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

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0
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1answer
11 views

Distribution of random variables when combined

I need help with this problem: If $X$ and $Y$ are two independent random variables and are both standard normal, what is the distribution of $\frac{1}{2}(X^2+Y^2)$? I think I start with ...
0
votes
1answer
20 views

Probability that sum of two uniformly distributed random variables is less than some constant

I am trying to find a way of determining the probability that the sum of two continuously uniformly distributed random variables is less than some constant $C$, formally: Let $A \sim ...
0
votes
1answer
23 views

Calculation of a characteristic function

Suppose $X_1, X_2, \ldots X_n \ldots$ are independent random variables with $$P(X_n = 1) = \frac{1}{2}$$$$P(X_n = -1) = \frac{1}{2}$$ Then $$\sqrt{\frac{3}{n^3}}\sum_{k=1}^n kX_k$$ converges to ...
1
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0answers
16 views

sum/product combination of random variables

Let $X$ and $Y$ be independent random variables. If I am asked about the distribution of random variable $XY+Y$, is it ok if I compute $XY$ first and then add the result to $Y$ (via convolution, or ...
0
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0answers
14 views

What is the nonlinear estimator for Gaussian Random variable?

I know that the best estimator is $g(x)=E\{Y|X=x\}$ and the conditional density for jointly Gaussian random variables is known to be Gaussian with mean and variance given by ...
0
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0answers
14 views

A stochastic process is generated as follows: we assign the value 1 to a head and the value 0 to a tail. Start at n=0, Compute Rxx(0,0) and Rxx(2,3)

I am kind of confused here, since autocorrelation describes the correlation between values of the process at different times, but for the first case, it is at the same time. I got that ...
3
votes
1answer
43 views

$X = E(Y | \sigma(X)) $ and $Y = E(X | \sigma(Y))$

Suppose $X, Y$ are random variables in $L^2$ such that $$X = E(Y | \sigma(X)) $$ $$Y = E(X | \sigma(Y))$$ Then I want to show that $X=Y$ almost everywhere. What I've done: By conditional Jensen ...
1
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0answers
16 views

Probability question involving drawing balls from an urn

Suppose there's an urn containing $r$ red balls and $b$ blue balls. At each trial, I'm drawing a ball at random from the urn, without replacement. Let $R$ denote the event of drawing a red ball, and ...
0
votes
1answer
27 views

Various modes of convergence of random variables

Let $\lbrace X_n \rbrace_n$ be a sequence of independent random variables such that $$P(\{X_n = \pm 1 \}) = \frac{1}{n}$$ $$P(\{X_n = 0 \}) = 1 - \frac{2}{n}$$ Is the sequence convergent: $1$) almost ...
1
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0answers
16 views

Mean Preserving PDF Spreading

I have a univariate discrete random variable and a histogram representing its PDF (which is asymmetrical). Is there a known way to increase/decrease the variance of the distribution (i.e. scaling it ...
0
votes
3answers
34 views

Product of random independent variables

What are the properties of the product of random variables? The book I have on probability and statistics only comments on their sum properties. I know that when you sum random independent variables ...
2
votes
2answers
24 views

Expectation value of absolute value of difference of two random variables

I do not really know how to prove the following statement: If E(|X-Y|)=0 then P(X=Y)=1. The main problem is how to handle the absolute value |X-Y|. If I say that |X-Y| >= 0 it follows that ...
-1
votes
2answers
52 views

How to compute the sum of geometric distribution [on hold]

How to compute the sum of random variables of geometric distribution $X_{i}(i=0,1,2..n)$ is the independent random variables of geometric distribution, that is, $P(X_{i}=x)=p(1-p)^{x}$, then how to ...
-1
votes
1answer
17 views

is it possible to implement random(0,1,..,m) with finite calls to random(0,1)? [on hold]

that is, is there a function $f$ that $Y=f(m,X_1,X_2,...,X_{n(m)})$ where $X_i\sim B(1,\frac{1}{2})$ and $Y\sim U\{0,m\}$? e.g. when $m=2^k-1$,$n(m)=k$ and ...
1
vote
1answer
36 views

An Elementary Convergence Problem in Probability

Suppose that $X_1,X_2,...$ are degenerate random variables such that $f_{X_n}$ denotes the mass function of $X_n$.$$f_{X_n}(x)=P(X_n=x)= \begin{cases} 1, & x=2+\dfrac{1}{n} \\ 0, ...
0
votes
0answers
21 views

Density of product of random variable

I would like to derive the product density of two independent continuous random variable in a measure theoretic framework. I am well aware of the result which can be found here: ...
1
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0answers
23 views

Cicurlar random-walk.

