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

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6
votes
1answer
60 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: $X, Y$ ...
1
vote
0answers
20 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
vote
0answers
21 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
votes
1answer
39 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 ...
0
votes
1answer
35 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
15 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) = ...
1
vote
1answer
54 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
41 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
54 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
50 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
20 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
45 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 ...
4
votes
1answer
40 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
60 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
24 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
28 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
18 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
vote
0answers
19 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
32 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
39 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
41 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
19 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
23 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
37 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
64 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
24 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
33 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
44 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
32 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
36 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
21 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
26 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
28 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
96 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 $$ ...
1
vote
0answers
21 views

Joint p.d.f $Y=x_1/x_2$ for two independent continuous random variables $X_1$ and $X_2$

The question reads like this: Two independent continuous random variables $X_1$ and $X_2$ have a joint p.d.f $f(x_1,x_2)$. Determine the p.d.f of $Y=X_2/X_1$, assuming $Y>0$. (That is $Y$ is ...
2
votes
2answers
105 views

Entropy upper bound inequality for Sub-Gaussian Random Variable

We say that the random variable $Z$ is $\sigma^2$-subGaussian if $\mathbb{E} \exp(tZ) \leq \exp(t^2\sigma^2)/2$. Define the $(x\log x)$-entropy (or simply the entropy) of a nonnegative random ...
1
vote
1answer
35 views

Infinitesimal Random Variable

I have been very confused by the idea of infinitesimal random variables, namely letting $\{Z(\omega,t)\}_{t\in\mathbb{R}}$ be a stochastic process. What do we mean by $dZ$. Is this meant by ...
2
votes
1answer
26 views

A variation of the coupon collectors problem

The problem goes as following: Let there be $n$ coupons, and $X_i$ be the random variable whose value is $1$ if coupon $i$ is collected during the first $n$ draws, and $0$ otherwise. What is the ...
0
votes
0answers
14 views

how to understand 'expected max approximation error'

The background is that: E() denotes the expectation and $y$ satisfies a certain probability distribution $g(y)$, then we independently sample $y_1,y_2$ from $g(y)$. It is assumed that $E(y_1-E(y))=0, ...
0
votes
1answer
43 views

Expected value question

My teacher gave the following question as a practice question for the exam... I was just wondering if someone could check if my answer is correct: A group of n ≥ 3 people is sitting at a round table, ...
0
votes
0answers
21 views

probability problem using Chebyshev's inequality

Suppose that a die has its "3" side changed to a "2". The problem is to first find a lower bound on the probability $P[3\leq X \leq 4]$ using Chebyshev's inequality. Then if we roll the die $n$ ...
1
vote
0answers
41 views

Roll a dice till consecutive sixes - Generating Function

Consider the following experiment: A fair dice is thrown until two consecutive sixes are rolled. Let $X$ be the number of rolls of the experiment. I need to find the probability generating function of ...
3
votes
1answer
68 views

Limit of a sequence of random variables

Suppose $Z_n$ is a sequence of independent random variables, which are uniformly picked from the interval $[1,2]$. Show that: $$ \lim_{n_\rightarrow \infty}P\left(\left|\sqrt[n] {Z_1 Z_2\cdots ...
1
vote
0answers
63 views

Conceptual Understanding of a Simple Random Process

I have a simple discrete time random process that with probability $0.5$ chooses a deterministic sequence so that $X(t) = -1$, for $t<1$ and $X(t) = +1$ for $t \geq 1$, similarly with probability ...
0
votes
1answer
83 views

Determine E(X) of X Where X Is Number Of Days Beer Is Drank On The Same Day? [duplicate]

Lindsay and Simon have discovered a new pub that has n different beers B1, B2, . . . , Bn, where n ≥ 1 is an integer. They want to try all different beers in this pub and agree on the following ...
0
votes
2answers
64 views

Determine E(X) of X with these conditions?

Let $n \geq 1$ be an integer and consider a uniformly random permutation $a_1, a_2, ... , a_n$ of the set $\{1, 2, . . . , n\}$. Define the random variable X to be the number of indices $i$ for which ...
-1
votes
3answers
33 views

Expected number of unique items between 2 subsets

Please help ASAP final in an hour and don't get this review question. There are two people at a donut shop that serves 10 different types of donuts. Both the people order a 5 donut subset (no 2 donut ...