find the value of c. I did a definite integral from 0 to 2 of cy^2. I get c = 3/8. Why would the book show an answer of c = 1/8? Is this an error on their part or am I missing something...