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

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2answers
11 views

Coupon Collector's Problem — Expected Value of each item

So I guess my problem is based on the famous coupon collector's problem, which is, if you should not be familiar with it, the following: Given N different coupons from which coupons are being drawn ...
1
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0answers
23 views

Probability of absorption of a biased random walk when the starting point has binomial distribution

Consider a random walk $\{0,1, ... , N\}$ with up probability $p$ and down probability of $p-1$ where $p \neq 1/2$. Suppose there are absorbing barriers at $0$ and $N$ and that the starting point, ...
0
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1answer
26 views

What is the probability that on a given day, the number of half gallon containers provided is enough?

In a grocery store 400 customers shop every day. The number of half gallons of nonfat milk bought by a randomly selected customer is a random variable X having P(X=0)=0.3, P(X=1)=0.5, and P(X=2)=0.2. ...
2
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0answers
32 views
+100

Mean value theorem for random variables

In a proof I am trying to understand a mean value theorem for random variables is used. It is stated that $$f(X+Y)=f(X)+f^\prime(X+\theta Y)(Y-X)$$ for real valued random variables $X$ and $Y$ and ...
1
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0answers
15 views

What is the pdf of $X$, where $dX_t = -aX_t + d N_t, N_t$ is a compound Poisson process?

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where (I'm not sure this notation makes sense, I'm not very familiar with the stochastic calculus ...
0
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1answer
46 views

Distribution of Bernoulli and Uniform Random Variable

Here's a problem I am stuck on: Let $X$ and $Y$ be independent random variables such that $X$ is Bernoulli-distributed with $p=1/2$, and $Y$ is uniformly distributed on the interval $[0,1]$. Then: ...
0
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2answers
31 views

Normal Distribution of Sums

I have two normally distributed random variables $X$ and $Y$. Then I know that the sum $X-Y$ is also normally distributed (i). However, I want to show (preferably by a counter example) that the ...
1
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0answers
10 views

On random functions taking values in the space of continous functions.

I upload the image of the passage that is unclear to me (Theoretical statistics by Keener): I do not understand how $W_1, W_2, \dots$ take values in $C(K)$. For every different realization of the ...
2
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1answer
27 views

How to represent $Prob(X_1+X_2 \leq a, X_2+X_3 \leq b, X_3 +X_4 > c)$ with mutually independent random variables?

There are four mutually independent random variables: $$X_i : \Omega \to \mathbb R$$ for $i= 1,2,3,4$ The cumulative distribution function of them is given as $F_i(x_i)$. How to represent ...
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0answers
13 views

Expected value of exponential random variable [on hold]

If an exponential random variable, X, has failure rate λ, what is E[X|X<λ]? I'm not sure how to start here. I know that E[X] = 1 / λ for an exponential random variable. Is the probability that X ...
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1answer
18 views

Expected Value: how to understand this expression?

So I have come across a question asked by my peers. Define: $$g:=\sqrt{E[|y_r(t)|^2]}$$ Given that $$y_r(t)=\sqrt{t}\cdot h+b+k+c,$$ where $h$, $b$, $k$, and $c$ are independent random variables. ...
0
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1answer
67 views

How to construct Poisson process as a random variable in order to evaluate integrals?

I want to find a probability space $\Omega$ that represents Poisson process as $$\Pi : \Omega \to \{A \in \mathcal{P}{(\mathbb{R^+})}\mid |A| = \aleph_0\}$$ Which is a mapping from $\Omega$ to all ...
0
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1answer
39 views

Proof conditional probability formula

a question for my homework for probability goes as follows: Given X,Y,U, three discrete random variables, prove the following: $$ p_{X|Y}(i|j) = \sum_{k=0}^{\infty}p_{X|YU}(i|j,k)p_{U|Y}(k|j) $$ The ...
0
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0answers
13 views

Posterior probability estimation in MAP model

I have a question about probability. I am using Bayes rule to determine which class the $x$ belong to. According to Bayesian formula, the MAP estimation is equivalently found by $$p(x \in \Omega_i|x)= ...
0
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3answers
26 views

Given the joint density function for X~Unf(0,2) & Y~Unf(0,3) find Pr(XY < 1)

I have two independent random variables, X~Unf(0,2) & Y~Unf(0,3). Their joint density function is f(x,y) = 1/6 if 0<=x<=2 and 0<=y<=3 else f(x,y) = 0. I'm supposed to find Pr(XY < ...
0
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1answer
32 views

What is the Difference in the Average and the Mathematical Expectation in the following Problem

Suppose that a school has 20 classes: 16 with 25 students in each, three with 100 students in each, and one with 300 students, for a total of 1000 students. The average class size is simply ...
0
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1answer
30 views

The probability of passing a probability course with limitations. [on hold]

John is attempting to pass a probability exam again and again until he has succeded. However, he is allowed to try only $n$ times. Suppose that the probability that John doesn't flunk is $p$ ...
7
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6answers
2k views

“Random” generation of rotation matrices

For a current project, I need to generate several $3\times 3$ rotation matrices for input into an algorithm. I thought I might go about this by randomly generating the number of elements needed to ...
0
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0answers
32 views

How can two random variables are continuous infers that their jointly random variable is continuous?

