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

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0
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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
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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
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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
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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
70 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 ...
1
vote
1answer
21 views

Convergence in distribution with exponential limit distribution

Let $X_1,X_2, \ldots$ be independent, identically distributed, positive random variables with probability density function $f$, which is continuous in $(0, \infty)$ and $\lambda :=\lim_{x \searrow 0} ...
0
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1answer
50 views

$\limsup$ and $\liminf $ of $\sum_{k=1}^n \frac{X_k}{\sigma \sqrt{n}} $

Suppose $(\Omega, \mathfrak{F}, p)$ is a probability space; $X_n$ are i.i.d. random variables defined on $\Omega$, with $E(X_i)=0$ and $Var(X_i)= \sigma$ for all $i$. Then $$ \limsup_n \sum_{k=1}^n ...
3
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0answers
40 views

Estimating the support of a probability density function

The inverse moment problem deals with the reconstruction of a probability density function (PDF) of a random variable (RV) by means of its statistical moments. In the special case of the Hausdorff ...
0
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0answers
24 views

Comparison of Expectation of two vector of random variable.

Let $\mathbf{x}\in \mathbb{R}^n$ and $\mathbf{y}\in \mathbb{R}^n$ be two vector of positive random variables where all components are independent (either uniform or positive Gaussian). If ...
0
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0answers
35 views

Probability distribution based on random events with increasing weights

Consider a weighting function, f(t), that is monotonically increasing with the time, t. [This may not be strictly necessary, as pointed out in the comments, but all of the cases I am actually ...
0
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1answer
21 views

Random Variables with Poisson parameters and limited number of trials

So I'm just unsure how I set up my answer to the following question, which is a homework question. " The number of surfaces blemished on any one chocolate is described by a Poisson distribution with ...
0
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2answers
71 views

A convergence question concerning the uniformly integrability

If $\{\xi_n\}$ is an uniformly integrable r.v.s., then $$\lim_{n\rightarrow\infty}E\left(\frac{1}{n}\sup_{1\leqslant k\leqslant n}|\xi_k|\right)=0.$$ Why?
3
votes
1answer
70 views

A necessary and sufficient condition for symmetry of a random variable

Prove that if $X$ is an integrable random variable, it has a symmetric distribution if and only if: $$E(X|X^2)=0$$ Can anyone check my solution? Firstly $$EX=0$$ Then we have: $$E(E(X|X^2))=EX=0$$ ...
1
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2answers
52 views

Let $U, V \sim U(0,1)$ be independent. What is $P(U \leq V)$?

Let $U, V \sim U(0,1)$ be independent. What is $P(U \leq V)$? My attempt: We are looking for $P(U-V \leq 0)$. For a given $t \in [0,1]$ this is equivalent to $$P(U \leq t, V \gt t)= P(U \leq t)P(V ...
1
vote
1answer
17 views

Convergences of square of difference in probability implies convergence in probability

Consider real valued random variables $X,X_n,n\in\mathbb{N}$. If $(X_n-X)^2\xrightarrow{P} 0$ then $X_n\xrightarrow{P} X$. I tried using Chebyshevs inequality (which seems to be the usual ...
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3answers
47 views

Graphic of the probability distribution function : How does it works?

Here is the graphic of the probability distribution function for a random variable $X$. How can I find the distribution of $Y=-X$? By definition the distribution function for a random variable is ...
0
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2answers
16 views

How many calls during a day? Number of exponential random variables during a fixed duration

I am trying to think about the summed duration of waiting times of i.i.d random variables with exponentially distributed wiating times, and particularly my question is how many such variables will ...
1
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1answer
40 views

Probability distribution function of $Z=\frac{Y}{X+1}$

I have two random variable $X,Y$ in the same space $(\Omega,\mathcal{F},\mathcal{P)}$. $X$ takes its values on $\Bbb{N}$, I denote $P(X=k):=p_k$ and $Y$ takes its values on $\Bbb{R}$? and ...
0
votes
1answer
36 views

On a limit of random variables.

This is a duplicate of this question that has not got an answer. I am going to try to improve my question that is probably missworded since I do not believe it to be difficult, even though I can't ...
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0answers
24 views

Is this an equivalent definition of probabilistic independence of random variables?

Let $X$ and $Y$ be random variables, $P(X=x) \neq 0$, $P(Y=y) \neq 0$ for any $x,y \in I\!R$, and suppose that \begin{equation} P(X=x ~|~ Y=y) ~~~=~~~ P(X=x ~|~ Y=z) \end{equation} holds for all ...
1
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1answer
34 views

Why does $P(X_i = X_j)= 0$ in this statement?

