Tagged Questions
5
votes
1answer
26 views
Prove that expected value of X is greater than Y, if given that $P(X\ge Y)=1$
I have to prove that $E(X)$ (Expected Value of a random variable X), is greater than $E(Y)$, if given that $P(X\ge Y)=1$.
my thoughts so far:
I know from the $P(X\ge Y)=1$ statement, that the values ...
0
votes
1answer
26 views
Examples of convergence of random variables
First, let's recall the definitions of 4 different types of convergence:almost surely, in $r$th mean, in probability and in distribution:
$X_n\xrightarrow{a.s.}X$ if $\{\omega \in ...
0
votes
1answer
26 views
Proof of Bienayme Inequality
I have a bit of trouble about the proof of Bienayme Inequality.
Bienayme Inequality is as follows:
If X has mean $\mu$ and variance $\sigma^2$, then
$$\mathbb{P}\left(\frac{|X-\mu|}{\sigma}\ge ...
0
votes
0answers
17 views
Marginal Pdfs for Continuous Random Variables
http://oi42.tinypic.com/ddyjph.jpg
this problem is confusing me, i know how to start it, we need to find $f_Y(y)$ so we integrate with respect to x and i get $-2e^{-x}e^{-y}|^y_0$ which then should ...
1
vote
2answers
49 views
What exactly does this physically mean?
Let X(w) be a real random variable on ($\Omega$ , P). The image X($\Omega$) the set of all the values X(w) can take ,written $\Omega^{X}$. For any set $ B \subset \Omega^{X}$ the probability of the ...
0
votes
2answers
30 views
Show that $-Z$ is also a standard normal random variable.
Show that $-Z$ is also a standard normal random variable; that is, show that $P[-Z < x] = P[Z < x] \,\forall x.$
1
vote
1answer
19 views
Poisson distribution question, tips needed!
A car dealership opens every day with a fresh stock of $A$ cars. Let $N$ be the r.v. corresponding to the number of purchases per day. Suppose $N$ is distributed according to the Poisson distribution ...
1
vote
2answers
30 views
Checking independence of random variables.
I'm revisiting the coupon collector's problem and I'm not sure how to prove that my variables are independent. Here's what I have:
Let $X$ denote the number of tries required to collect all the ...
0
votes
1answer
26 views
Lottery Competition
Suppose I want to hold lottery competition. I just want two people to win the lottery and there are 100 people buying my lottery. What should the maximum probability (of each player wining the ...
0
votes
1answer
58 views
Computing PDF of Products of Two Random Variables
I've been stuck on this problem for some days. I'm hoping someone would help by chipping in a few comments. I have two i.i.d. r.v.:
$$
f_X(x)=\frac{\left(1-e^{-\frac{x}{\alpha }}\right)^{\tilde{r}-1} ...
1
vote
0answers
26 views
P.d.f of a discrete fourier transform of binary variables
Let $\{a_n\}$ be a set of $N$ "binary" random variables uniformly distribuited in $\{-1,1\}$.
The discrete fourier transform is defined
$b_k=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} a_n e^{-2 \pi i k n ...
1
vote
1answer
39 views
Inequality concerning the pairwise correlation coefficients of three random variables
I was asked to prove:
The correlation coefficients, $\rho_{12}$, $\rho_{23}$, $\rho_{13}$ between three random variables $X_1$, $X_2$, $X_3$ obey
...
0
votes
0answers
37 views
Random Walk, Coin flip game [closed]
Consider the coin flipping game, where player $A$ pays $B$ \$1 for each Heads, and vice versa for each Tails. (The coin is unbiased here.) Let $X_1$ be the random variable recording the first time ...
3
votes
1answer
57 views
“Square root” of a normal RV?
Say $X_1,X_2$ are independently drawn from the same distribution (call it $X$) and that their product, $X_1X_2$ falls on a standard normal distribution.
Is it possible to get a pdf or cdf for $X$?
...
-4
votes
0answers
23 views
Symmetric random variable
If $X$ and $Y$ are two independent random variables where $Y$ is symmetric about $0$.Let $U=X+Y$ and $V=X-Y$ then $U$ and $V$ have the same distribution.
0
votes
1answer
26 views
How can I calculate the probality of to get a number with 15 digits lenght repeat
A number with nineteen digits is generated randomically by a particular system and it's guaranteed that every number generated is unique (by the system provider).
If I chunk this number and get the ...
0
votes
2answers
25 views
max of two random variables inequality
Let $X_1$, $X_2$, $Y_1$ and $Y_2$ be random variables.
Suppose we have $\mathbb{E}X_1 \geq \mathbb{E}Y_1$ and $\mathbb{E}X_2 \geq \mathbb{E}Y_2$, is it necessarily the case $\mathbb{E}\max(X_1,X_2) ...
