Questions tagged [random-variables]
Questions about maps from a probability space to a measure space which are measurable.
1
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0answers
13 views
Connecting two interpretations of the negative binomial distribution
In my probability course, my professor derived the negative binomial distribution by reasoning about the probability that the time of the $k$-th success, $T_k$, takes some value $n$. If $p$ is ...
0
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2answers
39 views
How to sample two values from a random variable X with the lesser to be a random variable Y?
The variable X has pdf $$f(x) = \frac18(6 - x)$$ for $$2 ≤ x ≤ 6$$
A sample of two values of X is taken. Denoting the lesser of the two values by Y, use the
cdf of X to write down the cdf of Y. Hence ...
0
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0answers
19 views
Convergence in probability of running maximum
Suppose we have a sequence of integrable random variables $(X_n)$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ such that $n^{-1}X_n\to 0$ in probability as $n\to\infty$. Suppose further ...
0
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1answer
27 views
find P( X$\lt$ Y) given X and Y are two independent exponential RV
The question is as follows:
You arrive at the post office and as you enter, each of the two clerks, Jim and Jack, starts serving a client. The amount of time needed by Jim to serve his client is X ...
1
vote
1answer
19 views
Is a Poisson r.v.'s parameter a rate $\mu$ or a count $\mu t$?
Let's say I want to model the arrivals of some quantity of interest, say customers coming to a store. I know that on average, $\mu$ customers arrive in on hour. My understanding is that if $N$ is the ...
0
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1answer
19 views
Let $(X,Y)$ be a uniformly chosen point of a region $A$, given $A$ compute $EX$
Let $(X,Y)$ be a uniformly chosen point of a region $A \subset \mathbb{R}^2$
Given we have the following joint pdf:
$$ f(x,y) = \begin{cases} \dfrac{a}{ \text{area of}~ A} & (x,y)\in A \\ 0 &...
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0answers
12 views
Characteristic function of exponential distributed random variable
Given:
$$f_X(x) = \lambda e^{-\lambda x},\; x\in X$$
Wanted: The corresponding characteristic function $\phi(ju)$.
\begin{align}
\phi(ju)&=\mathbb{E}(e^{j^2ux})\\
&= \lambda \int^{\infty}...
0
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1answer
25 views
let z1 and z2 be independent standard normal RV, find the pdf of $e^{3Z_1+2Z_2}$
Let Z1 and Z2 be independent standard normal random variables. Find the following
The probability density function of $e^{3Z_1+2Z_2}$
The given solutions is as follows:
But what doesn't make sense ...
0
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1answer
18 views
given exponential Cumulative distribution function, finding another Cumulative distribution function with functionl connection
There is given $X$ a random variable with exponential cumulative distribution such that $X~Exp(1)$ so the exponential Cumulative distribution function is:
$P(x\le t)= F_x(t)=(1-e^{-t} , 0\le t) \...
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0answers
14 views
How to minimize piecewise binary function with absolute value threshold
Let $f: \Bbb R^n \rightarrow \Bbb R$ and $\mathbf{g}:\Bbb R^n \rightarrow \Bbb R^n$. Let $\mathbf{a,b}$ be random variables with values in $\Bbb R^n$.
$$f\mathbf{(a,g(b))} =
\begin{cases}
...
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1answer
50 views
Joint distribution of absolute difference and sum of two independent exponential distributions
If $X\sim \rm{Exp}(1)$ and $Y\sim \rm{Exp}(1)$ are two independent random variables.
What is the joint distribution of $U = |X - Y|$ and $V = X + Y$?
I used the Jacobian transformation to obtain ...
0
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0answers
17 views
How do I evaluate the following combination of random variables? Is it martingale?
I'm about to analyse the following expression
$$Z_n:=\prod_{k=1}^n \left(\frac{\frac{Y_k}{\prod_{i=1}^k X_i}}{\sum_{j=1}^k \frac{Y_j}{\prod_{i=1}^j X_i}} \right),$$
where $Y_j$ for all $j\in \mathbb{...
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0answers
20 views
Division of two independent uniformly random variable [duplicate]
Given two independent random variable X and Y which both have uniform distribution over[0,1] I want to calculate PDF of $Z =\frac{X}{Y}$ and here is my solution:
$\int_{-\infty}^{\infty}zf_X(yz)f_Y(y)...
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votes
1answer
10 views
ls it possible to construct discrete r.v.s given expectation and variance?
Suppose there is a discrete r.v.s X, all we know is:
E(X) = 10 and VAR(X) = 2500
Any general way to find PMF of X?
Thanks.
4
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2answers
60 views
Special case of Bertrand Paradox or just a mistake?
I've been working on a question and it seems I have obtained a paradoxical answer. Odds are I've just committed a mistake somewhere, however, I will elucidate the question and my solution just in ...
