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

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-1
votes
0answers
8 views

Adding x+y gamma function

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
23 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
7 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
15 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
30 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
38 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
44 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
37 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. ...
1
vote
2answers
19 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
11 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
0answers
18 views

Find the PGF of two independent binomial random variables [on hold]

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 $E[X+Y]$
0
votes
0answers
24 views

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

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 $X+Y$ [on hold]

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
30 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
51 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 ...
0
votes
0answers
15 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
28 views

One of properties of Poisson random variables [closed]

Let $X$ be a Poisson random variable with parameter $\lambda$. How can I show that $\mathbb E[X^n]=\mathbb E[(X+1)^{n-1}]$? I've been stuck in calculation.
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votes
0answers
14 views

MLE of Binomial (checking presentation of answer) [closed]

I just wanted to check if my presentation of the final answer is acceptable. So we are given a Binomial random variable with parameters m, p i.e. X~Bin(m,p). Since m is known, we can take it to be a ...
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
62 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
22 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 ...
-1
votes
1answer
25 views

Second moment calculation [closed]

I have $n$ random indicator variables $X_i$ that are independent $P(X_i=1)=1/K!$ For every $i$. I would like to calculate $E[X^2]$ how do I do that in this case? Thank for your help
2
votes
1answer
84 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
18 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
89 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
33 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
25 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
38 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
18 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
31 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
57 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 ...