Questions tagged [random-variables]
Questions about maps from a probability space to a measure space which are measurable.
12,207
questions
1
vote
0
answers
23
views
Interchange of limit and discontinuous function
I encountered the following statement in some lecture notes and it does not seem right to me. I just wanted to have my thoughts verified or falsified.
$(X_n)_{n\in\mathbb{N}}$ is a sequence of random ...
- 5,725
0
votes
1
answer
29
views
Confusion in solving
An apartment building has 11 residents. There are 7 residents on the second
floor and 4 residents on the third floor. Six people use the elevator at random to go
to their apartments. What is the ...
- 141
0
votes
2
answers
20k
views
Find cdf given pmf
The Question
Let $p_X(x)$ be the pmf of a random variable $X$. Find the cdf $F(x)$ of $X$ and sketch its graph along with that of $p_X(x)$ if
a) $p_X(x) = 1, x = 0$, zero elsewhere
b) $p_X(x) = 1/3,...
- 141
1
vote
1
answer
219
views
How does the notation of second order derivative read in plain English?
How does the following formula read in the context of Random Variable and probability?
$$f_{XY}(x, y) = \frac{\partial^2 \cdot F_{XY}(x, y)}{\partial x \, \partial y}$$
Since, the plot of a ...
- 1,641
0
votes
1
answer
31
views
Randomly select an element from a list [closed]
Suppose I have a list that contains N elements, every time I randomly select an element from it and mark it as visited. What is the expected number of selection that I have to make until all the ...
- 3
1
vote
1
answer
178
views
Absolute convergence of a random variable in a countable space
I am reading Jacod and Protter's Probability Essentials and I am struggling to understand the following string of inequalities:
where $L^1$ is the space of real valued random variables on ($\Omega$, ...
- 57
3
votes
2
answers
779
views
Bivariate Poisson-Binomial distribution.
Suppose you have $100$ coins whose probabilities of obtaining the outcome "head" are $p_1,\ldots,\,p_{100}$. These probabilities are not necessarily equal each other. Consider the following random ...
- 131
1
vote
2
answers
93
views
How can I draw a distribution table for, say, 4 Random Variables?
I know that a bi-variate distribution table would be a matrix which is very easy to draw/represent.
Suppose, we roll 4 six-sided dies simultaneously, and we want to draw/represent a multivariate ...
- 1,641
4
votes
0
answers
259
views
Measurability of random variable wrt pullback $σ$-algebra
Three Polish spaces $A$, $B$ and $R$ are given, and are equipped with their Borel $σ$-algebras $\mathcal B_A$, $\mathcal B_B$ and $\mathcal B_R$. Let $f\colon A \to B$ and $g\colon A \to R$ be $(\...
- 41
0
votes
1
answer
368
views
Finding covariance using $E[X]$ and $E[Y]$
I have a table
where I have found $E[X] = 2.6$ and $E[Y] = 2.44$.
\begin{align*}
E[X] &= 2 \times 0.4 + 3 \times 0.6 = 2.6 \\
E[Y] &= 2 \times 0.56 + 3 \times 0.44 = 2.44
\end{align*}
I ...
6
votes
3
answers
3k
views
Question on coin tossing - probability
When considering an infinite sequence of tosses of a fair coin, how long will it take on an average until the pattern H T T H appears?
I tried to break the problem into cases where ultimately the ...
- 285
1
vote
0
answers
385
views
Evaluating conditional probability for a cipher text given a plain text message
I'm not posting this on Cryptography as I have not even understood the math for this, and this had nothing to do with the cryptographic definition inside. Caesar cipher should give enough context, if ...
- 175
5
votes
1
answer
569
views
An event with unit density still has zero information, despite not being an event that is guaranteed to occur.
My textbook says the following in a section on information theory:
The basic intuition behind information theory is that learning that an unlikely event has occurred is more informative than learning ...
