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

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Finding the probability using probability distributions.

A contractor is required by a county planning department to submit 1-5 forms in applying for a building permit. Let $Y$ be the number of forms required of the next applicant. The probability that $y$ ...
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1answer
25 views

Are functions of multiple independent random variables independent?

Suppose X, Y, and Z are independent geometric random variables with parameter $ \theta $. Now suppose V=G(X,Y) and U=F(Z). It seems intuitive that V and U would also be independent. The variation in ...
2
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1answer
61 views

Showing that $(1-u)z^2\leq P(uz\leq |X|)$ when $0<u<1, E(X^2)=1, $ and $0<z<E(|X|)$.

I am trying to show that $$(1-u)z^2\leq P(uz\leq |X|)$$ where $0<u<1, E(X^2)=1, $ and $0<z<E(|X|)$. I've been given a hint to consider Cauchy-Schwarz, however, I don't see where ...
0
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1answer
20 views

How can we derive cross covariance $R_\mathrm{xy}(t_1,t_2)=R_\mathrm{yx}^*(t_2,t_1)$?

In random process, cross covariance is nonnegative definite like $$R_\mathrm{xy}(t_1,t_2)=\mathbf{E}(\mathrm{X}(t_1)\mathrm{Y}^*(t_2))=R_\mathrm{yx}^*(t_2,t_1)$$ I'm wondering how it can be derived. ...
2
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1answer
20 views

How to find the probability of X=6, where X is the max of 3 6-sided dice

I'm fairly certain the answer is $91/6^3$ (confirming it through a script), but I'm not certain how to solve the problem in a mathematically sound way. I made an attempt that got to the right answer, ...
0
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1answer
37 views

The quantile function $F^{-1}(p) = c$ for all $p$ in the interval $(p_0, p_1)$ has the condition that $Pr(X=c) = p_1 - p_0$

Let $X$ be a random variable with c.d.f. $F$ and quantile function $F^{-1}$. Assume the following three conditions: (i) $F^{-1}(p) = c$ for all $p$ in the interval $(p_0, p_1)$, (ii) ...
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0answers
29 views

Generate two sets of (nonlinearly) dependent random numbers

I would like to find a method to generate two sets of (nonlinearly) dependent random numbers. Solution for linear dependence (that is, correlation). Generate two sets of uncorrelated random numbers ...
1
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1answer
32 views

Which has the least value?

Which of the following have the least value if $-1 < x < 0$? (A) $-x$ (B) $1/x$ (C)$-1/x$ (D)$1/x^2 $ (E)$1/x^3$ I'm not sure what to do, but I'll definitely try. We can ...
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0answers
19 views

Standard Uniform Distibution with Random Variable

Could someone help explain how to solve the following problem: From my understanding, this problem states that we have a function, Uniform(0, 1), that will generate a random value from 0 to 1 with ...
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0answers
39 views

what is the probability that there is a string of k consecutive heads?

A coin is flipped n times. Assuming that the flips are independent, with each one coming up heads with probability p, what is the probability that there is a string of k consecutive heads? An answer ...
-2
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1answer
54 views

Combination problem: random selection in a group

A scientific committee of 4 persons is to be randomly selected from a group consisting of 3 biologists, 3 physicists and 4 mathematicians. Let X denote the number of biologists, Y the number of ...
0
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1answer
27 views

Expected Value functions of two randon variables

$X$ and $Y$ are two independent random variables. $f$, $g$ and $h$ are 3 functions. Can the below expected value be calculated? $$E\left[ f(X)\sum_{k=0}^{\lfloor g(X) \rfloor }h(Y) \right]$$ ...
2
votes
1answer
39 views

What is the difference between $\mathbb E[Z|\mathcal G]=Y$ and $\mathbb E[Z|\mathcal G]\stackrel{\text{a.s.}}{=}Y$?

I'm somewhat confused by the definition of martingale: Let $(\Omega, \mathcal F, \mathcal F_n, \mathbb P)$ be a filtered probability space. We call $(X_n)_{n\in\mathbb N}$ martingale if for all ...
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1answer
32 views

prove a theorem about an upper bound of entropy of a random vector

There is a theorem that: if Z is any zero-mean, complex random vector with covariance $E[ZZ^H]=R_z$, then $H(Z)\leq \log|{\pi eR_z}|$, with equality holding if and only if Z has a circularly ...
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1answer
49 views

limsup of a sequence of random variables (definition)

Let $X_n$ be a sequence of random variables. First, $\limsup X_n=\inf_n\{\sup_{m\ge n}X_m\}$. So, $$\{\limsup X_n\le c\}=\bigcup_n\bigcap_{m\ge n}\{X_m\le c\}$$ Is it correct to say that if ...
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1answer
187 views

Proving Isserlis' Theorem for n=4

I have been trying very hard to prove Isserlis' theorem for n=4 case, i.e when we have 4 random variables that are jointly Gaussian variables with zero-means. ...
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0answers
31 views

Ratio Distribution of Two Dependent Chi-Squared without Joint Distribution

Assume vectors $\textbf{x}$ and $\textbf{y}$ are two independent Complex Gaussian random vectors with i.i.d elements. What is the PDF of $z$ as the ratio of norms in this form: \begin{equation} ...
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0answers
16 views

Entropy of bit position in a bit stream

8 bit strings are sent over a channel. First two bits are always 1. Last six bits can be either 0 or 1. Receiver randomly selects bit-position and reveals bit but not its position. If X is the random ...
0
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1answer
9 views

Why are A->C<-B conditionally dependent in a directed graph?

