Random variable Definition and 268 Threads

Hello. Can you help me solve it? ($F$ is a $\sigma $ algebra). Let $X$ be a rv over $(\Omega ,F,P)$. Set $Y:= min\left \{ 1,X \right \}$. What statement is TRUE? (1): $\left \{ Y=X \right \}\neq \Omega $; (2): $F_Y(x)=F_X(x)$ for every $x\epsilon \Re $; (3): $Y\leqslant X$ for every outcome...

Hello, I have this question, which I think I solve correctly, but I am getting the wrong answer. X represent the point that the computer chooses on a scale of 2 to 5 (continuous scale) in a non-uniform way using the density: f(x)=C*(1+x) what is the probability P(3<X<4|X>1) ? I solved the...

Hello all I have this question I am trying to solve. In an urn there are 6 balls, numbered: 1,2,3,4,5,6. We take 4 balls outs, without replacement. X - the minimal number we see Y - the maximal number we see I need to joint distribution. I understand that X is getting the values 1,2,3 while...

"Let X be a Bernoulli random variable. That is, P(X = 1) = p and P(X = 0) = 1 − p. Then E(X) = 1 × p + 0 × (1 − p) = p. Why does this definition make sense? By the law of large numbers, in n independent Bernoulli trials where n is very large, the fraction of 1’s is very close to p, and the...

Let ##X_i## are i.i.d. and take -1 and +1 with probability 1/2 each. How to prove ##\lim_{n\rightarrow\infty}{\sum_{i=1}^{n}{X_i} }##does not exsits (even infinite limit) almost surely. My work: I use cauchy sequence to prove it does not converge to a real number. But I do not how to prove it...

Homework Statement Suppose the distance X between a point target and a shot aimed at the point in a coin-operated target game is a continuous random variable with pdf f(x) = { k(1−x^2), −1≤x≤1 0, otherwise. (a) Find the value of k. (b) Find the cdf of X. (c) Compute P (−.5...

Homework Statement X ~ Uniform (0,1) Y = e-X Find FY (y) - or the CDF Find fY(y) - or the PDF Find E[Y] 2. Homework Equations E[Y] = E[e-X] = ∫0 , 1 e-xfx(x)dx FY(y) = P(Y < y) fY(y) = F'Y(y) The Attempt at a Solution FX(x) = { 0 for x<0 x for 0<x<1 1 for 1<x } fX(x) = { 1 for...

Here is a question about probability density. I am trying to work it out using a different method from the method on the textbook. But I get a different answer unfortunately. Can anyone help me out? Question: Let X be uniformly distributed random variable in the internal [ 0, 1]. Find the...

Given a Gaussian process X(t), identify which of the following , if any, are gaussian processes. (a)X(2t) solution said that X(2t) is not gaussian process, since and similarly Given Poisson process X(t) (a) X(2t) soultion said that X(2t) is not poisson process, since same reason above...

Homework Statement Random variable x is defined on interval (1,3) and it has probability mass function f(x) =A(x2 +1= a) Find PMF, g(y) for y=x2 b)Expectation of y c)Variance of y d)Distribution function of y. e)most probable value of y The Attempt at a Solution As far as a), i integrated from...

Hello, want to know if it's correct 1. Homework Statement X and Y two random variables iid of common density f and f(x)=x*exp(-x²/2) if x≥0 and f(x)=0 if x≤0 and Z=min(X,Y) Find -The density of Z -The density of Z² - E[Z²] Homework EquationsThe Attempt at a Solution 1.[/B] FZ(u) = P(min(X,Y...

Given the PDF: f(x) = 1/12 , 0 < x <= 3 x/18, 3 < x <= 6 0, otherwise find the expected value, E(x). I know how to find the expected value if there was only one interval, but don't how to do it for two.

Homework Statement Let X be a random variable with the following probability distribution X 0 1 2 3 4 f(x) 1/16 1/4 3/8 1/4 1/16 If another random variable Y = X^2 + 1 is formed, find the mean E[Y]. 2. Relevant equation...

I am trying to use a generated random sample in R to estimate the mean and variance for a Poisson random variable. The first one is a Poisson random variable with mean 5. To estimate the above I generate a random sample in R with the following code: P5 <- rpois(100,5) Given the above I want to...

Homework Statement Let X denote a continuous random variable with probability density function f(x) = kx3/15 for 1≤X≤2. Determine the value of the constant k. Homework Equations I'm not sure if this is right but I think ∫kx3/15 dx=1 with the parameters being between 2 and 1, The Attempt at a...

Homework Statement If X is uniformly distributed over (0,1), find the PDF of Y = |X| and Z = e^X Focusing on the |X| one Homework Equations Derivative of CDF is the PDF The Attempt at a Solution So I start by writing down the CDF of X, Fx(x): 0 for x <0 x for 0 ≤ x ≤ 1 1 for x ≥ 1 And I...

