Random variables Definition and 318 Threads

Homework Statement Two RVs X1 and X2 are continuous and have joint pdf f_{X_1,X_2}(x_1, x_2) = \begin{cases} x_1+x_2 &\mbox{for } 0 < x_1 < 1; 0 < x_2 < 1 \\ 0 & \mbox{ } \text{otherwise}. \end{cases} Find the pdf of Y = X_1X_2.Homework Equations I'm using the transformation "shortcut' that...

Homework Statement Let us choose at random a point from the interval (0,1) and let the random variable X_1 be equal to the number which corresponds to that point. Then choose a point at random from the interval (0,x_1), where x_1 is the experimental value of X_1; and let the random variable...

Homework Statement Homework Equations Y=1/2*(X1-X3)^2+1/14*(X2+2X4-3X5)^2The Attempt at a Solution For (a) part, I have only learned to find the moment-generating function of Y, but not finding the p.d.f. Moreover, the examples I have seen only involves random variables Xi to the power 1, but...

Homework Statement Three yearly losses. First: Exponential Second & Third: Weibull Losses are independent. Find the 95% VaR of the min loss Homework Equations The Attempt at a Solution My first thought was: Let L be total loss, A be first Loss, B be second loss, C be third...

i have a simple enough question Find the MGF of a continuous random variable with the PDF: f(x) = 2x, 0<x<1 I understand MGF is calculated as: $$M(S) = \int_{-\infty}^{+\infty} e^{Sx} f(x)dx$$ which would give me $$\int_{-\infty}^{+\infty} e^{Sx} 2xdx$$ but how would i compute this...

This is not a homework question but I project I am working on and need someone with more mathematical prowess than myself. I am using a computer program to draw random numbers from two independent distributions, x1 and x2, for two different cases and I want to establish a theoretical...

Homework Statement We are given the following table and need to find the E(XY) X|Y y = 17 20 23 35 48 p(x) x = -20 0.02 0.03 0.07 0.02 0.06 0.2 0 0 0.05 0 0.05 0.1 0.2 1 0.05 0.03 0.02 0.07 0.03 0.2 3 0.01 0.02 0.03 0 0.04 0.1 17 0.18 0.04 0.06 0.01 0.01 0.3 p(y) 0.26 0.17 0.18 0.15...

Are they always independent from each other so that you can multiply their E[X] together to form another E[X] with the same distribution and pmf or pdf?

Homework Statement Given a sequence of independent random variables {X_n}, each one with distribution Exp(1). Show that Y_n = \displaystyle\frac{X_n}{\log(n)} with n \geq 2 converges to 0 in probability but it doesn't coverges almost surely to 0. Homework Equations Density for each X_n...

Hi all, I would like to find the distribution (CDF or PDF) of a random variable Y, which is written as Y=X_1*X_2*...X_N/(X_1+X_2+...X_N)^N. X_1, X_2,...X_N are N i.i.d. random variables and we know they have the same PDF f_X(x). I know this can be solved by change of variables technique and...

hello! I'm trying to understand the following property: Let X and Y be independent random variables z: = X + Y. Then http://imageshack.us/a/img268/9228/71pe.png where fZ (z) is the probability mass function for a discrete random variable defined as follows...

the random variables X1,X2... are independent and they take 0 and 1 values and they have expected value 0 if we have Y=X1+X2+...+Xn and Z=X1+X2+...+Xn+Xn+1 what is the ρ(Y,Z) for n=46 i know that ρ(Y,Z)=cov(Y,Z)/(sqrt(var(Y)*sqrt(var(Z)) but i need some help on how to find the cov and vars...

Hi, Let's say I'm given X and Y identical independant continuous random variables. We pose Z =X/Y, I remember there is a way to find the density function of Z, altough I can't get to remember how to do it and my probability book is out of town.(And I'm not so sure what to look for in google)...

Hello. I have a problem with probability theory task. The task is: X and Y is independent random variables with same density function fx=fy=f. What will be probability of P(X>Y). This P(X>Y) reminds me a cdf: P(X>Y)=1-P(X<Y)=1-cdf of X. Cdf of x is equal to integral ∫f dx from -inf to...

Let x(1),...,x(N) all be independent uniformally distributed variables defined on (0,1), i.e. (x(1),...,x(N)) - U(0,1). Define the random variable y(i) = x(i)/(x(1)+...+x(N)) for all i=1,...,N. I’m looking for the pdf of the random variables y(1),…,y(N). Has anyone come across such random...

Homework Statement We have two independent, exponentially distributed random variables X and Y (with parameter a). Z = X/(X+Y) What is Z:s distributon function? Homework Equations The Attempt at a Solution I think I need some intuition to what I'm really doing with these, I'm having a...

Homework Statement Find correlation between random variables x and y in the following: $$P_{x,y}(x,y)=A \ xy \ e^{-(x^2)}e^{-\frac{y^2}{2}}u(x)u(y)$$ Homework Equations The co-variance ##\sigma_{xy}=\overline{(x-\bar{x})(y-\bar{y})}## or ##\sigma_{xy}=\overline{xy}-\bar{x}\bar{y}##...

I didn't post this in the probability section cause the questions I have are more regarded to communication system engineering. I haven't actually been able to wrap my head around these concepts mainly cause all the study material I use have these really ambiguous explanations of each...

Hi everyone, I have the following exercise. Fx(x)=0, x<-1 or x>1 Fx(x)=1/2, x=[-1;1] g(x)=x^2+1 --- this is the function of random variable I must calculate Fy which is the sum of solutions of g(xk)=y , Fy(y)=sumFx(xk)/|g`(xk)| g(x) is bijective on [-1;1] y=x^2+1=> x=+sqrt(y-1) or x=-sqrt(y-1)...

