Tagged Questions
0
votes
1answer
19 views
correlation of product with its normally distributed factors
If x and y are normally dist. with standard deviation of 10%, and they are independent, then their product X.Y is 71% correlated with Y (or X).
I can show this empirically, but how to I prove it in ...
0
votes
1answer
25 views
Uniform distribution joint $\to$ marginal
Let vector $(X,Y)$ have a uniform distribution on the set $N = \{ (x,y): x<1,y<1,1<x+y\}$. Determine distribution $X-Y$.
So far I've thought of this:
\begin{align}
P[X | Y=y] &\sim ...
0
votes
1answer
22 views
this is regarding exponentials distribution
In an office building, the lift breaks down randomly at a mean rate of 3 times per week. The random variable X represents the time in days between successive lift breakdowns.
(i) Calculate the ...
1
vote
1answer
19 views
What is the purpose to define different moments on a distribution?
What is the purpose to define different moments on a distribution?
The first moment is the expectation value of a function, what about the other?
0
votes
0answers
11 views
This is regarding hypothesis testing using F-distribution and Chi square distribution
Two ambulance stations, A & B, are in similar locations. Random samples of the response
times, in minutes, to emergency calls were recorded during a particular week. The
information is given in ...
0
votes
1answer
35 views
Acceptance sampling schemes for binomial distribution
Two acceptance sampling schemes, A and B, are proposed for deciding whether or not
to accept a large batch of items from a production process in which 5% of the items
produced are defective.
Scheme A: ...
0
votes
0answers
22 views
Proof that the radius is a sufficient statistic for a circle
How can I prove that the radius of a circle is the sufficient statistic for the probability of choosing random points in the area of the cricle?
0
votes
2answers
18 views
Probability of two variables of having the same value
Let $X$ and $Y$ be two random variables, whose PDFs $f_X$ and $f_Y$ are uniform. $f_X$ and $f_Y$ may overlap. For instance, they could represent two score distributions for two tuples $x$ and $y$ in a ...
0
votes
2answers
46 views
Probability Joint PDF
Every night Joe goes to the casino and takes with him an amount of money in dollars, X, that is distributed according to the pdf:
f(x) = Ax^2 for 0 < x < 10
where A is a constant that you need ...
0
votes
0answers
26 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
0answers
38 views
Problems sampling from a $pdf$ over $SO\left(3\right)$
I have a probability density function over $SO\left(3\right)$, which I am trying to sample from. The $pdf$ is given as a generalized fourier series:
$$ f\left(\omega,\theta,\phi\right)=\sum ...
0
votes
1answer
34 views
Probability exponential distribution.
May I please borrow your expertise or could anyone check if I'm on the right track please?
Consider customers arriving at a bank. The bank has $2$ types of customers - business and personal. On ...
1
vote
2answers
23 views
Closed form for Exponential Conditional Expected Value & Variance
I am wondering if there is a closed form for finding the expected value or variance for a conditional exponential distribution.
For example:
$$ E(X|x > a) $$ where X is exponential with mean ...
0
votes
0answers
48 views
Probability exponential distribution question
Could anyone please help me answer these questions? Or a little hint as to how I can answer them? It's for my assignment that's due tomorrow.. Really appreciate if anyone could help!
Consider ...
1
vote
1answer
32 views
Calculating the probabilities of different lengths of repetitions of numbers of length 4
I'm trying to calculate the probabilities of different lengths of repetitions of X length number however I know I'm doing it incorrectly since when I add all the probabilities together they don't ...
1
vote
2answers
30 views
Uniform distribution on the n-sphere.
I have the next RV:
$$\underline{W}=\frac{\underline{X}}{\frac{||\underline{X}||}{\sqrt{n}}}$$
where $$X_i \tilde \ N(0,1)$$
It's a random vector, and I want to show that it has a uniform ...
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 ...
3
votes
3answers
97 views
Compute value of $\pi$ up to 8 digits
I am quite lost on how approximate the value of $\pi$ up to 8 digits with a confidence of 99% using Monte Carlo. I think this requires a large number of trials but how can I know how many trials?
I ...
0
votes
0answers
27 views
Probability that a sub-sequence of i.i.d. zero-mean Gaussians is closer to a given point than the origin
I am given a sequence $X=\{X_1,X_2,\ldots,X_n\}$ of $n$ i.i.d. zero-mean Gaussian random variables $X_i\sim\mathcal{N}(0,\sigma^2)$, and a vector $\mathbf{y}=\{y_1, y_2, \ldots, y_m\}$ of $m$ real ...
