1
vote
1answer
21 views
Ratio of PDF to complementary CDF
Let $f(x)$ be a probability density function, and $F(x)$ be the cumulative distribution function of $f(x)$.
$$F(x) = \int_{-\infty}^{x}f(u)du$$
Then intuitively, what does the following ratio ...
2
votes
1answer
40 views
Find the limiting distribution
Find the limiting distribution for $n\rightarrow \infty \text{ of} \prod\limits^n_{i=1}X_i$. Given is that $f(x)=\frac{1}{2x\sqrt{2\pi}}e^{-\frac{1}{8}(\ln x-\theta)^2}, x\geq 0$.
0
votes
0answers
25 views
Calculating the probabilities of different lengths of repetitions of numbers of length 6
This question is similar to the question I asked here: Calculating the probabilities of different lengths of repetitions of numbers of length 4
except now I'm having problem with numbers of length 6.
...
1
vote
1answer
32 views
Calculating the probabilities of different lengths of repetitions of numbers of length 4
I'm trying to calculate the probabilities of different lengths of repetitions of X length number however I know I'm doing it incorrectly since when I add all the probabilities together they don't ...
0
votes
1answer
18 views
Expectation of function of stochast
I've got a general question regarding a certain sticking point I often encounter. When tackling questions where for example an UMVUE (uniformly minimum-variance unbiased estimator) has to found I get ...
1
vote
1answer
18 views
Joint distribution of multiple binomial distributions
In the picture below, how do they arrive at the joint density function? I understand how Binomial distributions work, but have never seen the joint distribution of them.
The original file can be ...
1
vote
1answer
28 views
Fisher information of a Binomial distribution
The Fisher information is defined as $\mathbb{E}\Bigg( \frac{d \log f(p,x)}{dp} \Bigg)^2$, where $f(p,x)={{n}\choose{x}} p^x (1-p)^{n-x}$ for a Binomial distribution. The derivative of the ...
2
votes
0answers
21 views
History of odds making in sports betting
Can anyone provide a reference to the history of odds making in sports betting? In many cases, certain odds are set and then adjusted as people make bets. However, I am having difficulty tracing the ...
5
votes
1answer
56 views
Applications of information geometry to the natural sciences
I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has ...
1
vote
0answers
29 views
Maximum likelihood in exponential family: $\partial_{\theta,\theta} \ln L = -\mathrm{Var}(T)$, $T$ sufficient for $\theta$
I'm confused on how the second derivative of the log-likelihood is computed in an exponential family.
There is a result which says that
If $T=T(X)$ is the natural sufficient statistic for ...
1
vote
1answer
34 views
The weighted distribution function for combination of two variables
For example, we have two random variables $a$ and $b$. And they have cumulative distribution function $F(x)$ and $H(x)$. We have number $0 < p < 1$.
Suppose, some machine get this random ...
5
votes
2answers
103 views
The probability of a drunk person/random walk
A drunk person wonders aimlessly along a path by going forward 1 step and backward 1 step with equal probabilities of $\frac12$.
a) After 10 steps, what is the probability that he has moved 2 steps ...
1
vote
1answer
47 views
Trouble understanding sum and product of probability distributions
Having trouble understanding where can we use the sum and product of probability distributions. Could someone please provide me with a real-life scenario? I think this is what I need to understand the ...
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 $X$?
...
1
vote
1answer
28 views
Definitions for an exponential family to be curved or flat?
I was wondering how a curved exponential family is defined? Also how is a flat exponential family defined?
Is "curved" or "flat" defined for a family of probability distributions, or for a
...
0
votes
1answer
47 views
Find the maximum likelihood estimator for $\theta$ when $f(x)=2\theta^{-2}x, 0\leq x \leq \theta$
Find the maximum likelihood estimator for $\theta$ when $f(x)=2\theta^{-2}x, 0\leq x \leq \theta$.
This should be a really easy question but I somehow cannot seem to get the right answer. My ...
0
votes
4answers
80 views
easy but hard probability task
I have 30 cards, and 4 red cards among them. I have 3 players, each player gets 10 cards. what is the probability of:
$A$ = player1 gets all 4 red cards
$B$ = each player gets 1 card.
I am ...
1
vote
2answers
46 views
Is a Riemannian metric positive definite or positive semidefinite?
From Wikipedia
The Fisher information matrix is a N x N positive semidefinite symmetric matrix, defining a Riemannian metric on the N-dimensional parameter space,
But a Riemannian metric is ...
