Questions about maps from a probability space to a measure space which are measurable.

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1answer
42 views

If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then $X_n\to\delta_0$ in distribution

If $P(a_n\leq X_n\leq b_n)\to1$ and $a_n\to0,b_n\to0$ then prove that $X_n\to\delta_0$ in distribution. Here $\delta_0$ is the degenerate random variable putting all its mass at the point $0$. I ...
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2answers
50 views

time it takes to service a car with exponential random variable with rate 1

Need help with this question here. Ill post exactly what it says then show my ideas so far. "The time it takes to service a car is an exponential random variable with rate 1. (a) If A brings his car ...
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1answer
25 views

Given the probability density of random variable $X$, what is the density of $Y=aX+b$?

I have a random variable $X$ with probability density $f_X$ and want to determine the probability density $f_Y$ of $Y=aX+b$ with $a,b \in \Bbb R$. How do I proceed here?
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1answer
41 views

Finding what distribution a random variable has.

Mike has a gold coin with fair probability, and a silver coin with $1\over 3$ probability for Heads and $2\over 3$ for tails. He tosses the gold coin 120 times and the number of heads is denoted N. ...
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2answers
50 views

Comparing two exponential random variables

Let $A$ and $B$ be independent random variables drawn from the exponential distribution with parameters $\lambda_A<\lambda_B$. What is the probability that $A<B$? I'm of course aware of the ...
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1answer
29 views

Confusion With “Nested” Random Variables

Problem The probability that a compnay's workforce has no accidents in a given month is $0.7$. The numbers of accidents from month to month are independent. What is the probability that the third ...
2
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1answer
27 views

Expectation of cumulative distribution function of a standard normal distributed random variable

Let $X$ be a normally distributed random variable with mean $0$ and variance $1$. Let $\Phi$ be the cumulative distribution function of the variable $X$. The find the expectation of $\Phi(X)$. I ...
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1answer
37 views

Density probability function and distribution of $Y=cosX$

I have the following problem. Let X be a random variable uniformly distributed on $[-1,1]$ and let be $Y=cos(X)$. a) Find the function of density and distribution of Y b) Find the espectation of Y ...
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0answers
9 views

Let $X \sim N(0,\sigma^2)$ and $Y$ a positive r.v. Possible to determine sign of $Cov(X,Y)$?

The problem is that $X$ and $Y$ are dependent r.v.s and are dependent via some complicated stochastic differential equation. I'm wondering if it's possible to determine if $Cov(X,Y) \geq 0 $ or ...
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1answer
29 views

Expectation of quotient of linear combinations of independent standard normal random variables

Let $a, b, c, d, e, f$ be complex numbers with nonnegative real parts and nonnegative imaginary parts, and let $X_{1}, X_{2}, X_{3}, X_{4}$ be independent standard normal random variables. How can I ...
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1answer
19 views

Are RV having same exp. value and covariance already have the same distribution?

Let $(X_1, ..., X_n), (Y_1, ... , Y_n)$ be random variables. $X_i$ has the same distribution as $Y_i$ for all $i$. $\forall i, j: Cov(X_i, X_j) = Cov(Y_i, Y_j)$ Do $(X_1, ..., X_n)$ and $(Y_1, .., ...
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1answer
33 views

Cross-correlation of identical sets: not getting expected result

I'm trying to work out the correlation coefficient of two sets using a given formula, but I'm not getting a perfect correlation when using identical sets. The correlation between a client’s ...
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2answers
64 views

The probability density of $X^2$?

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? ...
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2answers
32 views

Probability: Combinatorics and discrete random variables

I've managed part (i) fine but have no idea how to approach part (ii). I tried using Baye's theorem in order to calculate the conditional probability that the red team has size k given that it ...
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2answers
64 views

How big is the chance that a arbitrary man is taller than a arbitrary woman?

I'm a first year mathematics student, and I'm having trouble with computing the following: Assume that in a country the height $X$ of men is normally distributed, with $\mu_X = 180$ (the expected ...
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0answers
23 views

Does small perturbation in the denominator explode the expectation of a ratio of two random variables?

The puzzling thing I am facing is Suppose we have two random variables $X$ and $R$ such that $E(X^{-1}R)=1$. Now let $\tilde{X}=X+\mathcal{E}$ where $\mathcal{E}=X\epsilon$ and $\epsilon \sim ...
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2answers
93 views

How to prove it is a strictly stationary process?

$ξ(t) = z*sin(ωt + θ)$ where $z$ is a random variable and its distribution is unknown and $θ$ is another random variable that is independent of $z$ and $θ$ is uniformly distributed on $(0, 2\pi)$. ...
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0answers
33 views

What is expected value of three variables?