0
votes
2answers
46 views
Sample $x$ from $g(x)$
I got confused with all this randomness and probability functions. I was trying to implement the rejection sampling method which (apparently) is really simple. I was reading from Rejection Sampling in ...
1
vote
2answers
76 views
Independent and uniformly distributed on $(\frac{1}{2},1]$
I have two random variables $X,Y$ which are independent and uniformly distributed on $(\frac{1}{2},1]$. Then I consider two more random variables, $D=|X-Y|$ and $Z=\log\frac{X}{Y}$. I would like to ...
0
votes
1answer
38 views
Accept reject method to generate random numbers
The method says that having a proposal $g(x)$
Sample $X^* \tilde ~ g(x)$ and $U \tilde ~ Unif(0,1)$
Accept $X = X^*$ if $U ≤ f(X^*) / M g(X^*)$
Moreover, $M$ is constant that satisfies $Mg(x) ≥ ...
0
votes
2answers
27 views
Central limit theorem - std dev away from mean
I was reading about the CLT and found something that I think people use interchangeably. On one hand I found that 68% of the means are 1 standard deviations from away and 95% are 2 std dev. On the ...
1
vote
1answer
30 views
Compute the mean of a random variable
Imagine I have for a single individual some variable $X$ with mean $\lambda$ (for example the number of cars he has). Now I take a whole population of individuals. The parameter $\lambda$ for each of ...
1
vote
1answer
29 views
Showing it is a joint probability density function
I have two random variables $X,Y$ with a joint density function $f_{X,Y}(x,y)=x+y$ if $(x,y)\in[0,1]\times [0,1]$ and otherwise $f_{X,Y}(x,y)=0$
I want to analyze this case in different cases, first ...
1
vote
0answers
23 views
Bernstein type inequalities. Is there a standard list?
Suppose I have a sequence of iid random variables $X_i\geq 0$ with mean $\mu$ and $\mathbb E \left(e^{tX_i}\right) = G(t)$. Set $$S_n = \sum_{i=1} X_n.$$
For the purpose of this question the ...
1
vote
1answer
48 views
distribution function of time T
an ambulance station is located 30 miles from one end of a 100-mile road. the station services accidents along the entire road. suppose that an accident occurs. suppose that Suppose accidents occur ...
0
votes
1answer
13 views
Third central moment Bernoulli variable
I'm looking for a proof of the third central moment of a Bernoulli variable $X$ with probability $p$. I know it must be $p(1-p)(1-2p)$, but I'm looking for a way to show this. Any ideas? Thanks!
1
vote
1answer
61 views
Multivariate normal distribution density function
I was just reading the wikipedia article about Multivariate normal distribution: http://en.wikipedia.org/wiki/Multivariate_normal_distribution
I use a little bit different notation. If $X_1,...,X_n$ ...
0
votes
1answer
51 views
Markov Chain - Snakes and Ladders
A simple game of snakes and ladders is played on a board of nine squares. At each turn a player tosses a fair coin and advances one or two places according to whether the coin lands heads or tails. If ...
0
votes
1answer
32 views
How to calculate the pmf of $X_N$
How do I calculate the pmf of $X_N$, where $X$ is the number of people out of $N$ getting back their own hat after a random hat exchange?
How can I calculate it without listing all the possible ...
0
votes
0answers
27 views
distribution function and density function
A lion is standing $30$ meters from one end of a $100$-meter road. The lion will attack any zebra that appears on the road. Suppose that a zebra appears on the road, and suppose that the position at ...
1
vote
0answers
71 views
How to calculate probability with sigmoid output in feedforward neural network?
first of all I'm sorry for my not very skilled English, but I will do my best to explain my problem.
I'm trying to create a feedforward neural network with one hidden layer (with probably arctan ...
4
votes
1answer
23 views
How does the increase in overall number of events affect the peak (events/time)?
I have a (simple?) question that I hope someone will find interesting enough to help me out with.
A web site has a given number of subscribers who generate a certain amount of traffic on the web ...
1
vote
1answer
29 views
Is $\left(X_1,… ,X_n,\bar{X}\right)$ jointly normal distributed if $\left(X_1,… ,X_n\right)$ is?