Suppose you have a computer network with 5 code as following. Packet can arrive at any node and any other node can be its destination equal uniform probability. Determine the average number of ...
0
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0answers
16 views

Book on Convergence Concepts in Probability without Measure Theory

I am looking for a comprehensive book on Probability which discusses Convergence of Random Variables in detail, excluding portions of Measure Theory. Allan Gut's "Probability: A Graduate Course" seems ...
1
vote
2answers
21 views

Variance of Random Variable and Normal Variable

Let X be a random variable following normal distribution with mean +1 and variance 4. Let Y be another normal variable with mean -1 and variance unknown. If $$P(X\leq-1)=P(Y\geq2)$$ then standard ...
3
votes
0answers
37 views

Calculation of Conditional Expectation $\Bbb E[f(X)\mid Y]$

$\newcommand{\Cov}{\operatorname{Cov}}$and thank you for taking the time to read this. :) My question is about evaluating $\Bbb E[f(X) \mid Y]$ (a random variable in $Y$). There's plenty online (and ...
0
votes
1answer
19 views

Geometric distribution $G(p)$ for independent random variables $X$ and $Y$

Question: If the random variables $X$ and $Y$ are independent and each have the geometric distribution $G(p)$ - that is, $P(X=k)=P(Y=k)=pq^k$ for $k=0,1,2,\ldots$ (where $q=(1-p)$) show that: (I) ...
2
votes
2answers
42 views

How can we show that “almost surely” equal random variables have the same distribution?

How can we show that "almost surely" equal random variables have the same distribution? We know $X =\text{(a.s)} Y$. What I have so far: $$\begin{align*}\implies& P(X = Y) = 1 \\ ...
0
votes
0answers
13 views

Psudorandom number from diffrent generators.

Suppose I've N random number generators (uniform distribution) and I take 1 value from each one. Will this set of N variables be considered equivalent to N random numbers produced by a single ...
0
votes
0answers
22 views

Random-walk in a pentacle (5 nodes)

There are a total of 5 nodes at the edge of a pentagram At each node, you have a 4 choices which will lead you to either a destination node or non-destination node. Assume the decision of path is ...
0
votes
0answers
10 views

Wasserstein distance and maximization covariance

My question deals with the second order wasserstein distance $W_2$ on the set of measures, which is defined by: $W_2(\nu_1,\nu_2)^2= inf_{\Pi(X,Y)} E_{\Pi} (X-Y)^2$ where $\Pi$ is chosen such that ...
0
votes
1answer
34 views

Setting bound for an infinite expected value

Say $X=2^Z$ and $Z$ is a geometric random variable with $p=1/2$. It follows that, $E[X] = \infty$ So setting the upper bound by the markov inequality, $$P(X \geq t) \leq \frac{E[X]}{t} = ...
1
vote
1answer
11 views

How to show that increasing r.v. imply stochastic dominance?

How can one prove the following statement: If $X$ and $Y$ are random variables such that $X(\omega) \geqslant Y(\omega)$ for all $\omega$ then $\mathbb P(X>x) \geqslant \mathbb P(Y>x)$ ? I saw ...
1
vote
1answer
28 views

Kolmogorov 0-1 law

Initial question: $X_n$, $n \in\mathbb N$, are independent real-valued random variables. Let $S_n$ be defined, for each $n\in\mathbb N$, by the sum: $S_n = X_1+X_2+...+X_n$. Prove that either the ...
3
votes
2answers
41 views

Random points in spherical shell

I have a sphere of radius $R_1$, and a smaller, concentric sphere of radius $R_2$. Let them be centered at the origin $(0,0,0)$. I need to generate random points with uniform density in the volume ...
2
votes
1answer
46 views

Statistically Independent Random Variables

Problem: For the statistically independent ramdon variables X and Y with fX(x)=1, 1≤x≤2, and fY(y)=e-(y-1), 1≤y<∞, determine fZ(z) where Z=X+Y I couldn't find a ...
3
votes
2answers
34 views

Mathematical justification for incorporating a conditional event in expectation?

Let $X_1,X_2,\dots$ be independent and identically distributed random variables. Furthermore, consider the sum $$ Y = X_1 + X_2 + \dots + X_N $$ where the number of terms $N$ is itself a random ...
2
votes
1answer
22 views

Showing convergence by using Levy's continuity theorem

Let $X_n \sim \mathrm{Po} (n)$. I want to show that $Y_n = \displaystyle \frac{X_n -n}{ \sqrt{n}}$ for $n \rightarrow \infty$ converges in distribution to a random variable that is standard normal ...
1
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0answers
19 views

Sums of random variables

Good evening community, I need some help. Prove the followed theorems: (a) $X$ and $Y$ have the density functions $p_X$ and $p_Y$ in terms of the Lebesgue-Borel-measure and both are Independent, so ...
0
votes
2answers
18 views

Deriving probability densitys

How does one derive probability densitys involving fractions? For example, let $X^2$ and $Y^2$ be exponentially distributed random variables with parameter $\lambda = 1$. Determine the PDF for $Z = ...
2
votes
0answers
33 views

Finding the maximum expected value

Say we have $a_1, a_2, ..., a_m \in \{0, 1\}^n$ where the sum over the elements of each vector $a_i$ is $k$. Let $b \in \{0, 1\}^n$ be a random vector based on the uniform distribution. Also, let ...
0
votes
1answer
46 views

What is the probability at least one bulb is burnt out after time t?

