Random variables Definition and 318 Threads

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 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 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...

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...

Homework Statement D = (L + E) / S Where L, E, and S are mutually independent random variables that are each normally distributed. I need to find (symbolically), the conditional PDF f(d|s). Homework Equations The Attempt at a Solution Not sure what to do with so many...

I am stuck on the following problem: Five items are to be sampled from a large lot of samples. The inspector doesn't know that three of the five sampled items are defective. They will be tested in randomly selected order until a defective item is found, at which point the entire lot is...

Hi, I have the following question : How do we estimate the joint probability Pr(X_1, ... X_n) when n \rightarrow \infty ? Thanks a lot.

Homework Statement You have 7 apples whose weight (in gram) is independent of each other and normally distributed, N(\mu= 150, \sigma2 = 202). You also have a cabbage whose weight is independent of the apples and N(1000, 502) What is the probability that the seven apples will weigh more...

Homework Statement Let X and Y be independent random variables. Prove that g(X) and h(Y) are also independent where g and h are functions. Homework Equations I did some research and somehow stumbled upon how E(XY) = E(X)E(Y) is important in the proof. f(x,y) = f(x)f(y) F(x,y) =...

If the set of real numbers is considered as a sample space with the Borel sigma algebra for its events, and also as an observation space with the same sigma algebra, is a pdf or pmf a kind of random variable? That is, are they measurable functions?

Hoel: An Introduction to Mathematical Statistics introduces the following formulas for expectation, where the density is zero outside of the interval [a,b]. E\left [ X \right ] = \int_{a}^{b} x f(x) \; dx E\left [ g(X) \right ] = \int_{a}^{b} g(x) f(x) \; dx He says, "Let the random...

I'm wondering how conditional probability relates to concepts of sample space, observation space, random variable, etc. Using the notation introduced in the OP here, how would one define the standard notation for conditional probability "P(B|A)" where A and B are both subsets of some sample...

Hi, Why there is a half factor in the definition of the correlation of complex random variables, like: \phi_{zz}(\tau)=\frac{1}{2}\mathbf{E}\left[z^*(t+\tau)z(t)\right]? Thanks in advance

I'm studying for the probability actuarial exam and I came across a problem involving transformations of random variable and use of the Jacobian determinant to find the density of transformed random variable, and I was confused about the general method of finding these new densities. I know the...

Definition: Let X1,X2,... be a sequence of random variables defined on a sample space S. We say that Xn converges to a random variable X in probability if for each ε>0, P(|Xn-X|≥ε)->0 as n->∞. ==================================== Now I don't really understand the meaning of |Xn-X| used in...

In a recent federal appeals court case, a special 11-judge panel sat to decide on a certain particular legal issue under certain particular facts. Of the 11 judges, 3 were appointed by political party A, and 8 were appointed by political party B. Of the party-A judges, 2 of 3 sided with the...

I'm having a trouble doing this kind of problems :S Lets try this for example: The joint p.d.f of the continuous random variable X and Y is: f(x,y)= (2y+x)/8 for 0<x<2 ; 1<y<2 now we're asked to find a probability, say P(X+Y<2) I know i have to double integrate but how do I choose my...

Hi, Suppose X_1, X_2,\cdots be an independent and identically distributed sequence of exponentially distributed random variables with parameter 1. Now Let N_n:=\#\{1\leq k\leq n:X_k\geq \log(n)\} I was told that N_n\xrightarrow{\mathcal{D}}Y where Y\sim Poisson(1). Could anyone give...