- 4,182
1
vote
1
answer
46
views
Suitable distribution to make a problem tractable
Consider a set of random variables $x_i$ for $i\in\{1,\dots,n\}$. They are all drawn in an iid way from some distribution $F$. Now consider the function
$$
Y(x)=A\log\left(\sum_{i}B_{i}x_{i}^{a}\...
- 1,398
0
votes
2
answers
44
views
Normal random variable quiz
I must resolve a quiz about normal random variable.
"Let X be a normal random variable of mean 3 and variance 4. Prove that
$P((X-2)^2 > 4) =0.3753$
So, I made:
$Z = Norm(1,4)$
Using formula $...
- 17
0
votes
0
answers
32
views
What functions can be the limit of the CDF of the average of n IID random variables as n grows without bound?
Suppose that we have a probability distribution $D$. Define $F_n$ to be the cumulative probability distribution function of the average of $n$ independent random variables drawn from $D$, and define $...
0
votes
2
answers
1k
views
PDF of |X-Y| where X,Y ~ exp($\lambda$)
I need to find the PDF of $Z=|Y-X|$ given that $X,Y\sim exp(\lambda)$ and both independent.
What I did (I want to use the CDF and not convolutions):
$f_{X,Y}(x,y)=\cases{\lambda^2 e^{-\lambda(x+y)} &...
- 509
2
votes
1
answer
545
views
Expectation of product of more than two independent random variables
I am trying to determine whether independence of random variables changes when multiplied with other, potentially dependent variables. The question isn't in a measure-theoretic context.
In particular,...
- 991
0
votes
0
answers
69
views
Sum of random number of i.i.d. random variables
I encountered an unusual problem and am struggling to prove the following result.
Let $T$ be a positive random variable, and $\{x_i\}$ be a sequence of i.i.d. strictly positive random variables with ...
- 642
1
vote
3
answers
289
views
Inequality for the maximum of the absolute value of two normal distributed random-variables
I would like to show following statement:
For $M\geq 2,\ X_1,\dots,X_M\sim^{iid}\mathcal{N}(0,1)$ independent, it holds $P(\max_{i=1,\dots,M}\lvert X_i\rvert\geq y)\leq Me^{-y^2/2}$.
I think it ...
user592521
0
votes
0
answers
28
views
Simplification of \begin{equation} P(Z<\theta)=P(\sum_{i=1}^{m}(\max\{\log(a+cX_i),\log(a+cY_i)\})<\Theta). \end{equation}
Let $X_i$ and $Y_i$ two random variables with parameter $\alpha_i$ and $\beta_i$.
Let $m>0,c,b$ a positive numbers.
We define $Z$ a random variable by
$$
Z= \sum_{i=0}^{m}(\max\{\log(a+cX_i),\...
- 168
0
votes
1
answer
216
views
Sum of Moment Generating Functions determined by a Random Variable
I have a question that is stated as follows:
Two random variable $P$ and $Q$ have MGF's:
$$M_P(s) = \left(\frac{1}{3} + \frac{2}{3}e^s \right) ^{10}$$
$$M_Q(s) = \frac{\frac{1}{5}4e^s}{1-\...
- 47
1
vote
1
answer
420
views
Explanation of $\lim\sup$ of a sequence of random variables in measure theory
The definition I have been given of the $\limsup\limits_{n \to \infty} Y_n$ where the $Y_n$ are random variables is that it is another random variable defined as $(\limsup\limits_{n \to \infty} Y_n)(\...
- 1,138
1
vote
1
answer
201
views
Does there exist random variables such that any strict subset of $\{ X_1, X_2, ... X_n\}$ is independent but $X_1, X_2, ... X_n$ are not independent?
Let $n$ be any positive integer larger than $1$.
I want to construct random variables $X_1, X_2, ... X_n$such that any strict subset of $\{ X_1, X_2, ... X_n\}$ is independent but $X_1, X_2, ... X_n$ ...
- 861
3
votes
1
answer
346
views
Intuition behind a random variable being a measurable function with a specific example.