$P(A,B,C) = P(A)P(B)P(C|A,B)$. I understand how $A,B$ are marginally independent on $C$, but I'm confused as to how the $A, B$ are conditionally dependent on $C$. $P(A,B|C) = ...
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0answers
21 views

Show Y is location-scale if $\sigma > 0$ is unknown

Let X be a random variable having the gamma distribution with shape parameter $\alpha$ and scale parameter $\gamma$, where $\alpha$ is known and $\gamma$ is unknown. Let $Y= \sigma $ log $X$. Show ...
0
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1answer
28 views

Bounds for PDF of Sum of Two Dependent Random Variables

Assume $X$ and $Y$ are two dependent random variables and we do not have the joint distribution of these two. Is there an upper/lower bound for the PDF of $X+Y$? I found a paper which provides bounds ...
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2answers
48 views

Does $X_n \xrightarrow{\mathbb P} c \in \mathbb R$ imply $\phi(X_n) \xrightarrow{\mathbb P} \phi(c)$ in this case?

Let $X_n \xrightarrow{\mathbb P} c \in \mathbb R$ and $\phi: \mathbb R \to \mathbb R$ be bounded, continuous in $c$, and $\phi(c)=0$. Show that $\mathbb E\left[\phi(X_n)\right]\to0.$ I was going ...
0
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1answer
37 views

Joint Density function from marginal density functions

This there anyway to find the joint density function of random variables X and Y. Nothing is given about they being independent. So we have to solve by assuming that they are not independent.The ...
0
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1answer
16 views

Number of trials to observe all values of a uniform discrete random variable X with a probability of at least 1-q?

Let X take on p values with equal probability. If n trials are to be conducted to ensure that the probability of not observing any of these p values is less than or equal to q, what is the value of n? ...
3
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1answer
33 views

Probability of random assignment to form pairs

So the question goes: I have 100 individuals and 100 different buses, and I randomly assigned each individual to sit on a bus (each bus has equal probability of being selected). How many buses are ...
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0answers
13 views

Show equalities of random variables

In the text we showed that a geometrically distributed random variable W has the lack of memory property. Now assume that the range of W is {1,2,3...} and that P(W = j + 1|W>j} = p for j = ...
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2answers
44 views

Does $X_n \xrightarrow{\text{in distr.}} X$ and $|X_n|\leq Y$ imply $|X|\leq Y$?

We know that $$X_n \xrightarrow{\mathbb P} X \text{ and } |X_n|\leq Y \implies |X|\leq Y \text{ a.s.}$$ I was wondering if the same holds in case of convergence in distribution. So far, I've shown ...
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0answers
52 views

Why this is a martingale?

Setup: $W$ probability space $Z_i : W \to L_i $ random variables ($L_i$ finite, for example $\{0,1\}$) $f: Z_1 \times \ldots \times Z_n \to \mathbb{R}$ $X_i := \mathbb{E}[f \mid Z_1,..,Z_i]$ Why ...
0
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1answer
32 views

When is a random variable is said to be well-defined?

In the paper On the Bootstrap of the Sample Mean in the Infinite Variance Case by Keith Knight, on page 1170 at the bottom of the page before the theorem, the author mentions that the random variable ...
3
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1answer
35 views

How can I find the distribution of a stochastic variable X^2 if X is normal standard distributed? [duplicate]

I am considering a stochastic variable X that is standard normal distributed i.e. $$ F_X(x) = \int_{-\infty}^x\frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}}dt $$ How do I find out the distribution of $X^2$? ...
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0answers
33 views

Discretization of the standard uniform dist.