I have an individual $x=(x_1x_2x_3x_4)$ (for example $x_1=1111$). Now, given a set (4 individuals) of individuals such as ...
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0answers
37 views

List of random variables whose probability density function satisfies a certain inequality

For a continuous random variable $X$ with a probability density function (pdf) of $f(\cdot)$, consider the following inequality \begin{equation} \frac{f(x+y)}{f(x)}\ge \frac{f(x+y+z)}{f(x+z)}, ...
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1answer
22 views

Integrating Joint Random Variable Distributions i don't understand how to take the Integration Intervals

Have this problem, two random variables $X \sim Un(0,1)$ and $Y\sim Un(0,1)$. Need to find the distribution function of $Z= \frac{(X)}{(X+Y)}$, i have the solution as well, but i don't understand ...
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0answers
26 views

Notation: how can I say variable drawn from a distribution D lies in space X

Suppose I have a random distribution $D$ for which, if $x\sim D$, then $x\in X$. Is there a standard notation involving only $D$ and $X$? For example, let $N$ be the multinormal distribution with ...
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1answer
31 views

Direction on how to rightly compute a certain expected value.

Dan has 13 different cards, one of which is numbered with a unique number between 1-13. Every day, he pulls a specific card (which he never returns) with a uniform distribution between the cards left. ...
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0answers
25 views

Convergence in probability of function composition.

I need to show that $G_n \stackrel{P}{\to}_n F_0$, i.e. for any $\epsilon>0$ $$ P(|| G_n - F_0||>\epsilon) \to_n 0 $$ We know the following: $G_n$ and $F_0$ are a bilinear functions from ...
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1answer
29 views

Derivation of Likelihood Function for Random Effects Parameters

I initially posted this question in CV, but getting no responses or interest, I am deleting it there, and trying my luck in math.stackexchange, hoping that the math details of the following derivation ...
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1answer
44 views

Continuous random variable pdf question.

The continuous random variable X has pdf where $$f(x) = \begin{cases} \frac{25}{12(x+1)^2},\quad & 0\le x\le 4 \\ 0 & \text{otherwise} \end{cases} $$ $E(X+1) = 1\frac23$ and $E(X) = 2/3$ ...
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0answers
39 views

expectation value of independent random variables

In the statistics lecture that I'm attending, the professor once used the following: $X, Y$ random variables and i.i.d., then $$\mathbb{E}(XY) = \mathbb{E}(X)\mathbb{E}(Y)$$ I was trying to see an ...
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1answer
29 views

Exponential(1) distributed random variable convergence

I am stuck with convergency in probability... I have the following exercise: Let $(X_k)_{k\ge1}$ be a sequence of independent exponential-(1) distributed random variables. Show that $n^\alpha ...
0
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2answers
31 views

Find parameters so that random variables (connected to Brownian movement) are independent.

$W_t\sim\mathcal{N}(0,t)$ is Brownian movement, find values of parameters $a, b$ for which $aW_1-W_2$ and $W_3+bW_5$ are independent. I don't even know where to start, so any hint is highly ...
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1answer
17 views

Find the distribution of some random variable connected to Wiener Process. Please, check my solution.

I need to find a distribution of $ 5W_1-W_3+W_7 $, where $W_t$ stands for Wiener Process $W_t\sim\mathcal{N}(0,t)$. Is this solution right? $E(5W_1-W_3+W_7)=5E(W_1)-E(W_3)+E(W_7)=0$ and since ...
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2answers
41 views

Random variables with common mean, variance and pairwise correlation

Hi I'm currently working through past exam questions and am stuck with the following question: Random variables $X_1$, $X_2$ and $X_3$ are identically distributed, with common mean $\mu$, common ...
2
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3answers
80 views

Does $\operatorname{Cov}(X,Y) = 0$ mean $\operatorname{Cov}(X,\log Y) = 0$?

Suppose $X,Y$ are positive random variables with $\operatorname{Cov}(X,Y)=0$. Define $Z= \log Y$. Does it necessarily follow that $\operatorname{Cov}(X,Z) = 0$? I know it's true for linear ...
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2answers
45 views

Find pmf for $i=0,1,2,3,4$

I have a problem that I'm having trouble with. Here is the problem: "Five distinct numbers are randomly distributed to players numbered 1 through 5. Whenever two players compare their numbers, the ...
2
votes
4answers
88 views

What is the statistical equilibrium for this simulation of happy bubbles?

Happy Bubbles I hope this is not too specific or practical, but I just made a simulation of sorts and seem to have hit quite close to an equilibrium (by accident). Now I am wondering if and how you ...
1
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4answers
87 views

Probability formula, a value chosen at random is greater than another chosen value.