I'm just starting to learn measure theory and we have defined a random variable $X$ as a measurable function from some set $\Omega$ to some other set $E$ which is usually $\mathbb{R}$. We defined $\...
- 1,138
2
votes
1
answer
150
views
Measure Theoretic Probability - Inequalities involving multiple random variables meaning
I'm a little bit unclear on what certain inequalities of random variables refer to. For instance, if we have random variables $X,Y$ defined on the same probability space $(\Omega,\mathcal{F},\mathbb{P}...
- 489
2
votes
1
answer
2k
views
Upper bound of sub-gaussian norm of bounded random variable?
I am reading the High-Dimensional Probability by Dr.Roman Vershynin , where I stuck on some statement at page 28. where state as below:
Any bounded random variable $X$ is sub-gaussian with: $$\...
- 642
0
votes
2
answers
220
views
How accurate is it to generate random number using the inverse-cdf-method?
I am reading the following approach of generating a random number from a given probability distribution.
https://blogs.sas.com/content/iml/2013/07/22/the-inverse-cdf-method.html
I understand that ...
- 113
0
votes
1
answer
114
views
Calculating the support for the CDF: $Y = X^2, X \geq 0$ and $- 2X, X < 0$
I am trying to understand how to determine the limits of the integral for calculating the cdf of a fairly simple transformation, but struggling conceptually to understand the support.
Would it be ...
1
vote
1
answer
934
views
Time interval distribution between events
I am standing in front of a restaurant. Every time a customer goes in the restaurant, I want to make a prediction of dt (dt is ...
- 113
4
votes
0
answers
58
views
Distribution of implicitly defined random variable
Let $f$ be a function of 2 variables, $x$ and $a$. I do not specify a regularity class for $f$ as I don't want to restrict the question. We can assume that $f$ is as regular as we need (even analytic)....
- 831
2
votes
1
answer
146
views
Expectation $T(T-1)$ where $T$ is the sample mean of i.i.d. Bernoulli random numbers
I want the expectation of $T(T-1)$ where $T$ is the sample mean of i.i.d. Bernoulli random variables with parameter $p$.
Using the linearity property, we have
$$\mathbb E\{T(T - 1)\} = \mathbb E(T^2)...
- 313
4
votes
0
answers
78
views
Expectation of $\max_{j=1,\dots,M}X_j^2$ with normal distributed $X_j$
This is my first question here and I have thought about it for a long time. I found following Lemma in a paper:
Consider the independent random variables $X_1,\dots,X_M \sim \mathcal{N}(0,1)$. Then ...
user592521
0
votes
1
answer
80
views
Describe the induced probability $P_X(D)$ on the space $\mathbb{D}=\{0,1,2,3,4\}$ of the random variable $X$.
I know that $P_X(D)$ is determined if I find $P_X(\{d_i\})$ for each $d_i\in\{0,1,2,3,4\}$. So $P_X(\{0\})=P[X^{-1}(\{0\})]=\{\text{
the cards that are neither ace nor king nor queen nor jack} \}$, ...
- 485
1
vote
1
answer
1k
views
Find the probability that the passenger waits more than 10 min for the bus
Buses arrive at a bus stop at 15 min intervals starting at 7 am assume that a passenger arrives at the bus stop at random time X (given in minutes after 7 am ) with PDF
$$f_X(x) =
\begin{cases}
\...
3
votes
1
answer
134
views
Conditional expectation and optimization
Let $x$ and $z$ be two real random variables with unknown joint distribution but satisfying the following relations:
\begin{equation}
E\{x|z\}=\mu_0 \triangleq \mu(\boldsymbol{\theta}_0,z),
\end{...
- 319
2
votes
3
answers
186
views
If $E e^{\theta ^{2} X^{2}} \leq e^{c\theta^2}$ for every $\theta$, then $X$ is almost surely bounded
The original problem states as below:
Suppose some random variable $X$ satisfies $\DeclareMathOperator*{\E}{\mathbb{E}} \E e^{\theta ^{2} X^{2}} \leq e^{c\theta^2}$ for some constant $c$ and $\...
