Questions about maps from a probability space to a measure space which are measurable.
1
vote
1answer
18 views
$X_n \sim \text{Exponential}(\lambda_n)$, independent, $\sum 1/\lambda_n = \infty$, then, $\sum X_n=\infty$ a.s.
Let $\{X_n\}$ be a sequence of independent Exponential random variables with mean
$$
E(X_n)=\frac{1}{\lambda_n},
$$
where
$$
0 < \lambda_n < \infty.
$$
If
$$
\sum \frac{1}{\lambda_n} = \infty,
...
2
votes
1answer
22 views
How to find the variance of a “function”
Here's the problem:
I have $n$ i.i.d. rvs. $X_1,\ldots,X_n$. All coming from $ N(\theta, 1)$.
$$\bar{X} = \frac{1}{n} \sum_1^n X_i$$
$c$ is a constant.
How would I find:
...
1
vote
2answers
32 views
Can a Covariance matrix have negative elements?
I have a $N \times N$ covariance matrix $C$ of a multivariate Normal distribution. Can any of the elements of the Covariance matrix $C$ be negative for a real-valued distributions ?
0
votes
1answer
29 views
Check if discrete random variables are independent
I came across this probability question while checking the homework of one of the probability courses at my uni. It's easy but still interesting for very beginner in probability.
Suppose we have two ...
1
vote
1answer
39 views
Independence of conditional random variables
Assume I have two random variables $A$ and $B$, which are not independent. In my particular case they will be values of a stochastic process at two given points in time, where $A$ is observed at an ...
2
votes
2answers
78 views
+100
Confidence interval of a random variable with infinite mean. (St. Petersburg paradox)
Let $X_i$ be random (independent) discrete variables such that $$\forall k\ge 0 \quad P(X_i=2^k)=2^{-(k+1)}$$
$$\begin{array}{c||ccccccc}
v & 1 & 2 & 4 & 8 & 16 & 32 & ...
2
votes
0answers
23 views
Gambling Game: Martingales
This is a multipart question; if there's a strong preference for breaking this into separate questions I'll do that.
Imagine a game between a gambler and a croupier.
Total capital in the game is ...
1
vote
1answer
33 views
Bound of the variance of a random Variable
I am having trouble trying to prove that given a random variable $Y$ where $0 \lt m_1 \lt Y \lt m_2 < \infty$, where $m_1$ and $m_2$ are constants the
$\displaystyle Var(Y) \le \frac{(m_2 - ...
1
vote
0answers
24 views
convergence of discrete random variables with finite entropy
Let $Z$ be the set of discrete random variables on some probability space. Define the quantity $d(X_1,X_2)=h(X_1 \mid X_2)+h(X_2 \mid X_1)$ between two random variables $X_1, X_2 \in Z$. For $X \in Z$ ...
2
votes
1answer
31 views
Conditional Expectation With Respect to Filtration
I'm trying to solve this problem:
Let $\left(X_n\right)_{n\geq 1}$ be independent such that $\mathbb E\left(X_i\right)=m_i$ and $\mathrm{Var}(X_i)=\sigma_{i}^{2}$ for $i\geq 1$. Let $\displaystyle ...
0
votes
1answer
32 views
Question on $\lim\limits_{n\to\infty} P(|Y_n| \geq c) = \lim\limits_{n\to\infty} \frac 1n$
Consider a sequence of discrete random variables $Y_n$ with the following distribution:
$$P(Y_n = y) = \begin{cases} 1 - \frac 1n, & \text{for } y = 0, \\ \frac 1n, & \text{for } y = n^2, \\ ...
1
vote
2answers
40 views
Is it true that $\lim_{n\to\infty}E[X_n] = 0$ if $X_n\to 0$ in probability?
Is there any counter example that:
Let $X_1, X_2,\dots$ be a sequence of random variables that converge to $0$ in probability. That is, for any $c > 0$
$$\lim_{n\to\infty} P(|X_n - 0| > c) = ...
1
vote
1answer
26 views
probability: random variable
From the Ross book ex.13 chapter 4:
A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to sale with probability $0.3$ and the second will lead ...
0
votes
2answers
43 views
Probability distribution of a sum of uniform random variables
Given
$$X = \sum_i^n x_i$$
,where $x_i \in (a_i,b_i)$ are independent uniform random variables, how does one find the probability distribution of $X$.
0
votes
0answers
36 views
+50
Relation between factor graph and conditional probability distribution
First, I'm from computer science. I don't know how to say this problem in a mathematical way. So please bear with me.
The question
Let say I have a factor graph illustrated in the figure.
The ...
5
votes
0answers
58 views
What is the distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Gaussian but correlated?
If $Z = \sqrt{X^2+Y^2}$, and $X$ and $Y$ are zero-mean i.i.d. normally-distributed random variables, then $Z$ is Rayleigh distributed.
What is the distribution of $Z$ if $X$ and $Y$ are correlated ...
