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

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45 views

What's wrong with this random variable proof?

Let $X$ be a Binomial random variable $\sim B(p, n)$. Show that for $\lambda > 0$ and $\epsilon > 0$, $P(X - np > n\epsilon) \le \mathbb{E}\{\displaystyle e^{\lambda(X - np - ...
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2answers
71 views

If the sum of two independent random variables is in $L^2$, is it true that both of them are in $L^1$?

Let $X$ and $Y$ be two independent random variables. If $\mathbb E(X+Y)^2 < \infty$, do we have $\mathbb E |X| < \infty$ and $\mathbb E |Y| < \infty$? What I actually want is that $X$ and ...
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1answer
40 views

One problem on random vectors

$e_i$'s are $n$ - dimensional random vectors and any two different random vectors are uncorelated . I need to prove $$E[||\sum_{k=0}^{\infty} a_k e_k||^2] = E[\sum_{k=0}^{\infty} a_k^2 ||e_k||^2]$$
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1answer
25 views

Expectation of the square of the minimum of iid positive random variables

Let $X_1, X_2$ be i.i.d., positive random variables with $E[X_i] < \infty$ (but $E[X_i^2]$ might be $\infty$). $Y := \min \lbrace X_1, X_2 \rbrace$. I want to show that $E[Y^2] < \infty$. The ...
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2answers
40 views

Convergence of random variables in metric spaces

Let $S$ be a metric space equipped with a distance function $d$, and let $X_n,Y_n$ be sequences of random variables having values from $S$. Suppose that $X_n$ converges in distribution to some random ...
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0answers
25 views

Probability of divergence of a sum of random variables with constant positive expectation

I've encountered the following question: suppose $X_n$ is a sequence of positive random variables such that $\mathbb{E}(X_n)=1$ for all $n$. Does it follow that $\sum X_n$ diverges almost surely? ...
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1answer
31 views

Cauchy random values in a interval [a, b]

How do I generate random numbers following a Cauchy distribution in a given interval [a, b]. I tried using explained here Trucated distribution, but did not succeed
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1answer
56 views

Weak convergence: equivalence of definitions

Consider a sequence of random variables $(X_n)_{n\geq 0}$ and a random variable $X$. How to prove that the two following definitions of weak convergence are equivalent? Def 1 $(X_n)_{n\geq 0} ...
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1answer
47 views

Generate exponential random values in a given range [duplicate]

Need to generate random values ​​that follow an exponential distribution on an interval [a, b​​]. I tried using explained here Trucated distribution, but did not succeed
1
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1answer
34 views

Exponential Distribution, Statistics.

I know X has an exponential distribution with parameter $\theta =2$. I was asked to define $Y=lnx$ and determine the suppose of Y and the pdf for Y. Then let $X_1, X_2$ be two independent ...
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1answer
17 views

Covariance of two variables hat check

I can't do this, I give up. I am just not able to do it. I don't know what is wrong with me but I can't do it and I need help. Hat check experiment with 3 hats, outcomes 1,2,3|1,3,2|2,1,3|3,2,1 have ...
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2answers
42 views

Covariance of two random variables

I am trying to find the covariance of two random variables but I am not having any lucky. Just for simplicity lets say that my random variables are : X = value rolled by a die Y = 1 if even 0 if odd ...
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1answer
46 views

Generate random numbers following the exponential distribution in a given interval $[a, b]$

I know that to genarete ramdom variables following exponential distribution just do: $$X=-\frac{1}{\lambda}ln(U)$$ where $U\sim U(0,1)$ Now, to find a distribution restricted to the interval $(a, ...
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2answers
58 views

Operations on distributions

Say we have two r.v X and Y which are independent and differently distributed ( for e.g X follows a bell curve and Y follows an exponential distribution with parameter $\lambda > 0$ What are the ...
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0answers
19 views

Probability distribution of location of maximum of random process

I have the following problem: Given a complex function $H(x)$ at positions $x_1, x_2, x_3,..., x_n$ The function values at each position are independent random circularly Gaussian variables, this ...
2
votes
1answer
105 views

Can sum of two random variables be uniformly distributed

Say $X$ and $Y$ are two random variables where $X\in\{-\alpha,\alpha\}$, $Y\in\{-\alpha,\alpha\}$ and $Z=X+Y$. Is it possible to find two independent random variables with certain pdf (not necessarily ...
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1answer
34 views

Probability Density Function of non decreasing function

Can anyone please help me with this random variable question I've stumbled across. Recall from calculus that a function $h$ is called non-decreasing if $x\leq y$ implies $h(x)\leq h(y)$, for every ...
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1answer
23 views

Density functions of normal random variables.

