All Questions
112
questions
1
vote
0
answers
45
views
Is $f$ a probability mass function for a random variable $X$ if and only if it complies with these two properties?
I've read in multiple books that a function $F$ is a cumulative distribution function for some random variable $X$ if and only if it satisfies three conditions:
$F$ is non-decreasing.
$F$ is ...
- 5,166
1
vote
2
answers
164
views
Let $P(X_n=n)=1/n^a$ and zero otherwise. Let $Y_n=X_{n+1}\cdot X_n$. Find all $a$ such that $\liminf Y_n=0$ and $\limsup Y_n=\infty$ almost surely.
Remark: My attemps is WRONG as I fasly assumed that the $Y_n$ were independant. I ve corrected this in my own answer to this post. This answer is correct but not enough well wrotten, read the selected ...
- 719
0
votes
1
answer
42
views
$X$ a r.v. verifying $P(X=x)=0$&$F_X$ its repartition fct. Prove that $F_X(X)$ follows an uniform distribution law $]0;1[$.
Question:
Let $X$ a r.v. verifying $P(X=x)=0, \forall x \in \mathbb{R}$ and $F_X$ its repartition function. Prove that $F_X(X)$ follows an uniform distribution law $]0;1[$.
Answer:
1- We write $0 \leq ...
- 719
1
vote
0
answers
52
views
If $X$ is a random variable on the probability space $(\Omega,F,P)$. Compute $\lim_{n\to\infty} P(X=n)$
$$P(X=n)=1-P(X\neq n)=1-P(X<n)-P(X>n)=1-P(X>n)-P(X<n)=P(X\leq n)-P(X<n)$$
Therefore, it follows $$\lim_{n\to\infty} P(X=n)=\lim_{n\to\infty} P(X\leq n)- \lim_{n\to\infty} P(X<n).$$ ...
- 97
1
vote
2
answers
89
views
Extreme Value Theorem on Logistic Distribution
Let $π_1,π_2, β¦ , π_π$ be a random sample of size π from a distribution with cumulative distribution function $F(π₯) = \frac{1}{1 + \exp(-cx)}$, $β\infty < π₯ < \infty$, where $π > 0$. ...
- 41
0
votes
0
answers
30
views
The formula of probability
I have a very basic question about probability and the formula for calculating the probability of an event
Image an event (like A) we're using the ratio n(A)/n(S) to calculate the probability of this ...
0
votes
2
answers
41
views
Verifying a PDF derived from a CDF
If I have a CDF of a continuous random variable $x>0$, and derived its PDF by differentiating the CDF, how to verify that this PDF is correct is there a relation or a certain plot between the PDF ...
1
vote
2
answers
391
views
If $X_n \sim\text{Bernoulli} \left( \frac{1}{n} \right)$ and $Y_n = n X_n$ show $\left\{ Y_n \right\}_{n=1}^{\infty}$ converges in probability to zero
Let $X_n$ be a random variable that is Bernoulli distributed with parameter $(1/n)$ and let $Y_n =nX_n$, I want to show that $\left\{Y_n \right\}_{n=1}^{\infty}$ converges in probability to zero and ...
- 121
2
votes
1
answer
103
views
Show that $\mathbb{E}(\mathbf{1}_{\{X>t\}}|\mathcal{G})$ is $(\mathcal{B}(\mathbb{R})\otimes\mathcal{G},\mathcal{B}(\mathbb{R}))$-measurable.
Let $X$ be a non-negative integral random vairable on $(\Omega,\mathcal{F},\mathbb{P})$ and let $\mathcal{G}\subseteq\mathcal{F}$ be a sub-$\sigma$-algebra. I want to show that $$\mathbb{E}(X|\mathcal{...
- 5,930
2
votes
0
answers
68
views
Distribution function of $X-Y$, where $X,Y$ are uniformly distributed
I want to compute the distribution function of $X-Y$, denoted by $F_{X-Y}$, where $X,Y$ are indepedently uniformly distributed on $[-1,1]$.
\begin{align*}
&F_{X-Y}(z)=\begin{cases}
0,&z\leq -2\...
- 4,564
2
votes
2
answers
143
views
Find the density of $\cos(X)$ when $X$ is an exponential.
I want to find the density of $\cos(X)$ where $X$ is an exponential with density given by
$$
f_{X}(x) = re^{-rx}\mathbb{1}_{[0,+\infty}(x),\quad r>0
$$
My attempt is the following :
First we notice,...
- 1,381
1
vote
2
answers
44
views
$Gx$ is distributed uniformly in the set $\Bbb Z_2 ^n$.
I read the probabilistic proof of the Gilbert Varshamov bound in Coding Theory, and I came across the following argument, which I would like to discuss:
Suppose that $G$ is a random matrix in $M_
{nΓk}...
- 2,802
1
vote
1
answer
46
views
Determine, the distribution of $Z(w)=\left\{\begin{matrix} X(w)& \mbox{if}\ Y(w)\geq 0 \\ -X(w)&\mbox{if} \ Y(w)<0 \end{matrix}\right.$
Hey I want to check my solutions for this problem:
Let $X$ and $Y$ be standard normally distributed random variables on a probability space $(\Omega, \mathcal F, \mathbb P)$ and define
$Z(w)=\left\{\...
- 900
0
votes
0
answers
85
views
Problems on some exercises with characteristic function
Hey I have some questions about this two exercises:
Consider $(X_n){nβ\mathbb{N}}$, a sequence of independent and identically distributed real-valued random variables with an absolutely continuous ...
- 900
0
votes
1
answer
50
views
Poisson Approximation Book Pages
In a book of $500$ pages, there are $1000$ typographical errors. Assume that errors are
equally likely to be on any page (that is, the page number of each error follows the uniform
distribution), ...
user1071088