1
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2answers
35 views
PMF of function of random variables
Let X be a Geom($\frac{1}{2}$) random variable, and define Y=$X^{-1}$
What is the p.m.f. of Y ?
attempt:
pmf of a Geom RV in general form is $p(1-p)^{k-1}$
There is this similar question, not ...
0
votes
2answers
21 views
Shortcut to finding the distribution of a specific random variable
Question: A dice is rolled 3 times. Let X denote the maximum of the three values rolled. What is the distribution of X (that is, P[X = x] for x = 1,2,3,4,6)? You can leave your final answer in terms ...
-1
votes
0answers
22 views
Calculate Mean from the Moment Generating Function ( m.g.f / mgf) of Y = 2X +3
The question is as follows:
No calculators. Let X be a random variable with moment generating function
$M_{x}(t) = \frac{e^{(e^t-1)}}{2e^{-t} -1} \;\;\;\;\;for \;\; \;t<log(2)$
given Y = 2X +3
...
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0answers
18 views
Expressing quadratic form of normal variables in terms of chi-squared variables
On Wikpedia and in the references therein [1,2], it is stated that any quadratic form of normally distributed random variables can be expressed as the sum of many independent non-central chi-squared ...
0
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1answer
24 views
“Normalized” covariance matrix of a Gaussian random vector
Let $X\sim\mathcal{N}(0,I_{d})$. I would like to compute the the following quantity:
\begin{equation}
\mathbb{E}\bigg[\frac{XX^{\top}}{\|X\|_{2}^{2}}\bigg].
\end{equation}
Letting $B=\frac{XX^{\top}}{...
0
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0answers
9 views
Bayesian LASSO: A step within the Gibbs sampler
I'm intending to implement a Bayesian LASSO inside the Gibbs sampler I use to estimate a multivariate time-series model, but I have a doubt about how to draw this step.
The prior is a Double-...
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0answers
25 views
Quick terminology question - “Standardizing” Random Variables
This is a really boring question I apologize. I'm not sure what to call this process, I call it "standardization" however perhaps non beginners would think I'm being sloppy with terminology. For ...
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0answers
33 views
Intuition behind expected value formula? [duplicate]
I just learned the expected value formula for a positive random variable: $E[X] = \int_0^\infty P(X>x)dx$.
I can follow the proof but I'm a little unclear as to what the intuition behind this ...
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0answers
12 views
Variance analysis of the rank of groups of variables
So I am looking to analyse the variance in two groups of random variables.
There are 2 sets of 3 random variables A, B and C and X, Y and Z. If I calculate the numerical rank of these variables ...
0
votes
1answer
18 views
Is this Poisson Process problem worked out correctly?
Calls are received at a company call center according to a Poisson process at the rate of five calls per minute.
(a) Find the probability that no call occurs over a 30-second period.
(b) Find the ...
0
votes
0answers
19 views
Example of higher random vector moments
While reading about random vectors, I learned that...
$$
E\left[\vec{X}\right] = \left[\begin{array}{cccc}
E\left[\vec{X}_1\right] & E\left[\vec{X}_2\right] & \cdots & E\left[\vec{X}_m\...
1
vote
1answer
33 views
Geometric distribution/ Formula
I have a geometrically distributed random variable $X$. I want to calculate $P(X \geq k)$.
I get that $P(X \geq k)= (1-p)^{k-1}$.
Now I want to show that $P(X \geq k+n \mid X\geq n) = P(X \geq k)$.
...
1
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2answers
34 views
Expectation of non-negative random variable vs Truncated random variable
I want to show that E[X|X>t]>=E[X],
where X is discrete non-negative r.v, and t is positive deterministic value.
Some ideas?
Edit: >= instead of >
0
votes
1answer
15 views
Prove that a random variable defined in terms of lim sup of independent random variables is constant
I'm asked to do the following:
Let $(X_n)$ be a sequence of independent random variables. Show that the following random variable is constant almost surely:
$$X = \begin{cases}\exp\{-\limsup_{n \to \...
1
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0answers
17 views
Distribution/Variance of correlated squared normal random variables
If $X_{1}, X_{2}, \ldots, X_{N}$ are identically distributed normal random variables with mean $0$ and variance $\frac{(N+3)D\sigma^{2}}{N}$, then I want to calculate the distribution, or at least the ...
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2answers
28 views
Random variable and probability calculations
X is a random variable that follows a binomial property, with E(X) = 36 and $\sigma$(X) = 3.
Calculate p(X = $10$)
I know that the variance is V(X) = 9 and I know that I have to use the formulas of ...
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2answers
31 views
the difference between independent random variables and independent events, a strange instance
I know the conclusion that if event $A$ and $B$ are independent, $P(AB)=P(A)P(B) $. And furthermore, the two sigma algebra $\sigma(I_A)$ and $\sigma(I_B)$ should be independent, where $I_A:\Omega \...
2
votes
2answers
46 views
Inequality regarding sample mean
I was looking at the book "Asymptotic Theory of statistics and probability, DasGupta A., 2008" and in one point of a proof they use an inequality which I have not been able to understand.