$Z$ is a standard normal random variable. Find the density of $Y=\frac{Z^2}{4}$ I know the cumulative distribution function is $2*P$ ($0 <= Z <= 2root (y)$) And that this equals $2*phi(2root ...
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1answer
38 views

If speeds of two cars are Normal RV s, what is the distribution of the distance between them?

The speeds of two cars are random variables that follow $N(\mu_1,\sigma_1)$ and $N(\mu_2,\sigma_2)$ distributions.They start simultaneously. Let X be the distance between them after m hours. (Note ...
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0answers
27 views

Algebra of random variables

I am working on a project involving the implementation of a tool capable of numerically performing some basic algebraic operations (sum,product,inverse..) on independent random variables of different ...
2
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1answer
46 views

Fixed points in random permutation

Suppose two random permutations of the numbers 1 to n placed side by side. a) Calculate the expectation number of fixed points for $n = 5$. b) Find the value of expectation in the amount of fixed ...
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2answers
54 views

discrete random variable with expected value of $\pm\infty$?

I am currently studying discrete random variables and I was wondering if there is any probability space with two indipendent discrete random variables $X$ and $Y$ with expected values ...
2
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1answer
43 views

How to compute $P(X=1,Y=1,Z=-1)$

If $X$ and $Y$ are independent random variables, which take only the values $-1$ and $+1$, and $P(X=1)=a$, $P(Y=1)=b$ and a third random variable $Z$ is defined by $Z=\cos((X+Y)\frac{\pi}{2})$. Can ...
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1answer
22 views

Two Uniform Independent Random Variables: When is one greater?

You have two independent random variables: $X$ and $Y$, which are both uniformly distributed over $(0,1)$. Consider the inequality $X^2- 4Y < 0$. What percentage of the time is the inequality ...
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0answers
33 views

Proof also valid for discrete random variable?

In Theorem 2.2. in this paper you can find a proof of the one-sided Chebyshev inequality $$Pr[X \geq \mu +a ] \leq \frac{\sigma^2}{\sigma^2+a^2}$$ for a random variable $X$ with mean $\mu$ and ...
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1answer
52 views

Computing P(X+Y>0) for the joint pdf of X and Y.

Let X and Y be two jointly continuous random variables with the given joint PDF; $\begin{equation} \nonumber f_{XY}(x,y) = \left\{ \begin{array}{l l} \frac{1}{3}x^3+\frac{1}{5}y^2 & ...
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1answer
21 views

Variance of function of random variable - Probability

I've got a probability exam tonight and I'm just curious about an answer from a practice exam. Any tips/help would be much appreciated! Here's what I've got: X is a random variable Y = 4X + 1, is ...
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0answers
10 views

Maximum correlation of n variables

For $n>2$ variables, one cannot arbitrarily choose the correlations $\rho_{ij}$ because the resultant correlations must obey the law of cosines. Equivalently, the covariance matrix between them ...
5
votes
2answers
74 views

Expectation with square root

I don't know how to calculate the expectation when there is some square root in the expression. My problem is this: we have three real random variables $X,Y,Z$, independent and with standard normal ...
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1answer
50 views

example of cumulative distribution function being discontinuous in $\mathbb Q$

I am currently studying random variables. I know random variables whose cumulative distribution functions are continuous. But I was wondering if there is any random variable whose cumulative ...
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1answer
45 views

Probability of Independent Random Variables

Let $X$ and $Y$ be independent random variables, each of which is uniformly distributed between $0$ and $1$. Find the probability that $(X−\frac 1 2)^2+(Y−\frac 1 2)^2\leq \frac 1 9$. Give at least 8 ...
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2answers
76 views

Probability of the sum of exponential random variables

Let $X$ and $Y$ be independent random variables such that $X∼Exp(1)$ and $Y∼Exp(2)$. Find the probability that $3X+4Y≤5$. Give at least 10 correct digits after the decimal point. So the formula for ...
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1answer
23 views

Exponential Random Variables and Confidence

Assume that the amount of evidence against a defendant in a criminal trial is an exponential random variable X. If the defendant is innocent, then X has mean 1, and if the defendant is guilty, then X ...
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1answer
127 views

A fair dice is tossed 6 times. What is the probability that there is at least one pair of identical consecutive face values?