Say I have two number ranges, whole numbers only. Range 1: [-3,16] Range 2: [3,22] I choose randomly one number from Range 1 and one number from Range 2. Lets call them x and y. How do I find the ...
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0answers
28 views

computing the area of a region using Monte Carlo integration

Suppose that I am interested in estimating the area of $\Gamma \in \mathbb{R}^2$. I do not know the exact shape of $\Gamma$ but I have a sufficiently large number of sample points $(X,Y) \in \Gamma$ ...
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2answers
63 views

A random variable $X$ has support in $[a,b]$, and $\mathbb{E}X=b$. Prove that $P(X\geq\mathbb{E}X)=1$

Let $X$ be a discrete random variable such that $R(X) \subseteq I=[a,b]$, $-\infty<a<b<\infty$. Further let $\mathbb{P}$ be a probability measure. Is it possible to write for $x\in I$: ...
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1answer
42 views

Calculating probability distribution under given constraints

I recently asked a question about the construction of a random variable under given constraints (see: Construct a random variable under given constraints). The only answer to my question suggested a ...
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0answers
20 views

Can I have the following random matrix?

Let $A \in \mathbb{R}^{n \times n}$ a random matrix. Assume $E[A] = I$ and that all $A_{ij}$ are independent. Now let $U \Sigma V^{\top} = A$ be the SVD of A. Let $A'$ be the result of thin-SVD, ...
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0answers
16 views

Reference on the necessity proof of Kolmogorov's three series theorem wihout using central limit theorem.

I want a reference on the necessity proof of Kolmogorov's three series theorem wihout using central limit theorem. I understand some intuition behind taking independent copies of the random ...
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0answers
8 views

Propagating uncertainties in Gaussian fit

I'm doing an analysis where I have a set of random variables with some known uncertainties (the uncertainties are different for each random variable). The random variable is roughly Gaussian ...
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2answers
53 views

Construct a random variable under given constraints

In preparation for a probability examination, I am working on the following problem. Problem A box contains three white balls and ten black balls. Balls are drawn without replacement until all the ...
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1answer
14 views

Proving a.s. convergence by probabilistic convergence

Consider a sequence of random variable $\{X_n\}$. Let $$A_n = \sup\{|X_k - X_l|: k,l \geq n\}$$ $$B_n = \sup\{|X_k - X_n|: k \geq n\}$$ Now to prove a.s. convergence of $\{X_n\}$, I have seen in a ...
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1answer
30 views

Dominated convergence martingale and uniform integrability

For a fixed $t\in [0,1]$ I have a sequence $(X^t_n)_{n\geq 1}$ of normal distributed random variables which is a martingale and bounded in $L^2$. So by the martingale convergence theorem there exists ...
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1answer
22 views

Question about uncorrelatedness of random variables and distributions

I was wondering, if two random variables are dependent, does that mean that they must be correlated? does one imply on the other or vice versa? Also, if I know that a joint distribution of two ...
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0answers
20 views

Covariance matrix is positive definite, does the variable have density?

I know that if a random $n$-dimensional variable $Y= [Y_, ..., Y_n]^T$ has density $f$, then its covariance matrix $\Sigma $ is positive definite: $\forall x \in \mathbb{R}^n \setminus \{0\}: x^T ...
3
votes
2answers
67 views

Expected value of area of triangle

Here is the problem: Let $A$ be the point with coordinates $(1, 0)$ in $\mathbb R ^2$. Another point $B$ is chosen randomly over the unit circle. What is then the expected value of the area of the ...
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1answer
34 views

Deriving mass/density functions of variables $log(X)$, $X+Y$, $sgn(X-1/2)$.

Could you help me with the following question? Suppose that a point with co-ordinates $(X,Y)$ is chosen uniformly from the square $\{(x,y)\in \mathbb{R}^2: 0 \leq x,y \leq 1\}$. For each of the ...
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1answer
40 views

I.i.d. Discrete Random variables with mean 0 and variance 1 [closed]

Given a sequence of i.i.d. random variables $(X_i)_{i\geq 1}$ such that $\mathbb{E}(X)=0,\ \mathbb{E}(X^2)=1$ consider the sum $S_n=\sum_{k=1}^n{X_k}$. Is it true that independently of the ...
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0answers
80 views

Probability inequality given $E(X^2) = 1$ [closed]

This is Exercise 20 on page 198 of Resnick's "A Probability Path". I'm stumped; hints or full solutions appreciated. Suppose $E(X^2) = 1$ and $E(|X|) \geq a > 0$. Prove for $0 \leq \lambda \leq 1$ ...
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1answer
16 views

Variational series , probability.

A Variation series is a series of random variables $Y_1,Y_2...Y_k$ where $k$ represents the $k-th$ largest random variable between $X_1, X_2...$, $X_i$ (independent of eachother , equally ...