Given that $...
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0answers
33 views
Expected value-random variables
On $\Omega=[0,1]$ we consider $\sigma$-algebra Borel sets and Lebesgue measure. Let $Y(x)=x(1-x)$. Show that for any integrable random variable $X$:
$$E(X\mid Y)(x)=\frac{X(x)+X(1-x)}{2}\quad \text{...
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0answers
36 views
Let $X$ a random variable (such that $E(X)<+\infty$, proving : $\lim_{t \rightarrow -\infty }tP(X<t)=0$
Let $X$ be a random variable, and $E(X)<+\infty$.
I'm stuck on proving $\lim_{t \rightarrow -\infty }tP(X<t)=0$.
But I proved $\lim_{t \rightarrow +\infty }tP(X>t)=0$
and here's my method :...
1
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1answer
17 views
How would one go about proving that two random variables are uncorrelated but not independent? [duplicate]
Suppose $\theta$ is a Uniform random variable on [0, 2$\pi$]. Let $X$ be $cos(\theta)$ and $Y$ be $sin(\theta)$. We have to show that $X$ and $Y$ are uncorrelated but not independent.
My solution:
$...
0
votes
1answer
23 views
If $X$ is an exponentially distributed variable with mean $ \lambda$, $Y=−3\ln(X)$ has Gumbel distribution?
Let X be a random variable which follows an exponential distribution with parameter $\lambda$ ($\lambda>0$), find the distribution of the random variable $Y = −3\ln(X)$.
So this is my answer for ...
2
votes
1answer
50 views
Calculation of expectation of Poisson Process.
This is a problem related to Poisson Process where $\lambda = 2$.
$
E(N_3N_4) \\
= E[N_3(N_4-N_3 + N_3)] \\
= E[N_3(N_4-N_3) + N^2_3)] \\
= E[N_3 - N_0(N_4-N_3) + N^2_3)] \\
= E[N_3 - N_0(N_4-...
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0answers
21 views
Conditional distribution of two random variables.
Suppose I have two continuous random variables $X$ and $Z$, $X$ and $Z$ are not independent. While I know that $Z \sim \mathcal{N}(0, 1)$, I have no information for the distribution of $X$. Now one ...
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0answers
33 views
Are the two random variables independent?
Consider the following two random variables,
In first case you record the number of people arriving at a queue, for a random amount of time. Note, here the arrival of people in the queue is random (...
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0answers
16 views
A die roll and Binomial distribution.
Note: This is not an accurate picture. This picture is given to give a sense only.
Suppose, I assign probabilities to a die according to Binomial distribution. I.e. 1 and 6 have least probability of ...
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0answers
27 views
Estimate number of permutations with unknown number of variables
I need to estimate the number of possible permutations of a problem.
A way to do this, I suspect, is to randomly generate a decent amount of problems, and then find out how many were repeated. If ...
0
votes
1answer
24 views
What does it mean if someone says “this experiment has a X distribution”?
Kindly, help me to clear my understanding of probability distribution.
As far as I understand, probability-distribution means a table/list which lists the probabilities of each outcome of an ...
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1answer
32 views
What could be an example of 'time' in a stochastic process where 'time' is not a natural time?
I need any example of a stochastic process where 'time' is not a natural/real-world element what we generally call the time, and measure it using clocks/watches.
What could be an example of 'time' in ...
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0answers
25 views
What does it mean by 'time' in a random process?
I understand that
A random process is a series of random variables which is composed of
two components:
time
values
Both time and values can be either discrete and ...
1
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0answers
28 views
How to relate between Random Variable, Probability mass/density function and Probability distribution
While studying probability I came across these terms Random variable, probability function, and probability distribution. How to relate these these 3 terms in unfair coin toss problem where head ...
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0answers
19 views
Bernoulli random variables and two player games
With hockey playoffs upon us, I have been thinking about ways of modeling the results of athletic events.
Suppose team A wins 50% of its games against average teams, and team B wins 50% of its games ...
0
votes
1answer
32 views
Calculate the Mutual Information between $X$ and $X^2$, where $X$ is uniformly distributed.
Let us consider two random variables, $X$ and $Y$. Let $X$ be uniformly distributed in $[-1,1]$ and let $Y=X^{2}$.
Is it possible to calculate the Mutual Information between them? E what is the ...
0
votes
1answer
55 views
Proof of “Sum of random variables is a random variable”
There is already a question about it where in a proof is given to show that sum of two random variables X and Y, is a random variable. Calling Z=X+Y, the idea is to somehow show that Z is a measurable ...
1
vote
1answer
19 views
sum of 3 correlated jointly random variables
Suppose we have $3$ jointly normally distributed random variables $X_1, X_2, X_3$ with mean $0$ and variances $\sigma_1^2, \sigma_2^2, \sigma_3^2$. Suppose their correlations are $\rho_{12}, \rho_{23},...