For example, 231146 is a valid sample point but 131213 is not. This is a question on past exam that i have no idea to solve. Please help me!
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1answer
64 views

Poisson Random Variable

Let us model the number of winter storms in a given year as a Poisson random variable. Suppose that in a good year the average number of storms is 3, and in a bad year the average is 5. If the next ...
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2answers
40 views

Does X|Y = X formally, in the sense of RVs?

In Cover and Thomas' "Elements of Information Theory", the joint entropy $H(X,Y)$ is defined, but they state that this definition is nothing new if we consider that it is the entropy of a single ...
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0answers
14 views

Probability of subsequences of sequences of Bernoulli random variables [duplicate]

I have a sequence of iid Bernoulli random variables $X_1, X_2, \dots, X_n$ with $Pr(X_j) = p$, and I'd like to know (a lower bound on) the probability that there exists a consecutive subsequence of ...
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2answers
60 views

Calculating probability of pressing button three times and hearing sound at the third press

Consider the situation: There are two mysterious buttons in front of you. One of the buttons is harmless, whenever you press it, nothing happens. The other button is mostly harmless, when you press ...
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1answer
25 views

Calculating independence of two random variables

I have two random variables $X$ and $Y$. $X$ takes the values ${-1,0,1}$ and $Y = X^2$. I have to determine if these two are independent. I have already calculated that the covariance = 0 for these ...
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1answer
19 views

Gradient of draws from random variables

In the context of neural networks there has recently been some interest in differentiation incarnations of random variables. Example. Given a random variable $y \sim \mathcal{N}(\mu, \sigma^2)$. Now ...
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0answers
27 views

independence - joint distribution

Let $X,Y$ random variables with $Y$ real valued. I was wondering if any inequality of the type: \begin{equation} \mathbb{E}[f(X,Y)] \leq g \left[\sup_{\displaystyle s \in \mathbb{R}} ...
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0answers
10 views

calculating covariance of random variables.

Consider two random variables X and Y where X takes the values {-1, 0 1} with equal probabilities and Y = X^2. I have to calculate the covariance of X,Y. I know cov(X,Y) = E[XY] = E[X]E[Y] I have ...
4
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1answer
102 views

How to find $\mathbb{E}[X\mid\min(X,Y)]$?

Say I have two independent variables $X$ and $Y$ that are exponentially distributed with respective rates $\lambda_X$ and $\lambda_Y$. How do I compute $\mathbb{E}[X\mid \min\{X,Y\}]$?
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1answer
20 views

Poisson process independence

In a Poisson process with rate $\lambda$ we know that the number of arrivals from $t = 0$ to $t = 1$ is independent of the number arrivals $t = 1$ to $t = 2$. However, are $N(2)$ and $N(1)$ ...
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1answer
60 views

Conditional probability of random variables

Say I have random variables ~$Exp$ and lets call them $X$ with rate $\lambda$ and $Y$ with rate $\mu$. How do I find $\mathbb{P}\{X>Y|Y>4\}$? I know that $\mathbb{P}\{X>Y\} = \frac{\mu}{\mu + ...
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1answer
58 views

Expected value of multiple random variables, uniform distribution

Suppose that the random variables $X_1,\dotsc,X_n$ form a random sample of size $n$ from the uniform distribution on the interval $\left[0, 1\right]$. Let $Y_1 = \min\left\{X_1,\dotsc,X_n\right\}$, ...
5
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1answer
80 views

Closed formula for mean

Suppose we have the i.i.d. random variables $X_{11}, X_{12},\ldots, X_{nn}$, such that each $X_{ij}$ has standard normal distribution $N(0,1)$, with mean $0$ and variance $1$. Given some integer ...
3
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1answer
60 views

Kolmogorov's $0-1$ law and constant RV

Kolmogorov's $0-1$ Law: For any terminal event $A$ we have that either $\mathbb{P}(A)=1$ or $\mathbb{P}(A)=0$. Alternatively any $F_{\infty}$ measurable random variable (so basically a terminal ...
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2answers
33 views

Gaussian processes are determined by their mean and covariance functions.

A stochastic process $X_t$ is called Gaussian if the random vectors $(X_{t_1},...,X_{t_n})$ are multivariate normal. Why are the finite dimensional distributions of a Gaussian process determined by ...
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3answers
156 views

Maximum possible variance

From this biology article, end of page 4, the author talks about a random variable which never takes value outside the range [0,1] (0 and 1 included in the range). He says that the maximum variance ...