Given three lightbulbs whose lifetimes $X_i$ are exponential random variables with parameters $\lambda_i = i$, for $i = 1, 2, 3$ respectively. If all bulbs are switched on at time '0' what is the P(at ...
3
votes
1answer
30 views

Example expectation of an exponential function

Given a geometric random variable $Y$ with $p = 1/3$, I know that $E[Y] = 1/p = 3$. However, what is $E[e^{aY}]$ ? for a small value $a$. Thanks
0
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0answers
21 views

What is the mode of $Z=\sqrt{X-Y}+4$? Given $X$ and $Y$.

Given that $$\begin{array}{c|lcr} \text{X} & 2 & 6 \\ \hline \text{p} & 0.3 & 0.7 \\ \end{array}$$ and $$\begin{array}{c|lcr} \text{Y} & 1 & 2 \\ \hline \text{p} & ...
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votes
1answer
27 views

How to find the distribution of a function of multiple, not necessarily independent, random variables? [on hold]

If $Y$ is a random variable defined as $Y=g(X_1,X_2)$, where $X_1$ and $X_2$ are two different random variables whose distributions are known (say with pdf's $f_{X_1}$ and $f_{X_2}$), how do we find ...
0
votes
0answers
30 views

How to Combine Independent Variables

I want to know how to combine probabilities. I never had a chance to study this sort of thing in school so I don't even know the correct language to describe it. Suppose we have a grid of tiles with ...
1
vote
1answer
43 views

Variability of sum of independent random variables

I am trying to understand the Central limit theorem, especially the ${1\over \sqrt{n}}$ coefficient of a random variable $S_n = {1\over \sqrt{n}}(\sum_{i=1}^n {{X_i - \mu} \over {\sigma}})$. Lets ...
0
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0answers
31 views

Markov/Chebyshev - Returning homework to students

Homework was returned to students in a random way. What can we say about the probability that more than 10 students will get back the homework that they gave in in the beginning? Try to find the ...
4
votes
1answer
71 views

Solve the integral [closed]

Can anyone solve these two integrals . $$ \int_{0}^{ \infty } \frac{x^2 e^{-x^2/2 \sigma ^2}}{(x-a)^2+b^2} dx $$ and $$ \int_{0}^{ \infty } \frac{e^{-(\ln x - \mu )^2/2 \sigma ...
0
votes
1answer
23 views

Given common probability density function $f(x, y) = e^{-(x+y)}$ where $0 \le x ,y < \infty$, calculate $P(X < Y)$.

First time I'm getting to such question: Given common probability density function $f(x, y) = e^{-(x+y)}$ where $0 \le x ,y < \infty$, calculate $P(X < Y)$. What is the way to approach ...
3
votes
0answers
63 views

Duration of a Gambler's Ruin game against an opponent with infinite credit

A gambler enters the casino with $x\$$ in his pocket and sits on some table. At each iteration he bets $1\$$ and either wins $1\$$ with probability $p\leq\frac{1}{2}$ or loses $1\$$. Assuming that ...
0
votes
1answer
14 views

Formula for the correlation between two different variables

"Jon planted a plant. When the plant grew to 4 centimeters of height he decided to start to measure how much the plant grew each week. Here's the result Week 0: 4 cm. Week 1: 6 cm. Week 2: 10 cm. ...
1
vote
2answers
42 views

General sufficient condition for independence of these two random Variables.

I need to state and prove a general sufficient condition on(a,b,c) for independence of two random Variables. We have that $a,b$ and $c$ are real numbers and the random variables are below: $$ ...
1
vote
1answer
79 views

Find the characteristic function of combination of random variables

I have the same problem as here: Find characteristic function of random variable. Could you explain last equality? Can I get it without using law of total expectation? Update I have some idea, $E $$ ...
2
votes
0answers
36 views

Expected magnitude of a vector of $n$ i.i.d. random variables as $n\to\infty$

Suppose that $X_i$ are i.i.d. real valued random variables with probability distribution $f(x)$ for $i=1,2,3,\ldots$. Let $Y_n=\left(\sum_{i=1}^nX_i^2\right)^{1/2}$. Assuming that ...
2
votes
1answer
69 views

Showing that lim sup of sum of iid binary variables $X_i$ with $P[X_i = 1] = P[X_i = -1] = 1/2$ is a.s. infinite

Let $(X_i)_{i\in\mathbb{N}}$ be an i.i.d. sequence of binary random variables with $$P[X_i = 1]=P[X_i = -1] = \frac{1}{2}$$ and let $$S_n = \sum_{i=1}^{n} X_i.$$ I'd like to show that $$P[\lim ...