All Questions
18
questions
0
votes
1
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99
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Proving the general formula for conditional expectation of a Poisson Process
I am studying a course on Stochastic Processes and encountered the following proof exercise on Poisson Processes:
If $N$ is a Poisson Process with intensity $\lambda$, then for $0<s<t$ where $k ...
- 4,331
0
votes
0
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46
views
$P({X^{-1}(x)}) = p$ for all $x$ then Field is finite
Notation:
$\Omega$ denotes a $\sigma-$Field
$X$ is a function onto $\Omega$
$X^{-1}$ is pre-image
$P(\cdot)$ is a probability measure
$X^{-1}$ is also a $\sigma-$Field, as a consequence of ...
- 47
0
votes
1
answer
51
views
I have question about a proof in probability
I have a question about this proof:
Let $X$ be a random variable and $n>0$. Then we have that $P(|X|\geq
n) \leq \frac{E|X|}{n}$
Proof: Let $Y: \begin{pmatrix} 0 & n\\ P(|X|<n) & P(|X| \...
- 4,710
1
vote
1
answer
159
views
Durrett Theorem 3.2.8 (Are inverse distribution functions right continuous?)
In the book Probability: Theory and Examples by Durrett, on page 118 he claims the following (I have adapted the notation and verbiage slightly for simplicity and clarity):
If distribution functions $...
- 15.4k
2
votes
2
answers
90
views
A corollary of the Chernoff bound
During my Statistic course, we were asked the following question:
Let $ X_1, \ldots , X_n $ be a $n$ observations that are i.i.d and assume $ X_i \sim \mathcal{N} (0,\sigma^2) $.
Use the Chernoff ...
0
votes
1
answer
54
views
$ v(\omega) : = \inf \{ n \geq 2 : \omega \in A_n \} $ , proving that $v$ is a random variable
Let $ (X_n)_{n \geq 1} $ be a sequence of Bernoulli random variables of parameter $p$
$A_n : = \{ \omega \in \Omega : X_n(\omega) \neq X_{n-1}(\omega) \} $
$ v(\omega) : = \inf \{ n \geq 2 : \omega \...
- 409
0
votes
1
answer
51
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Probability theory - request for proof-verification f(further properties of characterisitc functions)
By the courtesy of the user @uniquesolution, I presumably managed to understand proof related to certain further property of characteristic function. I completed too big for me shortcuts in reasoning. ...
- 139
2
votes
1
answer
79
views
Prove that $\sum_{j=1}^{n} \mathbb{P}(I_j) \geq \mathbb{P}(I)$, when $\bigcup_{j=1}^n I_j \supseteq I$.
Let $I_1, I_2, \cdots, I_n$ be a finite collection of intervals on $[0,1]$, whose union contains an interval $I$, then $$\sum_{j=1}^{n} \mathbb{P}(I_j) \geq \mathbb{P}(I),$$ where $\mathbb{P}$ denotes ...
- 1,348
1
vote
2
answers
736
views
Proof of equivalence of alternative definition of $\lambda$-system
As the title says, I'm trying to prove that both definitions of a $\lambda$-system are equivalent to each other. It's maybe a bit longer than usual but I hope it's still within the scope of what's OK ...
- 335
1
vote
1
answer
292
views
Show that minimum $\text{min}(X_1,..,X_n)$ is geometrically distributed
$X_1,..,X_n$ are linearly independent, identical geometrically
distributed random variables with parameter $p \in (0,1)$, i.e. $$P(X_i=k) = p(1-p)^{k-1} \text{ with } k \in \mathbb{N} \text{ and } i ...
- 1,335
0
votes
1
answer
46
views
Prove that $X \sim N(\sigma, \sigma^2)$ is a scale family
Prove that $X \sim Normal(\sigma, \sigma^2)$ is a scale family. $\sigma>0$.
I'm not exactly sure how to approach this problem. I think I need to prove that the pdf of X is equal to $\frac1\sigma ...
user230601
1
vote
0
answers
598
views
Prove that Cauchy$(\mu, \sigma^2)$ is not an exponential family
Cauchy$(\mu, \sigma^2)$ is defined as $\frac1\sigma \frac1 {\pi(1+\frac{(x-\mu)^2}{\sigma^2})}$
I need to prove that this is not an exponential family. Here's what I've attempted.
I rearranged it to ...
user230601
0
votes
0
answers
359
views
On proving right-continuity of a distribution function
I am unsecure regarding my proof on right-continuity of the distribution function.
Let $\epsilon >0$.
By continuity from above of the measure $P$ we have $G_{n} \downarrow G \implies P(G_{n}) \...
user415535
1
vote
0
answers
89
views
Feedback of proof that $\lim \inf A_n \subseteq \lim \sup A_n$
I just wrote down the proof of the following easy proposition, and I was wondering about both the content (I would like to know if it is error-free), and the form of it.
Proposition: $\lim \inf A_n \...
- 4,113
2
votes
1
answer
386
views
Can I assume that random variables with exponential distribution are positive?
Let $(Y_n)$ be i.i.d random variables following exponential distribution with parameter $1$.
Let $X_n=\min(Y_1,\dotsc, Y_n)$
Prove that $ X_n \xrightarrow{P} 0$
It's easy to prove that $P(...
- 35.9k