Homework Statement Here's the problem with the solution provided: Homework Equations Fundamental Theorem of Calculus (FToC) The Attempt at a Solution So I understand everything up to where I need to take the derivative of the integral(s). Couple of things I know is that the derivative of...

We have a r.v. X with p.d.f. = sqrt(θ/πx)*exp(-xθ) , x>0 and θ a positive parameter. We are required to show that 2 θX has a x^2 distribution with 1 d.f. and deduce that, if x_1,……,x_n are independent r.v. with this p.d.f., then 2θ∑_(i=1)^n▒x_ι has a chi-squared distribution with n...

If one has a Bernoulli random variable W that is derived from a Variable T (Poisson λ), by the following rules W = (if T=0 then W=1 and if T>0 then W=0), I am having trouble finding the pf for W. Any suggestions about how to proceed forward?

With a Poission random variable, we know that \(E[X] = var(X) = \lambda\). By definition of the variance, we can the second moment to be \[ var(x) = E[X^2] - E^2[X]\Rightarrow E[X^2] = var(X) + E^2[X] = \lambda(1 + \lambda). \] The characteristic equation for the Poisson distribution is...

The discrete random variable K has the following PMF: p(k) = { 1/6 if k=0 2/6 if k=1 3/6 if k=2 0 otherwise } Let Y = 1/(1+K), find the PMF of Y My attempt: So, I am really confused about what this is asking. I took...

How do I estimate the pdf from a random variable \(X\) where \(X = U_1 - U_2\) and \(U_i\) are uniform random variables? In the code below, I used unifrnd(-5, 5, 1000, 1) which generated a 1000x1 vector of uniform random number between -5 and 5. How do I estimate the PDF for X? rng; X =...

Homework Statement please refer to the question, i can't figure out which part i did wrongly. i 'd been looking at this repeatedly , yet i can't find my mistake. thanks for the help! the correct ans is below the question. where the c= 283/5700 , q = 179/5700 Homework Equations The...

Homework Statement |zi - 3| = Pi Homework Equations Well, it clearly has to do with a circle but I do not believe there is a general equation for what I am asking about. The Attempt at a Solution There is no general solution not trying to solve anything. I want to know exactly...

Homework Statement Let ##Y_1,...Y_n## be independent standard normal random variables. What is the distribution of ##\displaystyle\sum_{i=1}^n{Y_i}^2## ? Let ##W_n=\displaystyle\frac{1}{n}\sum_{i=1}^n {Y_i}^2##. Does ##W_n\xrightarrow{p}c## for some constant ##c##? If so, what is the...

How to transform a random variable CDF to a standard normal Given F(x) = 1- exp (-sqrt x), for x greater that 0 Thanks.

Homework Statement If the probability density function(p.d.f.) of a random variable X is f(x) = 1/6 * e-|x|/3 where x is lying in (-∞,∞) and |-x| = x if x≥0, then what is the p.d.f. of the random variable Z = XY = X*|X| where Y = |X| ? Homework Equations Nothing special. The Attempt at a...

Homework Statement Find the conditional distribution function and density for the random variable X defined on R given that X is in some interval I = (a,b) where P(X in I) > 0. Assume that the density and distribution for the random variable X is known Homework Equations fX|X\inI =...

Hello I am trying to solve this problem: A coin is given with probability 1/3 for head (H) and 2/3 for tail (T). The coin is being drawn N times, where N is a Poisson random variable with E(N)=1. The drawing of the coin and N are independent. Let X be the number of heads (H) in the N draws...

Homework Statement A couple is expecting the arrival of a new boy. They are deciding on a name from the list S = { Steve, Stanley, Joseph, Elija }. Let X(ω) = first letter in name. Find Pr(X = S). Homework Equations The Attempt at a Solution Ok the answer is 2/3. How is it 2/3...

Assume X is a random variable under a probability space in which the sample space ?= {a,b,c,d,e}. Then if I am told that: X({a}) = 1 X({b}) = 2 X({c}) = 3 X({d}) = 4 X({e}) = 5 And that: P({a}) = P({c}) = P({e}) = 1/10 P({b}) = P({d}) = 7/20 Find the C.D.F of X, the density of X...

Hi, I have a quick question. Let R and S be two independent exponentially distributed random variables with rates λ and μ. How would I compute P{S < t < S + R}? I am a little bit confused because of the variables on either side of the inequalities. I have tried conditioning on both S and R...