Let $X:=\left(X_1,... ,X_n\right)\sim N_n(\mu,\Sigma)$, $\mu\in\mathbb{R}^n$, $\Sigma\in\mathbb{R}^{n\times n}$ symmetric and positive semi-definite and $\bar{X}:=\frac{1}{n}\sum_{i=1}^n X_i$ as ...
2
votes
1answer
48 views
Sum of Bernoulli random variables with different success probabilities
Let $X_{i} \in \{0,1\}$ be Bernouli random variable with probability of success $p_{i}$, i.e., $P(X_{i}=1) = p_{i}$ and $P(X_{i}=0) = 1-p_{i}$ and let $Y=\sum_{i=1}^{n}X_{i}$ for $n>0$. Is it ...
0
votes
1answer
48 views
Proof that a sequence of random variables have finite expectation
Let $X_n$ be iid non-negatives random variables. Prove that $\mathbb{E}[X_1] < \infty$ iff $P(X_n \ge n\text{ i.o.}) = 0$
I thought I would start like this for one direction
$\infty > ...
2
votes
1answer
21 views
Basic understanding of sampling from a continuous distribution.
For continuous distribution (on R) the probability of a single point is $0$.
So I'm not sure what does it mean to sample $M$ elements from a continuous distribution.
Let say there is a continuous ...
1
vote
3answers
44 views
Doubt about why I can't treat this as a Bernoulli process
I know the title is not descriptive enough, but I don't know how else to say it. I don't know why I can't use the Binomial distribution to get the result I'm looking for. The teacher solved it long ...
1
vote
3answers
28 views
prove that any positive integer-valued random variable with memoryless property has the geometric distribution for some $p$
How to prove that any positive integer-valued random variable with memoryless property has the geometric distribution for some $p$.
By memoryless property,
$$P(X=i+s | X>i)=P(X=s)$$
How to get ...
0
votes
1answer
26 views
Integrate over the uniform distribution on the simplex
Let $p=(p_1,\ldots,p_n)$ correspond to points in a simplex that add up to one, i.e. $p$ is a discrete probability distribution. I would like to compute an integral of the form $\int dp_1\ldots\int ...
0
votes
4answers
45 views
A basic doubt on the definition of a Poisson random variable
What is the significance of "large city" in the definition of the following Poisson variable :
"Number of phone calls placed during a ten second interval in a large city"
I guess either $n \to ...
0
votes
2answers
29 views
Expectation Values
Suppose that $\{X_n\}_{n\ge1}$ take values $-\dfrac{1}{2}$ and $\dfrac{1}{2}$ with probability $a$ and $1-a$ respectively and $0$ otherwise. Suppose further that they are independent and discrete and ...
1
vote
1answer
45 views
finding the probability density function of $ dY_t = - Y_t X_t dW_t$
Could someone point me to where I can learn how to derive the stationary distribution for the martingale $Y_t$ which itself has stochastic volatility drive by $X_t$:
\begin{align}
dY_t &= - Y_t\ ...
0
votes
1answer
58 views
Computing PDF of Products of Two Random Variables
I've been stuck on this problem for some days. I'm hoping someone would help by chipping in a few comments. I have two i.i.d. r.v.:
$$
f_X(x)=\frac{\left(1-e^{-\frac{x}{\alpha }}\right)^{\tilde{r}-1} ...
3
votes
3answers
64 views
$E[X]$ finite iff $\sum\limits_{n} P(X>an)$ converges
Show that:
$$\sum\limits_{n \in N } P(X>an) < \infty\ \text{for some}\ a > 0 \Rightarrow E[X] < \infty \Rightarrow \sum\limits_{n \in N } P(X>an) < \infty\ \text{for every}\ a > ...
1
vote
1answer
35 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 ...
1
vote
0answers
26 views
P.d.f of a discrete fourier transform of binary variables
Let $\{a_n\}$ be a set of $N$ "binary" random variables uniformly distribuited in $\{-1,1\}$.
The discrete fourier transform is defined
$b_k=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} a_n e^{-2 \pi i k n ...
0
votes
4answers
47 views
Probability of $X$ out of $N$ dice landing on $M$
The problem is as follows:
We have $N$ dice and we throw them on a table. What is the probability that $M$ will fall $X$ times?
Specific example:
We have $10$ dice and we throw them on a table. What ...
0
votes
1answer
24 views
Length of life of a fire detector
The length of life of a flame detector is exponentially distributed with paramater $\lambda=0.1/year$. Die number of events which activate the flame detector in an interval with length $t$ (heat, ...
