Random variable Definition and 282 Threads

Homework Statement Given a random variable X with a known distribution (e.g. a beta distribution), find the distribution of f(X) = X^2 + X The Attempt at a Solution I've tried the normal approaches: the standard transformation theorem; conditioning on X; Laplace transformation, etc...

Hi all, Could you please answer my following question related to an exponential random variable? Thank you. Let X represent the waiting times at a telephone in an office. Assume that X is an exponential random variable with parameter λ: P(X < t) = 1 - e^{-λ*t} At each time when the...

Hello, I have a really tough time understanding this concept although this isn't anything more complex than a composition of functions. I have a done example from my book that I am trying to interpret. p_\xi (x)=\frac{1}{\pi} for an interval between [0,∏] The...

Homework Statement If Z has a standard gaussian distribution then what is the distribution of Z2 and what is its mean? The Attempt at a Solution Let T = Z2 Then we can get that pdf T = e-T/2(1+1/T) x (1/√(2∏T) I am not sure if this is correct and don't know how to find the...

Hi all I'm looking for solving this problem to find the closed form solution if it is possible: Y=\frac{1}{X} Where X is uniform random variable > 0 I know the expected value for X which is \overline{X} is there a method to find the expected value of Y which is \overline{Y} in term of...

Hey, I have a pdf of a random variable Z given. I am being asked to calculate what the moment generating function of a r.v Y= Z + c will be where c is a constant in ℝ I tried to calculate it in the following way: \int^∞_0 e^{(z+c)t} f(z+c)dz where f(z) is an exponential pdf with...

For X ~ N(μ, σ), what is P[|X-μ] < σ] in terms of the Q function? I know that P[|X-μ] < σ] can be decomposed into P[X > -σ + μ] + P[X < σ + μ] I'm not sure what to do next. i know P[X < σ + μ] can be expressed as 1 - phi(σ + μ - μ / σ) = Q(1), but I'm not sure how to approach P[X > -σ + μ]. I...

Let X and Y be independent and uniform on {1, 2, ... M} Find P(X > Y) so i know that P(X = x) = 1/M and P(Y = y) = 1/M i don't understand how Find P(X > Y) = (M+1)/2M

Homework Statement The top-selling Red and Voss tire is rated 60000 miles, which means nothing. In fact, the distance the tires can run until wear-out is a normally distributed random variable with a mean of 70000 miles and a standard deviation of 5000 miles. A: What is the probability that...

Hi. I choose randomly a one word, and I decided to choose a word blue. Let random variable x be a length of the word blue. What is expected value and variance of a word blue? So, random variable x = 4. E(X) = Ʃ xi fX(xi) i:xi∈S x1 + x2 + x3 + x4 = 10. expected value =...

The random variable X assumes the values 1,2,3 and 4 with equal probability. Find the density functions of the following variates: Attempted solutions: X 1 2 3 4 Pr(X) 1/4 1/4 1/4 1/4 a) Y=1-2X Y -1 -3 -5 -7 Pr(Y) 1/4 1/4 1/4 1/4 b) Z= X/(X+1) Z...

Let X be a discrete random variable that can assume the values -1, 0,1,2,3,4 with the probabilities 1/6, 1/12, 1/6, 1/4, 1/12, 1/4. Find the probability densities of the following random variables: a) Z= X^2 + 1 h(y)= f(g^-1(y)) Attempted Solution X= -1 0 1 2 3 4 Z=...

Ok, since nobody answered my last problem, I simplify. :) Let Z = γ1X1 + γ2X2, where the gammas are just constants p(Z) = exp(Z)/(1 + exp(Z)) X1 and X2 are bivariate normal and put Y = α + β1X1 + β2X2 + ε where ε ~ N(0,σ). Now, we want to find f(p(Z)|X1,Y). In this case, is it legal...

Data is collected on the number of fish caught per day on a month long fishing expedition. It is hypothesised that the data are consistent with a negative Binomial random variable ,X , starting at 0, so that X~Neg Bin(k,p) where E[X]=k(1-p)/p and Var =k(1-p)/p^2 . However, before a hypothesis...

Hello everyone! I am stuck in my research with a probability density function problem.. I have 'Alpha' which is a random variable from 0-180. Alpha has a uniform pdf equal to 1/180. Now, 'Phi' is a function of 'Alpha' and the relation is given by, Phi = (-0.000001274370471*Alpha^4) +...