I've found this statement: $X_1\, \ldots X_k$ are independent random variables defined on the probability space $(\mathbb{R}, \mathfrak{B}, p)$, each with the same density $f$. Thus $(X_1\, \ldots ...
2
votes
2answers
45 views

Expectation of product of two random variables

Let $X,Y$, two random variables which are indicators. Lets assume $P(X=1)=p$ and $P(Y=1)=q$ for some $0 \le p,q \le 1$. I've understood that: $E[XY] = P(X=1, Y=1)$. How to show it? $$E[XY] = ...
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0answers
15 views

PDF of three random variables simplification

It is well-known that the PDF of two RV of the form $Z=\frac{X Y}{X+Y}$ can be represented by $\min(X,Y)$. and the corresponding equivalent CDF is therefore can be obtained by ...
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1answer
55 views

Interpreting The Weak Law of Large Numbers

Given the usual setup, the weak law of large numbers states that for any $\epsilon > 0$ $$ \lim_{n\rightarrow \infty}P(|M_n - \mu| > \epsilon) = 0 $$ According to this author, the ...
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2answers
22 views

For random variable X, is there any general result for satisfying $E[X] = E[X^k]$ for $k>0, k \neq 1, k \in \mathbb{R}$?

For random variable $X$, is there any general result for satisfying $E[X] = E[X^k]$ for $k>0$, $k \neq 1$, $k \in \mathbb R$ and where $k$ is given? I do not assume any distribution here, but if ...
1
vote
1answer
25 views

Understanding Random Walks

So I was reading the random walk article from wiki, and was currently focusing on the Lattice Random Walk section: There, I understand the definition of $Z_1,Z_2...Z_n$, which defines $Z_i$ as a ...
1
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0answers
31 views

better expression for simple random walk

Let $P_{k,j}$ be the probability that the probability that simple symmetric random walk starting from the origin reaches the point $k \in \mathbb{N}$ precisely in $j$ steps without ever returning to ...
3
votes
1answer
113 views

Find characteristic function of random variable

Let random variables $X, Y, Z$ independent. $X$ with uniform distribution on $[-a,a]$, $Y$ with Poisson distribution with parameter $ν$, $Z$ with Bernoulli distribution with parameter $p$. Find the ...
3
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1answer
64 views

Is there a closed form for this distribution (maximum difference between successes and failures in i.i.d. Bernoulli flips)?

Consider a series of i.i.d. coin flips: $$X_1,X_2,\ldots, X_n\sim \begin{cases} 1 &\text{w.p. } p \\ -1 &\text{else} \end{cases} $$ We define $$Y = \max_{i\leq ...
1
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2answers
43 views

If $X, X_1, X_2, \ldots $ are positive and $X_n\stackrel{P}\to X $ and $E(X_n) \to E(X)$, then $X_n \stackrel{L_1}\to X$

Let $X, X_1, X_2, \ldots $ be positive random variables. Prove that if $X_n\stackrel{P}\to X $ and $E(X_n) \to E(X)$, then $X_n \stackrel{L_1}\to X$ My attempt: I tried to truncate $E(|X_n-X|)$ ...
0
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0answers
26 views

Conditional Probability of Poisson random variables. Am I doing this right?

So I'm just unsure if I'm doing this right . I'll give question. the initial set up of my question and what I evaluated it to be. More interested if I got the set up right than exact figures as I'm ...
1
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1answer
40 views

What is the chances of a duplicate in this equation

I'm not very good at math; However I have a scenario where I'm trying to find the chance of duplicate for randomly generated data. In a nuttshell I have a "bag" with 62 different items, lets say a ...
1
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1answer
64 views

A question about independence of sigma algebras (generated by random variables)

Let $X_1, X_2, \ldots$ i.i.d random variables. Is it possible that $$\{X_{n+1} \in B\} \in \sigma({X_1, \ldots, X_n})$$ for some $B$? Why yes/not? I want to show that $\sigma(X_{n+1})$ and ...
0
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2answers
50 views

Random variables and Linearity

I have an equation $Y = 5 + 3\times X$ and I assume that $X$ is a random variable taking values from a uniform distribution. Can I consider that also $Y$ is a random variable which takes values from a ...
0
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0answers
22 views

An example of memoryless yet non-independent random process?