Homework Statement Let X_1, X_2 have the joint pdf h(x_1, x_2) = 8x_1x_2, 0<x_1<x_2<1 , zero elsewhere. Find the joint pdf of Y_1=X_1/X_2 and Y_2=X_2. Homework Equations p_Y(y_1,y_2)=p_X[w_1(y_1,y_2),w_2(y_1,y_2)] where w_i is the inverse of y_1=u_1(x_1,x_2) The Attempt at a Solution We can...

Hi, I'm having a bit of a problem with a probability question. The question is Let X be a normal random variable with mean \mu and variance \sigma^{2}. Find E[(X -\mu)^{k}] for all k = 1,2,... I'm not really sure what to do and need some help to confirm how to approach the question...

We selected X point from interval (-1,2). If X=x, we selected point Y from (-1,x^2). Define the function of density of the random variable Y.

Homework Statement What's the expected value of this problem (random variable)? X: represent the result of dice number 1 - result of dice number 2 example dice 1 first roll = 2; second roll = 3 dice 2 first roll = 1; second roll = 2 X = 2+3 -(1+2) = 2 what's the expected value...

Hi, I'm an economics graduate student doing some work on a nested logit model. I am trying to generate random variables that follow the following CDF: F(x_1, x_2) =\textrm{exp}[ -(e^{-2x_1}+e^{-2x_2}) ^{1/2}] (This is an extreme-value distribution) With a single random variable, I...

Homework Statement A man wants to travel to four cities (A,B,C,D) but he has such a bad memory that he can't remember the cities that visited, therefore, if he travel to city A he can choose between (B,C,D) and if he then travel to B he can choose between (A,C,D). Find v, If v it's the...

If X is a continuous random variable and g is a continuous function defined on X (Ω), then Y = g(X ) is a continuous random variable. Prove or disprove it.

If X is a random variable distributed uniformly in [0, Y], where Y is geometric with mean alpha. i) Is this definition valid for uniform distribution ? ii) If it is valid, what is the pdf of the transformation Y-X?

I'm having a bit of a problem proving the second condition for a martingale, the discrete time branching process Z(n)=X(n)/m^n, where m is the mean number of offspring per individual and X(n) is the size of the nth generation. I have E[z(n)]=E[x(n)]/m^n=m^n/m^n (from definition E[X^n]=m^n) =...

Problem: Suppose that $X \text{ ~ Exp}(\lambda)$ and denote its distribution function by $F$. What is the distribution of $Y=F(X)$? My attempt: First off, I'm assuming this is asking for the CDF of $Y$. Sometimes it's not clear what terminology refers to the PDF or the CDF for me. $P[Y \le y]=...

This is something that when I see the work done it makes sense, but I find it difficult to do myself. I'm also aware there is an explicit formula for doing this but that involves Jacobians and a well-defined inverse, so I think it's more intuitive to do it step-by-step. Problem: Suppose $X...

Professor Roberto has to take an oral examination. The grading scale is as follows: 5: = best and 1: = worst. At most he only gives the note 4. Each student under review is questioned if he is a Lakers fan. The student's grade is based on his answer (is a fan / not a fan) and on the language in...

Homework Statement X is uniformly distributed over [-1,1]. Compute the density function f(y) of Y = 2X2 + 1. Homework Equations The Attempt at a Solution FY(Y) = P(Y < y) = P(2X2 + 1 < y) = P(X < +\sqrt{1/2(y-1)} = FX(+\sqrt{1/2(y-1)}) We have that f(x) = 0.5 for -1 < x <...

Homework Statement Give a method for generating a random variable with distribution function F(x) = 1/2(x+x^{2}) 0<x<1 The Attempt at a Solution From what i can tell i am supposed to do something like: Let U be a uniformly distributed random variable over (0,1). U =...

The cumulative distribution function of a continuous random variable is given as follows: 0 0 ( ) 0 5 5 1 5 X if x x F x if x x           a. Determine and name the density function of . [02] b. Use both and ( ) X F x to find P(X  3) . [05] c. Find the variance of ...

f(x)=1, θ-1/2 ≤ x ≤ θ+1/2 Given that Z=(b-a)(x-θ)+(1/2)(a+b) how would you show that Z has a continuous uniform distribution over the interval (a,b)? Any help would be much appreciated.

Homework Statement X is a normal random variable with mean 1, variance 4. 1. Find P( X(X-1) > 2 ) 2. Find a value 'a' for which P(|X| > a ) = .25 The Attempt at a Solution I had no idea how to start 1. For 2, i got this far then got stuck: P(|X| > a) = 1 - P((X-1)/2 <=...

First of, I apologize for the vague title, I didn't know how to summarize this issue. Homework Statement Suppose that the interest rate obtained in month i is a random variable Ri with the uniform distribution on [0.01, 0.03], where R1,R2, . . . are independent. A capital of 1 unit...