Hello everyone, I'm having a little trouble with a probability problem with three parts; I think I'm having trouble wrapping my head around just what's going on here. If anyone could give me a starting point, I'd appreciate it. Here's the problem (Billingsley 5.1) (X a random variable)...

Let X be a continuous random variable. What value of b minimizes E(|X-b|)? Giv Homework Statement Let X be a continuous random variable. What value of b minimizes E(|X-b|)? Give the derivation The Attempt at a Solution E(|X - b|) E[e - \bar{x}] = E(X) E(|E[e - \bar{x}] - b|)...

Homework Statement The problem is to find sufficient and preferably also necessary conditions on random variable X such that its characteristic function g(x) satisfies the limit property: \lim_{t\to0}\frac{1-g(\lambda t)}{1-g(t)}=\lambda^2 I may assume X is symmetric around 0, so the...

Hi all, I am having trouble with the concept of joint pdf's. For example - a set Z1,Z2,...ZN are each gaussian rv. Let Z1~N(0,1), let X be +1 or -1 each with probability 0.5. Z2=Z1X1, so Z2 is ~N(0,1). (I assume this to be As Z2 is just Z1 multiplied by a simple factor, an instance...

Homework Statement There is a parking lot outside a building with n evenly spaced parking spaces that are numbered 1 through N with parking space #1 being closest to the entrance of the building, and parking space #n being the furthest away from the entrance to the building. A driver enters...

A random variable is defined as a function from one set, called a sample space, to another, called an observation space, both of which must be underlying sets of probability spaces. But often when people talk about a random variable - as in definitions of a particular, named distribution, such...

Homework Statement I'm wondering how I go about calculating the expectation of a random variable? Is it a different process for a discrete and a continuous? Can you show me an example? Say Poisson and expoential? Also, in the formula Var(X) = E[X^2] - (E[X])^2 how does one...

Homework Statement part iv confuse me,especially the limits for y please look to my answer for this part and comment Homework Equations The Attempt at a Solution i) I got c = 1/3 ii) P(X^2 >=1)=P(X>=1) + P(X<= -1) = 7/9 iii) P(X-1>=-1/4) = P(X>= -1/4+1)=37/576 iv) we find...

What is the PDF of the exponential of a Gaussian random variable? i.e. suppose W is a random variable drawn from a Gaussian distribution, then what is the random distribution of exp(W)? Thank you!

Homework Statement Lets say I roll 2 fair dice and take the sum of the square of each dice. What formula will be the variance? Homework Equations var(x)=e(x^2)-e(x)^2The Attempt at a Solution For dice A; E(A)=3.5 E(A^2)=91/6 ^ same for dice B. VAR(A^2+B^2)=E(A^4)-E(A^2)^2+E(B^4)-E(B^2)^2 ?

Hi Experts, I'm working in industry and have an application requiring some expert knowledge on statistics/probability. I have a probability distribution function (PDF) for a Gaussian random variable. I know the standard deviation of the PDF. I also know total number of experiments conducted...

The probability density function of the time customers arrive at a terminal (in minutes after 8:00 A.M) is f(x)= (e^(-x/10))/10 for 0 < x c) Determine the probability that: two or more customers arrive before 8:40 A.M among five that arrive at the terminal. Assume arrivals are...

Hi everyone, I'm confused about Durrett's formal definition of a random variable, as well his formal notions of probability spaces in general. I always try to make abstract definitions concrete through simple examples, but I can't wrap my head around this one: Durrett defines: X is a random...

Hi all, I have a question relating to the sum of continuous random variable probabilities that I hope you can help to answer. In any probability density function (pdf), dealing with discrete or continuous random variables, the sum of the probabilities of all possible events must equal 1...

For an exponential random variable X with rate u What is E{X|X>a} where a is a scale value from searching in internet I found that E{X|X>a}=a+E{x} but I can not prove it Help please

Homework Statement For an exponential random variable X with rate u What is E{X|X>a} where a is a scale value Homework Equations The Attempt at a Solution

Homework Statement 10 face cards are face down in a row on a table. Exactly one of them is an ace. You turn the cards over one at a time, moving from left to right. Let X be the random variable for the number of cards turned before the ace is turned over. What is the probability function...

Homework Statement Let the random variable X represent the length of the side of a square. It has a uniform distribution over the interval (0, 5). What is the cumulative distribution function for the area of the square, Y? Homework Equations F(x) = 0.2x (the cdf of the side). The...

