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

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0
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1answer
25 views

How can you find $P(\frac{X}{Y-X}<0)$ if $X\sim Geometric(p)$ and $Y\sim Bernoulli(p)$

Let the independent random variables $X\sim Geometric(p)$ and $Y\sim Bernoulli(p)$, I want to prove that $P(\dfrac{X}{Y-X}<0)=(p-1)^{2}(p+1)$. Do I need the joint probability mass function for ...
0
votes
2answers
26 views

To use or not Bernoulli trials

I was asked to model the following experiment: Consider the n-th toss of a fair coin, and the event $E$ = '$k$-th toss results in heads'. I find easier to model the experiment using n random ...
0
votes
0answers
13 views

Support for a linear combination or transformation of random variables

Let $X, Y \sim iid U(0,1)$ and $c_1, c_2 \in \mathbb{R}$. In the linear combination $Z = c_1X+c_2Y$, we know that the probability density function of $Z$ depends on the relationships of $c_1$ and ...
0
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1answer
9 views

“cover the unit sphere by c-fine grid” to prove the vector length preserved by random projection?

The below figure is extracted from the paper http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4031351 . I did not understand the techniques used in the proof, namely, 1."cover the unit ...
1
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0answers
25 views

Distribution Function Of a Random Variable X - Question

This is a homework question pretty much but I do not understand how to approach it. The distribution function of the random variable X is given: F(X) = 0, x < 0 x/2, 0 <= x < 1 2/3, 1 ...
0
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1answer
24 views

How to calculate the probability distribution function (PDF) and the cumulative distribution function (CDF)?

Sorry I'm a novice to both functions and just didn't get a clue how to solve this problem (having been reading the theories for the whole day but still ...) The problem is: We have now two investment ...
0
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0answers
13 views

Martingale difference sequence [on hold]

Show that the sequence of random variables $w_t = (u^2_t - \sigma^2)x^2_{t-1}$ is a martingale difference sequence.
2
votes
0answers
18 views

Max cut problem

I've just looked at the standard proof using the probabilistic method stating that the max cut problem has a lower bound of $|E|/2$ for any graph $G=(V,E)$. More specifically if $X$ is the random ...
6
votes
1answer
47 views

Trying to understand the behaviour of i.i.d.

In a course called introduction to probability theorem we are covering now i.i.d. (independent and identically distributed random variables). I already know when two variables are independent: $X, Y$ ...
0
votes
0answers
14 views

Characteristic function of an asymmetric Laplace distributed random variable

What is the characteristic function of a random variable with density $$f_X(x) = \frac{1}{2} [ 1_{x>0} \, a e^{-a x} + 1_{x<0} \, b e^{b x} ], \; \; \; \quad a,b > 0 \quad \quad ? $$ My ...
1
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0answers
17 views

When is a coupling ''natural''?

The definition of coupling is written below. In some articles, I found the term "natural coupling". When is a coupling said to be ''natural''? Definition of coupling between two random variables: Let ...
0
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1answer
30 views

p.d.f. of a position variable from stochastic velocity p.d.f.

I have a stochastic process, $v(t)$, that represents a velocity, and has a known probability distribution function $f(x,t)$ which is time-varying. I am interested to acquire a probability ...
-1
votes
0answers
18 views

Adding x+y gamma function [on hold]

I know how to set up a independent variable when it's normal. And I'm sure they are somehow related but I'm not sure where to start on this problem. It's not covered at all in the book I have and he ...
0
votes
1answer
34 views

Non-standard question about random variables

I am not sure which subbranch of mathematics this is, so I cannot give a precise tag. I am doing research, and this suddenly popped out of no where. So, please hear me out. $x$ is a variable that ...
0
votes
1answer
11 views

new bounds for transformed random variable

Let $Y \sim U\left ( 0,1 \right)$, I have already determined the new pdf for the transformation $Z=Y^2$. I used the cdf technique for this. So the new pdf for $Z=Y^2$ is $f_Z(z) = ...
2
votes
0answers
32 views

Convergence in probability, expected value

I have problems with the following two sequences of random variables: We assume that $X_1, X_2, ... $ are iid. Let $m=EX_i$ The first one is: $$ \alpha_n := \frac{1}{n} \sum_{i=1}^n (X_i - m)^2$$ I ...
1
vote
1answer
37 views

Probability of tail event using Kolmogorov's 0-1 law

If $X_1,X_2,... $ are independent random variables and $X=\sup_nX_n$ then $P(X<\infty)$ is either 0 or 1. I think that if we prove the event to be a tail event then the result will follow. But I ...
0
votes
1answer
40 views

Variance and Expected value of internet connection

I am working on a probability/statistics problem! The problem is as follows: Your internet connection is very poor. It constantly alternates between being functional for x minutes and being down for ...
1
vote
1answer
48 views

How long would it take to a lottery number repeat?