-1
votes
0answers
34 views
Problem about The Strong Law of Large Numbers
The fortune $X_n$ of a gambler evolves as $X_n = Z_n X_{n−1}$,
where the $Z_n$ are independent identically distributed random variables with PMF
$$
p_Z(z) = 1/3 \text{ for } z = 3, \quad p_Z(z) = 2/3 ...
-2
votes
1answer
24 views
Problem about Convergence in Probability (3)
Let $X_1,X_2,\dots$ be a sequence of random variables that converge to $0$ in probability.
That is, for any $\varepsilon > 0$, $\lim\limits_{n\rightarrow +\infty} Pr(|X_n-0|>\varepsilon) = 0$
...
-1
votes
2answers
30 views
Problem about convergence in Probability (2) [duplicate]
Let $X_1,X_2,\dots$ be a sequence of random variables with
$$
\lim_{n\rightarrow+\infty}E\left[\left|X_n\right|\right]=0
$$
Is it true or false that the sequence $X_n$ must converge to $0$ in ...
0
votes
1answer
33 views
Convergence to a constant in probability but not almost surely
Please give a example that a sequence of random variables that converge to a
constant $c$ in probability but fail to converge to $c$ with probability $1$.
Thanks very much.
0
votes
0answers
29 views
Random Poisson Sum of Random Variables conditioned on the Poisson upper limit
I am trying to get a closed form expression for the expected value of the following summation of RVs:
$$
\sum_{y=1}^{Y} f_X(x|n)*P[n=y]
$$
where Y is Poisson distributed with parameter λ and ...
0
votes
1answer
29 views
Random Poisson Sum of Random Variables with known distribution
I am trying to get a closed form expression for the expected value of the following summation of RVs: $\sum_{i=1}^{Y} X_{i}$, where $Y$ is Poisson distributed with parameter $\lambda$ and $ X_{i} $ ...
0
votes
1answer
44 views
Conditional Density, Additive Gaussian
A signal, X, is a random variable with the following density function:
$$f_X(x) =\begin{cases} \frac{3}{25}(x-5)^2, &0 \le x \le 5\\0, &otherwise \end{cases}
$$
The signal is transmitted ...
1
vote
2answers
38 views
Must the sequence $X_n$ converge to $0$ in probability?
Let $X_1, X_2,\dots$ be a sequence of random variables with
$\lim_{n\to +\infty} E[|X_n|] = 0$.
Is it correct or wrong that the sequence $X_n$ must converge to $0$ in probability?
0
votes
0answers
31 views
a sequence of random variables that converge to a constant c in probability but fail to converge to c with probability 1?
Any example that a sequence of random variables that converge to a
constant c in probability but fail to converge to c with probability 1?
0
votes
2answers
57 views
Proof of analogue of the Cauchy-Schwarz inequality for random variables
The Cauchy-Schwarz inequality tells us that for two vectors $u$ and $v$ in an inner product space,
$$\lvert (u,v)\rvert \leq \lVert u\rVert \lVert v \rVert$$
with the equality holding iff one vector ...
1
vote
3answers
25 views
Standardizing A Random Variable That is Normally Distributed
To standardize a random variable that is normally distributed, it makes absolute sense to subtract the expected value $\mu$ , from each value that the random variable can assume--it shifts all of the ...
0
votes
1answer
29 views
Using Chernoff bound to analysis the Lazyselect algorithm
It's my homework of the course of randomized algorithm. In the textbook (Randomized Altorithm by Rajeev Motwani et.al.), the author analyzed this algorithm using Chebyshev bound, but are there any ...
1
vote
1answer
27 views
Convergence of sum of random variables
Let $X_n$, $n\geq 0$, be i.i.d. random variables such that: $\mathbb E(X_1)=0$, and $0<\mathbb E(|X_1|^2)<\infty$. Given that $\alpha >\frac{1}{2}$, I need to show that ...
1
vote
1answer
34 views
Uniform convergence of a series through induction principle
I'm working on this paper. Can you please explain me the following passage of the proof that the series (2.9) converges uniformly?
Given:
$$
|\delta_{k+1}| \le ...
1
vote
1answer
41 views
maximum of exponentials
I am really having difficulties to prove the following: consider $X_1,\dots, X_n$ all exponentially distributed with rate $\lambda$ (i.e. $X_i \sim exp( 1/\lambda)$). Then argue that we can write
...
0
votes
1answer
39 views
Expected value of minimum of exponentials
I am not sure of the following. I have $(i-1)$ exponential random variables with rates $\theta$ and $\mu$ and I want the expected value that the particular $\mu$ random variable is the minimum. Think ...
0
votes
2answers
28 views
Random Number generation
I need to find a method to solve this problem:
give a method for generating a random variable having distribution function
$$F(x) = \frac{1}{2}(x + x^2) \hspace{1cm} 0 < x < 1$$
0
votes
0answers
21 views
Skewness of a sum with a positive summand
Let $X$ and $Z$ be two random variables with finite third moment, and let $Z>0$. Is it true that the skewness of $X+Z$ is greater or equal than that of $X$? Such a relation clearly holds for the ...