In an analog to digital conversion and analog waveform is sampled, quantized and coded. A quantized function is a function that assigns to each sample value x a value y from a generally finite set of predetermined values. Consider the quantized defined by g(x)=[x]+1, where [x] denotes the...

Homework Statement [/b] There are two problems I need help with, which are posted below. Any help is appreciated. 1)Let X have a Poisson distribution with parameter λ. If we know that P(X = 1|X ≤ 1) = 0.8, then what is the expectation and variance of X? 2)A random variable X is a sum of...

Homework Statement Z is a random variable. P(X=a) = p1 P(X=b) = p2 P(X=c) = p3 P(X=d) = p4 Find the variance. Homework Equations Var(X) = E(X2) - E(X)2The Attempt at a Solution Okay so for the E(X2), I am currently very confused. My professor gave us this formula where E(X^2) =...

Homework Statement random variable of X has pdf: f(x) = (3/16)*x2 from interval (-2,2). Also, E(X) = 0, and E(X2) = 12/5. Find the pdf of Y = X2 Homework Equations The Attempt at a Solution I don't really know where to start.

Hi, I am interested in the product of a Gaussian random variable with itself. If X is Gaussian then what is X^2? We know that the resultant variable of the product of two independent Guassian variables is still Gaussian but I am afraid that this is not true when you multiply it with itself. Is...

X is a random variable with a continuous distribution with density f(x)=e^(-2|x|), x e R How would you calculate E(e^(ax)) for a e R? Will it be right to take a certain range of a? And also, can you take the bounds for the integral to be between -Infinity and Infinity?

Hi there, As many texts' discussion, we usually use a variable x for any value randomly picked. For a Bernoulli trials, i.e. each random variable x can either be successful or fail. If the probability of success if p and that of failure is q=1-p, then the expectation value of x would be...

Homework Statement http://s359.photobucket.com/albums/oo40/jsmith613/?action=view&current=MathQUUU.pngHomework Equations The Attempt at a Solution So I know that k = 1 But if F(>3) = 1 then why does F(3) also equal 1 Thanks

Hello, I am given a random variable X with a p.d.f. fX(x;\theta) (depending on a certain deterministic parameter \theta) and I want to consider N sampled observations of that variable: x1,...,xN. Is it correct to consider each observation as a separate random variable xi with the same pdf...

Homework Statement find the characteristic equation of a binomial variable with pmf p(x) =\frac{n!}{(n-k)!k!}*p^{k}*(1-p)^{n-k}Homework Equations characteristic equation I(t) = \sump(x)*e^{tk}The Attempt at a Solution I(t) = \sum\frac{n!}{(n-k)!k!}*(p^{k}*(1-p)^{-k}*e^{tk})*(1-p)^{n} i am...

Homework Statement From past experience, a professor knows that the test score of students taking a final examination is a random variable with mean 65. Give an upper bound on the probability that a student's test score will exceed 75. Homework Equations None that I know of. The...

Homework Statement A trial consists of tossing two dice. The result is counted as successful if the sum of the outcomes is 12. What is the probability that the number of successes in 36 such trials is greater than one? What is this probability if we approximate its value using the Poisson...

Hello, given a continuous random variable x with a known PDF, how can we determine in general the PDF of the transformed variable f(x) ? For example f(x)=x+1, of f(x)=x2 ... ? Also, if we have two random variables x,y and their PDF's, is it always impossible to determine the PDF of f(x,y)...

Homework Statement Hi, at the moment I am trying to revise for my Probability exam, and a couple of the questions on the past paper are as follows, however I can find nothing in our notes that is of any use! Any help would be greatly appreciated, thankyou. i) Two random variables X and Y...

Homework Statement Let X be an RV of the continuous type, and let Y=g(X) be defined as g(x)=1 if x>0, and =-1 if x<=0. Find the distribution of Y. Homework Equations P(Y <= y) = P(X belongs to g^(-1)(-inf, y]) The Attempt at a Solution I'm really not too sure what to do here so...

Howdy guys. Given that X is a random variable how would you prove |X| to be one too? Thanks for any suggestions!

Homework Statement Let X and Y be jointly absolutely continuous Random Variables. Suppose X~Exponential(2) and that P(Y>5|X=x)=e-3x. Compute p(Y>5).Homework Equations X~Exponential(2) means that its a exponential distribution integrated from -inf to inf, then sub lambda as 2. The Attempt at...