0
votes
0answers
42 views
Generalized Likelihood Ratio Test and Hypothesis Testing
Below is a question from a review sheet on an upcoming final that I am really struggling with. Any help is greatly appreciated!
Let $Y_1, Y_2,...,Y_8$ be a random sample from the uniform ...
3
votes
1answer
49 views
Find version of conditional expectation
I'm struggling with the concept of conditional expectation. We didn't cover it in my probability theory class, yet it's required for my statistics course.
I'm basically having no idea how to solve any ...
1
vote
1answer
39 views
Verify a distribution that is not exponential family
I understand that if the support of a distribution depends on the parameter $\theta$, it is not exponential family even if its pdf can be written in the form $ f(x | \theta) = h(x)c(\theta) \exp\left( ...
0
votes
1answer
51 views
Rat in Maze Probability
I am trying understand what am I missing in my way of solving rat in maze problem...
The question and solution is given in this link
http://www.ams.sunysb.edu/~jsbm/courses/311/rat-in-maze.pdf
...
1
vote
0answers
23 views
What is the probability that we get more than $\frac{n}{2} + 2\sqrt{nln(n)}$ heads? [duplicate]
Toss $n$ coins. What is the probability that we get more than $\frac{n}{2} + 2\sqrt{n[\ln(n)]}$ heads?
How do I apply Chernoff Bounds to this?
I really need help understanding Chernoff Bounds.
0
votes
1answer
26 views
Question regarding Type II Error in Hypothesis Testing
The following is a homework problem and I am not really sure where to begin or how find what the question is asking.
Suppose that one observation from the exponential pdf $f_{y}(y)=\lambda ...
2
votes
1answer
46 views
Using empirical density function as an estimator of a given probability density
We know empirical distribution function is defined as $F_n(x)=\frac{1}{n}\sum\limits_{i=1}^nI(X_i \leq x)$. Then define empirical density function as $ f_n(x) = \frac{F_n(x+b_n)-F_n(x-b_n)}{2b_n} $ .
...
0
votes
1answer
71 views
How to sample uniformly from an $\epsilon$ ball?
Given a real rectangular matrix $X$, I would like to uniformly sample from the set of real rectangular matrices $\mathbb{M}$ that satisfy $||X-S||\leq \epsilon, \forall S\in\mathbb{M}$ and for a fixed ...
1
vote
0answers
59 views
Probability distribution with maximal entropy on $[0,1] \cup \{2\}$
For given closed set $F$ on $\mathbb R$ one can think of probability distribution $\mathbb P^\ast_F$ with support on $F$ and with maximal entropy. It is well known that
If $F=[0,1]$ then $\mathbb ...
1
vote
0answers
41 views
asymptotic order of the variance of the maximum of iid standard Gaussian
Suppose that $X_1,\cdots,X_n$ are iid standard Gaussian. $X_{(n)}$ is the maximum of $(X_1,\cdots,X_n)$, how can I find the asymptotic order of $VAR[X_{(n)}]$?
The density function of $X_{(n)}$ can ...
1
vote
1answer
31 views
Hypothesis Testing a small sample for the binomial parameter p
The following is a question from a homework set that I truly do not understand how to even begin.
The following is a Minitab printout of the binomial pdf $p_{x}(k) {9 \choose k}(0.6)^k(0.4)^{9-k}$, ...
0
votes
0answers
19 views
Power Curves from Normal Distribution.
The following is a homework problem that I cannot figure out because I am having trouble finding the Type II error.
Construct a power curve for the $\alpha = 0.05$ test of $H_0:\mu = 60$ versus $H_1: ...
0
votes
0answers
49 views
something about property of Bernoulli random variables
Let $b_i, i=1, \ldots, n$ be Bernoulli random variables with probability $P(b_i=1)=2k/n,$ where $k\leq n.$
Show the following:
Let $\chi$ be an indicator function that $k$ out of $n$ of $b_i$ are ...
0
votes
2answers
50 views
Central Limit Theorem VS Normal Model
I just had a quick question regarding the Central Limit Theorem and Normal Model. I am in an elementary probability course and we have learnt that the CLT is as follows:
$$ Z = \frac{X_1 + X_2 + X_3 ...
2
votes
1answer
42 views
Integral of bivariate normal distribution function with respect to itself
Define $F: \mathbb{R}^2 \rightarrow \mathbb{R}$ by
\begin{align*}
F(x,y)=\int_{-\infty}^{x} \int_{-\infty}^y \frac{1}{2\pi \sqrt{1-\rho^2}}
exp\left(\frac{-u^2-v^2+2\rho uv}{2(1-\rho^2)}\right) ...