0
votes
1answer
53 views
Conditional expectation $E[X|Y<y]$
Let $X:\Omega \to \mathbb{R}$ and $Y:\Omega \to \mathbb{R}$. Consider the joint pdf $f_{XY}(x,y)$ and univariate pdfs $f_X(x)$ and $f_Y(y)$.
Is it true that $E[X|Y < y]$ equals:
$$ \displaystyle ...
2
votes
2answers
29 views
Finding sequences such that function of sum of r.v's is martingale
Let $\left(X_n\right)_{n\geq 1}$ be independent such that $\mathbb E\left(X_i\right)=m_i$ and $\mathrm{Var}(X_i)=\sigma_{i}^{2}$ for $i\geq 1$. Let $\displaystyle S_{n}=\sum_{i=1}^{n}X_i$ and ...
2
votes
1answer
40 views
Nonnegative Superharmonic Function is Constant for $d>2$?
I have to do the following:
Let $\alpha>0$ be fixed, $(X_i)_{i\geq 1}$ be i.i.d., $\mathbb R^{d}$-valued random variables, uniformly distributed on the ball $B(0,a)$. Set $\displaystyle ...
2
votes
2answers
33 views
A question involving Invariant Set in ergodic theorem
I have a question about the invariant set in the ergodic theorem, I am wondering if anyone could give me some help or hint. In the measurable space (X, $\Sigma$) and consider a measurable self map T, ...
0
votes
1answer
40 views
Is the relation of having positive covariance well behaved with respect to taking the inverse?
Let $X$ and $Y$ be two random variables, $X$ strictly positive. Assume that Cov$(X,Y)>0$. Does this imply that Cov$(1/X, Y)<0$?
I know that being positively correlated is not a transitive ...
1
vote
1answer
34 views
Are the multiplications of i.i.d random variables , i.i.d?
If we know that $X_1$ and $X_2$ are i.i.d random variables, and $Z_1$ and $Z_2$ are also i.i.d random variables, can we say $X_1Z_1$ and $X_2Z_2$ are i.i.d random variables too? suppose that $X_1$ ...
0
votes
0answers
16 views
Is $f_{\Theta|Z}(\theta|z)$ Gaussian when $Z = \theta^3 + V$, and given that $\Theta$ and $V$ are Gaussian?
$\Theta$ and $V$ are zero mean Gaussian random variables with variances $\sigma_\Theta^2$ and $\sigma_V^2$.
A third random variable $Z$ is defined as:
$$
Z = \Theta^3 + V
$$
Is ...
2
votes
1answer
11 views
Equivalent condition of uniform integrability of a sequence of random variables
Here's the definition I have for a sequence of random variables to be uniformly integrable:
$(1)$ A sequence of random variables $X_1, X_2, \ldots$ is uniformly integrable (U.I.) if for every ...
0
votes
1answer
21 views
PMF and variance
two fair three sided dice are rolled simultaneously. let X be the sum of two rolls. calcilate PMF(probability mass function) and variance of X.
If any body has solved examples link on the topic of ...
1
vote
0answers
38 views
About Strict Stationary of AR(1) Sequence
The usual Auto regressive process considers the time t from negative infinity and positive infinity, but what if we restrict our time to strict positive space, do we still have our stationary result?
...
0
votes
1answer
18 views
discrete random variable PMF
suppose You rented a house and realtor gave you 5 keys, one for each of the 5 doors of house. unfortunately all keys look identical. so to open the front door, you try them at random.
=> Find the ...
2
votes
2answers
44 views
Finding variance given expected value
How would one find the variance of a random variable, $X$ given that it is composed of say two dependent random variables $Y_1$ and $Y_2$ (so $X = Y_1 + Y_2$), each with expected value of .5 and ...
1
vote
1answer
28 views
Is $\{ r \mapsto X_{r} \text{ is continuous for all } s < t \} \in \sigma(X_s : s \leq t)$?
If $(X_t)_{t \geq 0}$ is a stochastic process, is $\{ r \mapsto X_{r} \text{ is continuous for all } $s < t$ \} \in \sigma(X_s : s \leq t)$?
I'm particularly interested in the case where $X_t$ is ...
1
vote
0answers
15 views
Strong Law of Large Numbers a.s. sense implies $L^1$ sense?
I have to show that the strong law of large numbers in the almost sure sense implies the strong law of large numbers in the $L^1$ sense. I'm not sure what's being asked, can anyone give me a hint or ...
1
vote
1answer
42 views
Question regarding exchangeable sequence of random variable
I have a question regarding the exchangeable random variable
consider ($x_{m}$) be a (infinite) sequence of random variable, if ($x_{m}$) is stationary, does it implies that ($x_{m}$) is ...
0
votes
1answer
36 views
Convergence in L 1
Prove that if the $X_l$'s are i.i.d. and in $L^1$, then $(n^{-1} \sum^n_{k=1} X_k)_{n \ge 1}$ is uniformly integrable .
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