In Professor Stewart’s Cabinet of Mathematical Curiosities the following is asked: You have $1000$ songs on your MP3 player. If it plays songs ‘at random’, how long would you expect to wait ...
0
votes
3answers
45 views

Probability of success in $n$ trials

I'm stuck on my statistics homework and would appreciate your help. Question: Repeated independent trials of a certain experiment are carried out. On each trial the probability of success is $0.12$. ...
1
vote
0answers
17 views

Distribution of a r.v. with the same mean and variance is abs. cont. with resp. to the normal distr.

I have a question concerning the Kullback-Leibler divergence or relative entropy. In a book I found the following definition of the KL-divergence: Let $(\Omega, \mathcal F)$ be a measurable space. ...
2
votes
2answers
24 views

Joint Random Variable: Given f(x,y), find P(X>Y)

There are 2 continuous random variables, X and Y. Say the joint pdf of (X,Y) is f(x,y). How do you find the P(X>Y) generally? Like I am not sure where to start with.
1
vote
1answer
13 views

diagonalizing a matrix with random elements

Consider the matrix $A = \begin{pmatrix} cY & 0 \\ 2 & 1\end{pmatrix}$, where $c \in \mathbb{R}$ and $Y$ is a random variable that is uniformly distributed over $[0,1]$ (That is, $Y \sim ...
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0answers
18 views

Find the PGF of two independent binomial random variables [closed]

Let $X$ and $Y$ be independent binomial random variables with parameters $(n_1,p_1)$ and $(n_2,p_2)$ respectively. Find the PGF $\phi_{X+Y}(z)$, find the expectation $E[X+Y]$
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0answers
24 views

Let X and Y be jointly continuous random variables, and let $A$ be an arbitrary subset of $\mathbb{R^2}$. [closed]

Let X and Y be jointly continuous random variables, and let $A$ be an arbitrary subset of $\mathbb{R^2}$. I want to calculate the probability that the random vector $(X, Y )$ lies in the set $A$. ...
-2
votes
0answers
19 views

Continuous random variable $X+Y$ [closed]

The support of a continuous random variable is the set of the outcomes such that $f(x) > 0$. If $X$ has support $[a, b]$ and $Y$ support $[c, d]$, what is the support of $X + Y$?
4
votes
1answer
31 views

Almost sure convergence of a sequence of Gaussians with vanishing variance

Let $(X_n)_{n\geq 1} $ a sequence of independent random variables. We assume that $X_n \sim \mathcal{N}(0,\sigma_n^2)$ and that $(\sigma_n)_{n\geq 1}$ is a vanishing sequence of positive numbers. Let ...
2
votes
1answer
52 views

Show that a Markov Chain is ergodic

Let $Y_n$ be iid random variables with values 1,2,3..n so that $P[Y_i=j]=p_j>0$, where $i\leq1$ and $1\leq j\leq n$. I think I managed to show that $Y_n$ is a Markov chain using the definition, ...
0
votes
0answers
27 views

Applying chain rule in probability?

Let $X,Y$ be random variables with distribution functions $F_X(x)$, $F_Y(y)$. Let $W(u,v)=max\{0,u+v-1\}$. why can we take the following limits "inside" $W$? $lim_{(x,y)\to ...
1
vote
0answers
22 views

Does normalization of a random vector, destroy uniformity?

If I have a random vector in Rn that has a uniform distribution in the domain [a,b]n, a<0, b>0. Is uniformity lost or preserved (in the unit sphere) if I normalize the vector (using the euclidean ...
1
vote
0answers
25 views

Prove that the variance of a discrete random variable increases with a parameter

I have an infinite number of known probability density functions $f_1(x),f_2(x),f_3(x),...$. The PDFs $f_k(x)=\sum_{j=1}^k v(A+j-1)e^{-v(A+j-1)x}\binom{k}{j-1}q^{j-1}(1-q)^{k-j-1}$. Let ...
0
votes
0answers
14 views

perfect coin is tossed n times. Let Sn denotes the number of heads obtained. What is the expectation of Sn?

The Problem is: A perfect coin is tossed n times. Let Sn denotes the number of heads obtained. What is the expectation of Sn? I got to E(Sn) = $\sum_{n=1}^{+\infty} \space\space\space Sn ...
1
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0answers
18 views

Convergence of vectors

Recently I've read a paper and there is one moment I cannot fully realise on my own. It states as follows. There is a vector of estimates $\hat{\mathbf{X}} = (\hat{X}_1, \dots, \hat{X}_N)$ (N is ...
0
votes
1answer
31 views

The solution to this joint distribution problem is too terse for me to understand.