I am new to random process. I know that independence indicates memorylessness yet the memorylessness is not necessarily independence. There are abundant examples of independent random process (like ...
3
votes
2answers
62 views

A question about sum of n random variables

Let $X_1, \ldots, X_n$ be random variables. We know that $X_1, \ldots, X_n$ are $\sigma(X_1, \ldots, X_n)$ - measurable. But how about $X_1 + \cdots + X_n$? Is it $\sigma(X_1, \ldots, X_n)$ - ...
2
votes
1answer
28 views

Finding the PDF from the CDF where the CDF is not differentiable at some point

I got the following problem: Let $X$ be a continuous random variable with $CDF$ denoted $F_X$ defined as follows: $F_X(x)= \begin{cases} 1-x^{-4/3}, & x\in[1,\infty) \\ 0, & x\in ...
0
votes
0answers
19 views

Find probability density function of random vector

Random vector has continuous distribution of setA={(u,v), v>=0, u+v<=1, v-u<=1}. I need to find joint probability density function of this vector. In my ...
1
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1answer
23 views

Probability: Expectation: indicator RV, what is 1-((N-1)/N))^n?

Say there are N coupon types, you collect n coupons, and what's the expected number of types of coupons? My question is specifically about $1-(\frac{N-1}{N})^n$, the probability of getting a coupon ...
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2answers
120 views

Probability that there is sub-sequence of exact length

Can you help me to solve the following: Find probability that in sequence of N random uniformly distributed numbers there is increasing sub-sequence of exact length L.
4
votes
1answer
53 views

Almost surely vs expectation

Let $X_1, X_2, X_3 \dots$ be a sequence of random variables. In the limit as $i \rightarrow \infty$ we have $$ X_i \rightarrow 0 \text{ almost surely} $$ Does it follow that In the limit ...
0
votes
1answer
37 views

The smallest filtration for which a sequence of random variables is adapted

Let $X_1, ..., X_n$ be a sequence of random variables. Show that $\hspace{60pt}$ $\mathcal{F}_n$ = $\sigma(X_1, ..., X_n)$ is the smallest filtration such that the sequence $X_1, ..., X_n$ is ...
0
votes
2answers
13 views

Showing that $n1_{ \lbrace U<1/n \rbrace}$ converges to $0$ almost surely

Let $U \sim \text{Uniform}[0, 1]$ and $X_n = n1_{\lbrace U< 1/n \rbrace}$. I want to show that $X_n$ converges to $0$ almost surely. My attempt: I use Fatou's Lemma with the reasoning that if I ...
0
votes
0answers
23 views

How to Justify the exclusion of some samples?

I am calculating the asymptotic cumulative distribution of $M_n = \max(X_1,X_2,\dots,X_N)$. My problem is $X_1,X_2,\dots X_p$ and $X_k,X_{k+1},\dots,X_N$ have non identical CDF for $p<<k$ and ...
4
votes
2answers
79 views

If $X_i$ are iid $U(0,1)$ random variables, $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$

I want to show $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$ as $n \to \infty$, where $X_i$ is an i.i.d sequence of $[0,1]$-uniformly distributed random ...
0
votes
1answer
17 views

Random Variable probability summation tweaking

I can't seem to figure out what they do to get to the bottom
1
vote
1answer
70 views

Approximate normal distribution

Let $ X \sim N (0, 1)$. For $x$ large enough, the tail of the distribution of $X$ may be approximated as $$P(X > x) \sim e^{-x^2/2}/(x\sqrt{2\pi})$$ Consider a sequence of independent r.v. all ...
1
vote
1answer
19 views

Positive component of a submartingale is a submartingale

I am trying to prove the Doob's Upcrossing Lemma and the first step requires to prove that: If $X$ is a submartingale, then $(X-a)_+$ is a submartingale. I found it intuitive but i failed to prove. ...
0
votes
1answer
14 views

Can I sum variances to a compound variance?

Say I have three locations A,B,C and I have a person going from A to B and measure the time it takes. Same for B to C. Let the variance of the time it takes for the path AB be a and for BC b. Is it ...
3
votes
2answers
50 views

Limit law of real-valued independent random variables

Let $X_n$ and $Y_n$ be real-valued independent r.vs, each of whose limit law is $X$ and Y, resp. i.e $X_n \overset{d}{\to} X$ and $Y_n \overset{d}{\to} Y$ for some r.vs $X$ and $Y$. Then, are $X$ ...