Hi Everyone! I have two normally distributed random variables. One on the x axis, the other on the y axis, like a complex normal random variable. I'm trying to find the pdf of the angle between a fixed point on the x-y plane(let's say point 1,0) and the vector formed by combining the two...

Oke this is a simple question but it has me a bit stumped. Given a random variable X with a uniform probability distribution between [0,2]. What is the probability distribution function (pdf) of X^2 ?

Homework Statement Let X and Y have the joint probability density function f(x,y)=k(1-y), if 0<x<y<1 and 0 elsewhere. a)Find the value of k that makes this pdf valid. b) Find P(X<3/4,Y>1/2) c) Find the marginal density function of X and Y d) Find the expected value and variance of X and...

Homework Statement The joint PDF (probability density function) ##p_{X,Y}(x,y)## of two continuous random variables by: $$ p_{X,Y}= Axy e^{-(x^2)}e^{\frac{-y^2}{2}}u(x)u(y)$$ a) find A b) Find ##p_X (x), \ p_{y}, \ p_{X|Y}(x|y), and \ p_{Y|X}(y|x)## Homework Equations The first...

Homework Statement A binary information source produces 0 and 1 with equal probability. The output of the source, denoted as X, is transmitted via an additive white Gaussian noise (AWGN) channel. The output of the channel, denoted as Y, satisfies Y = X + N, where the random noise N has the...

[FONT=Courier]Classify the following as discrete or continuous random variables. [FONT=Courier] [FONT=Courier](A) The number of people in India [FONT=Courier](B) The time it takes to overhaul an engine [FONT=Courier](C) The blood pressures of patients admitted to a hospital in one day...

Homework Statement Let X ~ Exponential(3) and Y ~ Poisson(5). Assume X and Y are independent. Let Z = X + Y. Compute the Cov(X,Z).Homework Equations I know Cov(X, Z) = E(XZ) - E(X)E(Z). But how do I compute E(XZ) and E(Z) ?? Since for E(XZ), I would need the pdf/pmf (Exp is abs cts, while...

X and Y are independent, exponentially distributed random variables - with possibly different parameters Determine the density func. of Z = X / Y How to attack ?

Here is the question: Consider the experiment of rolling two tetrahedra that are unfair in the sense that each has the following probabilities for each of the four faces: P{1 dot}=1/10 P{2 dots}=2/10 P{3 dots}=3/10 P{4 dots}=4/10 Let X be the total of the outcomes in the two...

Homework Statement Let X,W,Y be iid with a common geometric density f_x(x)= p(1-p)^x for x nonnegative integer and p is in the interval (0,1) What is the characteristic function of A= X-2W+3Y ? Determine the family of the conditional distribution of X given X+W? Homework Equations...

Homework Statement f(x,y)= (4/5)(x+3y)exp(-x-2y) for x,y, >0 Find E[Y|X] Homework Equations E[Y|X] =integral y *f_xy (x,y)/ f_x (x) dy The Attempt at a Solution f_x (x) = integral [o,∞] [4/5](x+3y)exp(-x-2y) dx = (2x+3)/(5exp(x)) When taking the integral of y[(4/5)(x+3y)exp(-x-2y)] /...

Homework Statement let y_1 and y_2 be iid bin(5,1/4) random variables let v=y_1+2*y_2 and u = 3*y_1 -2y_2 find f_uv (u,v) and the cov(u,v) Homework Equations f_y (y) = (5 choose y) (1/4)^y (3/4)^5-x for x=0,1,2,3,4,5 covariance=E(uv)-E(u)E(v) The Attempt at a Solution...

In the last question in this link: http://pages.uoregon.edu/csinclai/teaching/Fall2009/files/hw8.pdf 1) I did not understand how they got the region for y1, y2, and y3... 2) How would the solution be different (or not possible) if X1, X2, and X3 were not iid? Thanks in advance

We have two independent random variables X and Y whose pdfs are given as f(x) and f(y). Now when you multiply X and Y you get a random variable say Z. Now what is the resulting pdf f(z)? I mean how is that related to the pdf of f(x) and f(y)? From what I read it looks like f(z)=f(x) *...

Hi, I need your help, Say we have two random variables with some joint pdf f(x,y). How would I go about finding the pdf of their sum?

Hello Let's say we have some continuous i.i.d random variables X_1, \ldots X_n from a known distribution with some parameter \theta We then place them in ascending order X_{(1)}, \ldots X_{(n)} such that X_{(i)}, < X_{(i+1)}. We call this operation T(\mathbf{X}) where \mathbf{X} is our...

Homework Statement Take Ω = [0, 1] and P the uniform probability. (a) Give an example of two random variables X and Y defined on Ω that both have the same Bernoulli distribution with parameter θ = 1/3. (b) Give an example of such random variables that are independent, and not independent...

Homework Statement The probability of being dealt a full house is approximately 0.0014. Find the probability that in 1000 hands of poker you will be dealt at least 2 full houses Homework Equations I can use binomial distribution. The Attempt at a Solution The probability of getting...

Homework Statement I'm trying to show that U(X+Y) = X in distribution, where X and Y are independent exp(λ) distributed and U is uniformly distributed on (0,1) independent of X+Y.Homework Equations The Attempt at a Solution X+Y is gamma(2,λ) distributed. But I can't figure out how to deal with...

I have seen the following "extension" of discrete random variables definition, from: pediaview.com/openpedia/Probability_distributions (Abstract) "... Equivalently to the above, a discrete random variable can be defined as a random variable whose cumulative distribution function (cdf)...

[FONT=arial narrow]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...

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...