0
votes
1answer
41 views
Different interpretations of indicator random variable
In a Probability book by Karr, the Indicator random variable is defined as follows:
Indicator random variable. The indicator function of an event A is a random variable: for each B,
{$1_A\le t$} = ...
1
vote
4answers
69 views
What does it mean to integrate with respect to the distribution function?
If $f(x)$ is a density function and $F(x)$ is a distribution function of a random variable $X$ then I understand that the expectation of x is often written as:
$$E(X) = \int x f(x) dx$$
where the ...
2
votes
2answers
30 views
Variance of derangements
Suppose I choose a random permutation on n numbers. It is easy to prove that the mean of the number of fixed points (i.e. the numbers that get mapped to themselves) is 1. Is there an easy (constant) ...
2
votes
3answers
67 views
Homework Question. Joint Probability Distribution.
Here is the question.
The joint PDF of X and Y is given by $f_{XY}(x,y) = {\frac 14} e^{-|x|-|y|}$. Find $P(X \le 1 ,and, Y \le 0)$
Solving the problem I first found the marginal probabilities of X ...
0
votes
0answers
30 views
Homework Help. Probability Density Functions.
$X$ is $N(10,1)$. Find $f(x|(x-10)^2 < 4)$
This is a homework question. I can only figure out that X is normally distributed with mean 10 and variance 1.
Can you please explain what is meant to ...
0
votes
1answer
39 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 ...
2
votes
1answer
49 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 ...
4
votes
2answers
133 views
Confidence interval of a random variable with infinite mean. (St. Petersburg paradox)
Let $X_i$ be random (independent) discrete variables such that $$\forall k\ge 0 \quad P(X_i=2^k)=2^{-(k+1)}$$
$$\begin{array}{c||ccccccc}
v & 1 & 2 & 4 & 8 & 16 & 32 & ...
1
vote
1answer
34 views
Bound of the variance of a random Variable
I am having trouble trying to prove that given a random variable $Y$ where $0 \lt m_1 \lt Y \lt m_2 < \infty$, where $m_1$ and $m_2$ are constants the
$\displaystyle Var(Y) \le \frac{(m_2 - ...
0
votes
1answer
33 views
Question on $\lim\limits_{n\to\infty} P(|Y_n| \geq c) = \lim\limits_{n\to\infty} \frac 1n$
Consider a sequence of discrete random variables $Y_n$ with the following distribution:
$$P(Y_n = y) = \begin{cases} 1 - \frac 1n, & \text{for } y = 0, \\ \frac 1n, & \text{for } y = n^2, \\ ...
1
vote
2answers
43 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 $c > 0$
$$\lim_{n\to\infty} P(|X_n - 0| > c) = ...
1
vote
1answer
27 views
probability: random variable
From the Ross book ex.13 chapter 4:
A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to sale with probability $0.3$ and the second will lead ...
-1
votes
0answers
48 views
Problem about The Strong Law of Large Numbers
The fortune $X_n$ of a gambler evolves as $X_n = Z_n X_{n−1}$,
where the $Z_n$ are independent identically distributed random variables with PMF
$$
p_Z(z) = 1/3 \text{ for } z = 3, \quad p_Z(z) = 2/3 ...
-2
votes
1answer
28 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
32 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
41 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} $ ...
1
vote
2answers
44 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
102 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
47 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
35 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
45 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
0answers
29 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
43 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
38 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
1answer
24 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 ...
2
votes
2answers
50 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
48 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
55 views
An example that shows weak law of large number fails and a question about it.
$X_n$ is a sequence of independent random variables with $\mathbb{P}[X_n=n^2-1]=n^{-2},\mathbb{P}[X_n=-1]=1-n^{-2}$ and $Var[X_n]$ is unbounded. Set $S_n=X_1+...+X_n$. Prove that $\frac{S_n}{n} ...
1
vote
4answers
54 views
Distribution of a random variable
$X_1$, $X_2$, $X_3$ are independent random variables, each with an exponential distribution, but with means of $2.0, 5.0, 10.0$ respectively. Let $Y$= the smallest or minimum value of these three ...
1
vote
1answer
26 views
Rademacher random variables in terms of Bernoulli
I've found out that Rademacher random variables and Bernoulli random variables plays an important role in Probability theory. I am wondering how they are connected. For example,
Let $r_i, i=1, ...
1
vote
2answers
51 views
Definition of atomic $\sigma$-field.
Reading an article in probability theory I faced with phrase atomic $\sigma$-field. I tried to search for the definition, but google doesn't give any meaningful result. As a result I'm looking for the ...
3
votes
0answers
60 views
Expected value with a kronecker product and Gaussian distributional assumption
What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $ is a random variable? The kronecker product ...