- 642
0
votes
1
answer
70
views
Two dependent random variables with $E[XY]=E[X]E[Y]$ while $E[X] \not = 0$ and $E[Y] \not = 0$. [closed]
Please, help me find two dependent random variables with $E[X] \not = 0$, $E[Y] \not = 0$ and $E[XY] = E[X]E[Y]$.
- 11
1
vote
2
answers
422
views
Understanding vector interpretation of random variables.
I am reading some notes on Conditional Expectations as Orthogonal Projections (Page 60 - Section 3.5.2.3). There, they have an example of two biased coin, which I paraphrase.
Let $X$ denote a ...
- 2,618
0
votes
2
answers
489
views
Player 1 and Player 2 compare their numbers, and the player with the higher number wins the round.
Five distinct numbers are distributed to players numbered 1 through 5. They play a game as follows:
Player 1 and Player 2 compare their numbers, and the player with the higher number wins the round. ...
- 145
0
votes
1
answer
234
views
Expected value of power of sum of square of dependent Gaussian
So, I have a set of dependent Gaussian RVs $\{X_k\}_{k=1}^{N}$ with known joint PDF (zero mean and given covariance matrix). I'm interested in whether we can compute the quantity:
$$ \mathbb{E}\left[\...
- 131
1
vote
1
answer
55
views
Probability of seeing $n$ buses in time $t$
Suppose you go to a bus stop. The inter-arrival time between successive buses is ${\rm Exp}(\lambda)$. You arrive at time zero and leave at time $t$. Let $Y$ denote the number of buses you saw. Find ...
- 1,081
-2
votes
1
answer
169
views
Alternate proof for variance of a Poisson random variable?
We know that Poisson random variable is the limiting case of a binomial random variable with parameters n and p, where n $\rightarrow \infty, p \rightarrow 0$ and $np = \lambda < \infty$ ($\lambda$...
0
votes
2
answers
387
views
Verify that X^2 is a continuous random variable..
Let X be a continuous random variable. Verify that the following function is continuous random variable.
$X^2$.
From the definition of a continuous random variable, I know that $P(X=x)=0$ for all $x \...
0
votes
1
answer
249
views
Probability exponential RV belongs to interval $[n, n+1]$ with $n$ odd
Let $X$ be an exponential random variable with parameter $\lambda > 0$. Calculate the probability that $X$ belongs to the interval $[n,n+1]$ with $n$ odd.
From the solution (Q1) I see we need to ...
- 169
0
votes
1
answer
77
views
What is a measure of the rapidity of change in a random variable?
This question appeared in the GATE exam 2006 ICE paper.
The solution given is,
I don't see how that the integral is the SD of $f(t)$ even if we assume mean is 0 and it's defined over the entire ...
- 578
-1
votes
1
answer
34
views
Given $ X ~ Beta(4,2) $ , How can I calculate $P(X<0.5)$? [closed]
Given $ X ~ Beta(4,2) $ , How can I calculate $P(X<0.5)$ (the probability that X is lesser than $0.5$) ?
- 187
1
vote
1
answer
232
views
What is the distribution of $x^p$ if x is chi-squared distributed?
I am thinking about the powers of random variables that have a skewed distribution to construct confidence intervals for these.
So, consider x follows a chi-squared distribution. What is the ...
- 13
2
votes
0
answers
46
views
Convergence of expectations of product of r.v.
I am trying to prove (or disprove) this proposition:
Let $(X_{n})_{n \ge 0}$ and $(Y_{n})_{n \ge 0}$ be two sequences of random variables, defined on some probability space $(\Omega, \mathcal{F}, \...
- 121
1
vote
1
answer
165
views
Large Deviation Principle
I have been reading Amir Dembo's book, and at the very beginning, I found this result that came across and unfortunately, I cannot derive it by myself. So, I'm looking for some help.
It happens that ...
- 538