I need some help. Sorry for the poor use of LaTEX... a) $U_{n} = \lfloor{nU}\rfloor/n$ prove that $\lim_{n \to +\infty} {U_{n}} = U$ where $U \sim unif[0,1]$ and thus that $lim_{n \to +\infty} ...
1
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1answer
35 views

What is the variance of this random variable: number of items

Let us assume that we have a capacity $n$ which tends to infinity. We have an infinite number of random variables $X_1, X_2, \dotsc$, where each $X_i$ is independent and identically distributed with ...
0
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1answer
68 views

Expected value and Variance calculation

Suppose $f$ is an uniformly distributed random variable with parameters $-1,1$ and $g$ is a Poisson-distributed random variable with parameter $\lambda >0$. We assume that $f$ and $g$ are ...
0
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1answer
38 views

Corollary of Kolmogorov zero-one law [duplicate]

Here is another corollary of the theorem: Kolmogorov zero-one law given in my textbook (Probability path). How can I apply the said theorem given that if $X_n$ are independent random variables, ...
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0answers
14 views

Z transform of a random variable

Let's say we have: $\tau(k) = k\times r$ where $r \sim N(0,\sigma ^2)$. Therefore, in Z domain we have: $\tau(z) = -z\frac{dR(z)}{dz}$. But what is the Z transform of a Gaussian random variable ...
0
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1answer
57 views

Difficult probability problem [closed]

I am stuck in this question $33$ miners are trapped in a mine. There´s an elevator that takes $10$ minutes to go down and another $10$ minutes to pick up a miner and return to the surface. One of ...
0
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1answer
67 views

Z~U[0,1] and X=f(Z) and f is:

I have found the f(z): Now, I need to find pdf of X. And I can see that 0< f(Z)=X<1, I don't know how I am going to get f(X), I just can see that f(X)=0 when X<0 and x>1, but I can see a ...
3
votes
1answer
52 views

Construct independent $X_n$ such that $\sum_{n=1}^\infty Var(X_n)=\infty$

How to construct an independent random variable $(X_n)_{n\in\mathbb{N}}$ such that $\sum_{n=1}^\infty X_n$ converges and $\mathrm{Var}(X_n)$ is uniformly bounded by some constant C, but ...
3
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0answers
49 views

Distribution of $\frac{X}{|Y|}$, where X and Y are standard normal r.v.'s

Let X and Y be independent standard normal random variables. What is the distribution of $\large \frac{X}{|Y|}$? Attempt: Let $\large U = \frac{X}{|Y|}$ and $ V = |Y|$. This transformation is not ...
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1answer
55 views

Exponential random variable

Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked is an exponential r.v. with parameter 1/20. Smith has a used car that he claims ...
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1answer
60 views

Proving Negative of Standard Normal is Standard Normal

Let X be standard normal random variable $N(1, 0)$ prove that $-X$ is also standard normal. I think I am stuck on a technicality but here is my attempt: Let $Y = -X$ P(Y $\leq$ u) = P($-X$ $\leq$ ...
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0answers
13 views

Question regarding piecewisely defined random variable

If I have two independent random variables X and Y with known PDF. And another random variable defined piecewisely as $Z=0.9\times(X-Y)$ if $X-Y<c1$ $Z=0.4\times(X-Y)$ if $c1<=X-Y<=c2$ ...
2
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2answers
124 views

Would Evaluating a polynomial at uniformly random points outputs random values?

I`m wondering if we evaluate a polynomial on many points picked uniformly at random. Can we say the output values Y's are uniformly at random?
2
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2answers
41 views

Definition of multivariate random variable

Let $(\Omega,\mathcal E,P)$ be a probability space and $X_1,\dots,X_n$ random variables on $(\Omega,\mathcal E, P)$. Then I can define the vector $X=(X_1,\dots,X_n)\colon \Omega \to \mathbb R ^n$ and ...
0
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2answers
49 views

Uniform Random Number

Two uniform random numbers are chosen one after the other. what is the probability of second number second random number greater than first number? I tried this way Please correct me if I am wrong. ...
0
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1answer
39 views

Discrete RV problem

A flight control system uses four independent computers working in parallel. At each critical step, the computers “vote” to determine the appropriate step. The probability that a computer will ask for ...
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0answers
25 views

Convergence of random saddle point

Let $y_n^*$ be the solution of $$ y = g_n(y) $$ where $g_n(\cdot)$ is a random function. Suppose that for fixed $y$ $$ g_n(y)\to h(y) $$ almost surely and pointwise as $n\to\infty$. Is there any ...
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0answers
44 views

Find the type of noise a random variable should be contaminated with, such that certain properties are satisfied.

Let $\mathbf{x}$ be a random column vector in $\Bbb{R}^n$, given as follows $\mathbf{x}=(x_1,\ldots,x_n)^\top$. We require $\mathbf{x}$ to follow a multivariate Gaussian distribution with mean vector ...
0
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2answers
18 views

Marginalization question $\Pr[a] = E_X[\Pr[\ a|X\ ]\ ]$

I'm reading explanation of a theorem, and there's one step that I can't understand. I know it should be simple enough, but I just can't think of the reasoning atm. The step says, According to ...
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2answers
16 views

$L^1(P)$ random variable limit.

for random variable $X$ in $L^1(P)$, What is $\lim_{a\to \infty}a * P( |X| > a )$ ? I think its value equals 0. But I can't solve this problem. I used markov's inequality But I can not ...