I was wondering if I could get clarification on the following problem: We know that $\sum_x\sum_y f(x, y) = 1$. Then $4\theta_1 + 6\theta_2 = 1$. I understand that $P[X = 1] = ... = P[Y = 4] = ...
1
vote
1answer
38 views

Is c parameter or constant (random variable X with given density)

problem: is c constant or parameter solution for this is to $ \int_{1}^{2} cx^2 dx = \frac{7c}{3} $ $ \int_{2}^{3} cx dx = \frac{5c}{2} $ Until now I understand what is going on; next (I am ...
1
vote
1answer
31 views

Calculating inter-arrival times and arrival times of a Poisson process

For a practice exam in stochastic processes I have to answer the following questions. Let $\{N(t): t\geq 0\}$ be a poisson process with rate $\lambda$. Let $T_n$ denote the n-th inter-arrival time ...
0
votes
1answer
17 views

probability of joint PDF

I found $k = 4$ and yes, the are independent. But for the last one I know how to find the probability if they are like $x$ from $0$ to a number and $y$ from $0$ to a number so the limit of double ...
3
votes
1answer
22 views

Moment generating function of Random Sums

I am unsure of a particular step in the supplied solution of this problem. Problem: We are given $X_{i}$, for i = 1,..., n, is a sequence of iid Geometric Random Variables. N ~ Geometric(p), and ...
0
votes
1answer
32 views

Probability of receiving a correct packet of N bits

When a packet is transmitted on a communications link, the probability that a bit in packet is received in error is p. Assume that the packet has N bits. Suppose the packet length is random i.e. N is ...
2
votes
1answer
63 views

Proving the inequality $\frac{1}{k!}+\frac{1}{(k + 1)!}+\frac{ 1}{ (k + 2)! }+…\leq {(\frac{e}{k})}^k$

In the first part of the question we showed that $P(X \geq k)\leq E(e^{tX}e^{-kt})$ for all $t \geq 0$ and real $k$ by the use of Markov's inequality. This wasn't too bad. Now, in the second part, ...
3
votes
1answer
23 views

Inner product on random variables

Let $(\Omega, \mathscr{F}, P)$ be a probability space and let $L^2$ denote the space of real-valued, discrete random variables with finite variance that map $\Omega$ to a set $Q$. Define ...
2
votes
1answer
27 views

Simulating r.v.'s from a joint density by using rejection sampling in R

I wish to sample variables $v$ and $w$ from the joint density $$(v+w)e^{-\frac{(v+w)^{2}}{2x_{0}}-2\mu v-(\mu -\lambda )w},$$ where $x_0$, $\mu$ and $\lambda$ can be seen as positive constant. Since ...
0
votes
2answers
38 views

Probability Question About Uniform Random Variables and Median

Let U, V, W ∼ Uniform(0, 1) be independent. Find the probability that the median (i.e., the second smallest) of these three random variables lies in the interval (1/4, 3/4). I cannot figure out what ...
0
votes
1answer
28 views

Question on the proof of the upper bound of girth in dense graph.

I have trouble understanding the proof of the following theorem from Upfal's Probability textbook pg 134 Theorem: For any integer $k \geq 3$ there is a graph with n nodes, at least ...
2
votes
1answer
31 views

Campbell's theorem variance

From Wikipedia, For a Poisson point process $N$ and a measurable function $f: \textbf{R}^d\rightarrow \textbf{R}$, the random sum $$\Sigma=\sum_{x\in {N}}f(x)$$ [...for complex value ...
0
votes
1answer
19 views

Expected no of flips before a TT comes, using series sum

To find out the expected no of flips of a coin to get a TT, i want to find it out using a series of probability multiplied with their values. In a similar question using sum of series the expected ...
0
votes
1answer
55 views

Variance of the number of r.v summed to fill certain capacity

Let us assume that we have a certain capacity T. We have an infinite number of random variables $X_1,X_2,\dots,$ where each $X_i$ is independent and has a particular pdf $P_i(X)$. And we have that ...
1
vote
1answer
25 views

Convergence of random variables in $L^1$

So $g$ is a continuous real-valued function and are given that the sequence of random variables $Y_n$ converges to $Y$ in $L^1$, $E[|g(Y_n)|]<\infty$ and $E[|g(Y)|]<\infty$. Show that $g(Y_n)$ ...
0
votes
1answer
25 views

Covariance of two random variables (one is squared)

I have problem figuring out the solution for this task: X1 and X2 are independent random variables with normal distribution ~N(2,1). What is a covariance of $X_1 − 4X_2^2$ and $X_1 + X_2$. So far ...
2
votes
1answer
85 views

Almost sure convergence of the series of independent random variables

Let $\{X_n:n\ge1\}$ be i.i.d. random variables with $\operatorname EX_1=0$ and $\operatorname E|X_1|^p<\infty$, where $1<p<2$. Let $\{b_n:n\ge1\}$ be a real sequence. Does the series $$ ...