We assume that $\forall a,b$ such that $a2+b2>0$, $aX+bY $ is continuous random variable. But we don't assume that $X$ and $Y$ are independent. My question is the following: Under which ...
0
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1answer
5 views

Variance function is variance stabilising

Y has mean u and variance function V(u). If $V(u) = \alpha.u^v$ then $h(y) = y^{(2-v)/2}$ is variance stabilising which means that Var(h(Y)) is approximately constant. I tried to prove it computing ...
2
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1answer
33 views

Sequence satisfies weak law of large numbers but doesn't satisfy strong law of large numbers

Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables such that $$P(X_n=n+1)=P(X_n=-(n+1))=\frac{1}{2(n+1)log(n+1)}$$ $$P(X_n=0)=1-\frac{1}{(n+1)log(n+1)}$$ Prove that $X_n$ ...
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0answers
19 views

How to compute the average power of an ergodic process?

Rxx(0)=3 is the average power and if i take limit as t goes to infinity i will get the (E[x])^2 to get variance you subtract 3-2 = 1 is this correct ? and can someone tell the difference ...
0
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1answer
50 views

Random Variable $X= U:(0,4)$ is given. Find the CDF of $\min \{ |X-1|, 5-X\}+1$

Random Variable $X= U:(0,4)$ is given. Find the CDF of $Y=\min \{ |X-1|, 5-X\}+1$ X has uniform distribution. So we know that $$Y\in (1,6)$$, therefore $$y\leq 1 \ \ \ F_y(y)=0 ; y\geq6\ \ \ ...
1
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1answer
23 views

How to calculate the probility of 2 independent events of having same value?

We are learning to calculate the probability of sums and difference of random numbers. Here is the problem: One athlete knows from past experience that the distances of his javelin throws follow a ...
0
votes
3answers
44 views

Finding Variance and Expectation of Boolean Variable

Below is the joint distribution of Boolean random variables X1, X2 and X3. How do I find variance and expectation of X2? I understand that variance is "average of squares of difference from mean ...
1
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1answer
14 views

Probability of event in join sample space of X & Y

From what I understand, the answer should be $(0.1+0.35+0.05)$, since the given points have probability summation $1$. Am I correct? By the way, the correct answer unknown.
1
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3answers
340 views

Distribution of the first passage time of a Gaussian random walk

Does anyone know the distribution for the first passage time of a Gaussian random walk i.e. $$ S_n = \sum_{i=1}^n X_i $$ where $X_i$ are iid normally distributed random variables. The first passage ...
0
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0answers
12 views

Finding an interval estimate for $\mu$ given a sample size and variance

I'm in a statistics class and am doing a problem for homework about confidence intervals. I don't really know what it's asking though or when I've even reached a valid solution. The problem says: ...
2
votes
2answers
27 views

Expectation over multiple variable?

So I have come across a question asked by my peers. Define: $$g:=\sqrt{E[|y_r(t)|^2]}$$ Given that $$y_r(t)=\sqrt{t}\cdot h+k,$$ where $h$ and $k$ are independent random variables with variance ...
1
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2answers
30 views

Uniform distributed variable X:U(-9,9) is given. Find the CDF of Y if Y is..

$$Y= \begin{cases} 4X,\ \ \ |X| \leq 3 \\ 0,\ \ \ \ \ \ |X|>3 \end{cases}$$ My take on it: $$F_Y(y)=0; y\leq-12;F_Y(y)=1; y\geq 12;$$ $$F_Y(y)=\{Y < y \}$$ In class in a similar task we ...
1
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2answers
25 views

Probability distribution of number of columns that has two even numbers in a chart

We distribute numbers $\{1,2,...,10\}$ in random to the following chart: Let $X$ be the number of columns that has two even numbers. What is the distribution of $X$? My attempt: ...
4
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0answers
102 views

A Fibonacci like Stochastic process

Let $X_0, X_1$ and $\{a_n,n\geq0\}$ are i.i.d. $\sim $Bernoulli$(1/2)$ taking values in $\{0,2\}$. Let us define $X_n$ for $n>1$ as below$$X_{n+1}=a_nX_n+a_{n-1}X_{n-1},\ n\geq1 $$ Then it follows ...
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0answers
24 views

Distribution for Arithmetic Mean of n Geometrically Distributed Random Variables

For the evaluation of an algorithm I implemented for work, I need to find the distribution function for the arithmetic mean of $n$ independent, geometrically distributed random variables. Let ...
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0answers
7 views

specific examples of random variables satisfying a given condition.