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$ ...
1
vote
0answers
26 views
The field generated by the empirical distribution function
The Gist
I've got three question related to the field generated by the empirical distribution function ("the empirical field").
Is the empirical field identical to the symmetric field?
Are all ...
1
vote
1answer
39 views
Finding Rao-Cramer Lower bound
Third part of a homework problem and I understand the material, but I can't integrate this "thing"...
$f(x;\theta)=\frac{1}{\theta^2}x^{\frac{1-\theta}{\theta}}, 0<x<1, 0<\theta<\infty.$
...
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, ...
4
votes
1answer
240 views
Partial sum of numbers
My TA gave today this question as a nice question to think about. He said its involves standard ideas of Probability theory and numbers. But, I don't even know how to start.
Let $x_1, \ldots, x_n$ ...
0
votes
2answers
139 views
$E(X\mid X\lt x)$ with $X\sim\text{Exp}(a)$
$X\sim\text{Exp}(a)$.
How do I calculate $E(X\mid X\lt x)$?
Workings:
\begin{align}E(X|X<x)&=\int_0^txf(x|x<x)dx\\
&=\frac{\int_0^txP(x,x<x)dx}{\int_0^tP(x<x)dx}\\
...
2
votes
1answer
98 views
Condition Expectation of Difference between Two Poisson processes
$P_t$ and $Q_t$ are poisson processes with rates $a$ and $b$.
How do I calculate $E[(P_t-Q_t)]^2|Q_t=m-P_t]$?
1
vote
1answer
105 views
Conditional variance of arrival times
Given a poisson process $P_t$ with rate $r$, with arrival times $S_n$
How do I calculate the Variance of $S_2-S_1|P_t=2$?
0
votes
1answer
97 views
Conditional CDF of Poisson process
$X_t$ and $Y_t$ are poisson processes with rates $a$ and $b$ (independent processes)
$n = 1,2,3...$
Find the conditional CDF $F_X{}_t{}_|{}_X{}_t{}_+{}_Y{}_t{}_={}_n(x)$
I get an answer of ...
2
votes
0answers
56 views
Covariance of two random variables with monotone transformation
Suppose I know that, for two random variables $X,Y$, we have
$$Cov(X,Y)\neq 0.$$
What happens if we take a monotone transformation of $X$; will the inequality persist? That is, say $f(.)$ is a ...
2
votes
0answers
44 views
Consider a bipartite graph and Deduce that there exist infinitely many bipartite graphs
Consider a bipartite graph $G=(X \cup Y, E)$ with both sides of equal size; we let $n$ denote $|X|=|Y|$
We are also given an integer $d\ge 3$ and we wish each vertex in $X$ to be adjacent to at most ...
1
vote
1answer
57 views
estimation of a moment for the sum with Bernoulli random variables
Let $x\in R_+^n$ and let $b_i, i=1, \ldots, n$ be $(0,1)$ Bernoulli random variables with $P(b_i=1)=p$. Denote $S=\sum_{i=1}^n x_ib_i$. For $q\geq 2$ estimate from above
$$
E\left|S\right|^q
$$
2
votes
0answers
69 views
Does weak convergence of two independent sequences of random processes imply weak convergence of the couple sequence?
I consider sequences of random processes in the Skorohod space. Let $F_n$ and $G_n$ converge weakly to F and G, respectively, in $D[-\infty,\infty]$ endowed with the supnorm. I suppose that this ...
0
votes
0answers
45 views
Continuous-time stochastic process that is left-continuous predictable process - why? [duplicate]
Predictable processes are basically deterministic processes - and I am wondering why continuous-time processes that are left-continuous are automatically predictable processes. To my eyes, ...
0
votes
3answers
62 views
Probability: Permutations
Consider the experiment of picking a random permutation $\pi$ on $\{1,2,...,n\}$, and define the associated random variable $f(\pi)$ as the number of fixed points of $\pi$, i.e, the number of $i$ ...
2
votes
2answers
52 views
Can we simplify an expression of random variables? (can we treat random variables as real numbers?)
Suppose that we have an expression of random variables including $X-X$ or $2X-X$ or $XY-XY$ and so on. can we treat random variables as real numbers? That is, can we delete $X-X$ or replace $2X-X$ by ...
0
votes
0answers
47 views
Find a sample size for population in proportion
Can you help me with this question?
Sometimes we perform experiments that compare the probability of success against an external standard, instead of comparing two probabilities. For example, a ...