Theorems such as the central limit theorem only says random variables satisfying certain conditions have some properties. Now, what I am curious about is the existence of such random variables. For ...
0
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2answers
27 views

transformation of single random variables with absolute value ??

integral I got the final answer to be fy(y)= 1 0< y < 1 I am not sure could anyone correct me if its wrong !
1
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3answers
44 views

What allows us to divide a random variable into multiple ones?

I can't wrap my head around the solution presented for this problem: Suppose a trial has a success probability $p$, let $X$ be the random variable for the number of trials it takes to stop at $r$ ...
6
votes
1answer
39 views

Divergent series of random variables

I've been trying to prove that given a sequence of independent random variables with identical distribution $\{X_n\}_{n \in \mathbb{N}}$ such that $P(X_1 \neq 0)>0$, so also $P(X_i \neq 0) >0 \ ...
1
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1answer
39 views

PDF & CDF of a Sum of Weighted Independent Random Variables $Z=aX+bY$

From this question here, I learned that the Cumulative Distribution Function (CDF) of $Z=X+Y$ is: \begin{eqnarray*} F_Z \left( z \right) & = & \int F_X \left( z - y \right) dF_Y \left( y ...
5
votes
2answers
328 views

The condition for $Y$ to make $\mathbb{E}[\max\{X_1+Y,X_2\}] > \mathbb{E}[\max\{X_1, X_2\}]$

I would like to know the condition for a random variable $Y$ in order to make $\mathbb{E}[\max\{X_1+Y,X_2\}] > \mathbb{E}[\max\{X_1, X_2\}]$, where $X_1$ and $X_2$ are iid. Any help would be ...
1
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1answer
348 views

Expected value vs using method of indicator

I am having a hard time understanding the difference between getting the Expected value by finding the mean E(X)=np and using the method of indicator to find the expected value. For example if we ...
1
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1answer
32 views

Expected value of doubling or halving a number with equal probability

I have this question that you start with a value say c. At each step, you either double or half the value with equal probability. Let $X_i$ be the value of c at ith-step, I need to find the expected ...
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0answers
22 views

problem about graph of auto-correlation for wide-sense stationary process?

I have the answers but I don't understand the idea and how it can be solved ? please clarify and help me to understand it thank you all
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1answer
27 views

find the power of a random process?

I know all the steps expect the last step i don't know how to evaluate the integral can someone show me the step that lead to the answer to be A^2/2
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1answer
51 views

If X and Y are correlated random variables and Z is independent of X and Y, then are X+Z and Y independent? [closed]

I am always blocked by the following question. Conditions: 1) $X$, $Y$ and $Z$ are all random variables; 2) $Z$ is independent of $X$ and $Y$; 3) $X$ and $Y$ are correlated. Question: are $X+Z$ and ...
1
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0answers
185 views

Conditional expectation, quadratic function, absolute value

We are given two random variables defined on $[0,1]$: $$X(\omega) = 2 \omega -1 + |2 \omega -1|$$ $$Y(\omega) = 1-|2 \omega^2 -1|$$ I am supposed to find $\mathbb{E}(X|Y)$ which by definition is a ...
1
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1answer
28 views

Does the following sequence of random variables converge?

Let $X_1,X_2,...$ be independent random variables with $P[X_n=0]=1-1/n$, $P[X_n=1]=1/2n$, $P[X_n=-1]=1/2n$ Does $X_n$ converge almost surely? , Does $X_n$ converge in probability? I just started to ...
1
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1answer
8 views

What can we say about the concentration around 0 linear transformation of Gaussian random variables?

I have a matrix $X \in \mathbb{R}^{n \times m}$ such that each $A_{ij}$ is a Gaussian with mean $0$ and variance $1$. We have $m > n$. I also have a vector $v \in \mathbb{R}^m$ such that $||v||_2 ...
0
votes
1answer
20 views

Maximum of two independent uniform random variables

Let $x$ and $y$ be uniformly distributed, independent random variables on $[0,1]$. What is the probability that the maximum between $x$ and $y$ is less than $1/2$ and greater than $1/3$?
2
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0answers
20 views

Probability of having at least one coupon out of N types

I'm facing a question regarding random variables: A coupon website has N distinct kinds of coupons. Each selection of a coupon is equally likely and selections are independent. Let $T$ be a ...
0
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1answer
36 views

Probability of most frequent occurrences of suits/values when drawing 4 cards from 52

Draw 4 cards from a card deck with 52 cards (4 colours and 13 values for each colour) one after the other -- none is put back. Let's have two discrete random varaibles X and Y. X counts the maximum ...