0
votes
0answers
9 views

Error of a Serial Processs

Give random variable X and two processes A, B . Assume that $ Y_{1}, Y_{2}$ are estimated versions of X by using processes A, B respectively, with probability: $P\left \{ \left | X-Y_{1} \right ...
1
vote
1answer
14 views

Easy way to compute $Pr[\sum_{i=1}^t X_i \geq z]$

We have a set of $t$ independent random variables $X_i \sim Bin(n_i, p_i)$. We know that $$Pr[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$ But is there an easy way to ...
0
votes
0answers
11 views

Can a biased physical random source be post-processed to control the bias?

Let $X_i$ with $i\in\mathbb N$ be a sequence of independent 6-ary random variables with distribution $\operatorname{Pr}(X_i=e)=p^e_i$ where $e\in\{1,2,3,4,5,6\}$ and $\sum_{e=1}^6p^e_i=1$. Let's ...
0
votes
0answers
14 views
1
vote
0answers
11 views

derivation law from the call option formula

i am reading a article about the option pricing. and i got stuck with some typical statement. $C(K)=\int (x-K)^+\mu(dx)$ is given. here, $\mu$ is implied law of asset price and C(K) is the price ...
2
votes
0answers
15 views

$X$ and $Y$ are i.i.d random variables with finite second moments. $X+Y$ and $X-Y$ are independent, show that $X$ is Gaussian.

$X$ and $Y$ are i.i.d random variables with finite second moments. $X+Y$ and $X-Y$ are independent, show that $X$ is Gaussian. Without loss of generality we may assume that $X$ and $Y$ are ...
0
votes
0answers
13 views

Does it represent a joint density function

Let equation of a plane be $ax+by+cz = 0$, where $a,b,c,x,y,z$ are random variables. I know the distribution of $x,y,z$, and I need to find the joint distribution of the coefficients $a,b,c$. Let the ...
1
vote
3answers
36 views

Covariance of Binomials

I'm doing some basic error, but I just can't see where... Let $X_i\sim \mathrm{Bin}(\theta_i,n)$, and $X_j\sim \mathrm{Bin}(\theta_j,n)$ I want to find $\mathrm{Cov}(X_i,X_j)$. So, ...
3
votes
1answer
52 views

[Probability]need help to understand the following expression

So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively. assume c is a constant. I am having huge trouble to understand the integration of the following ...
0
votes
0answers
13 views

Urn model over continuous variables

I have a urn containing $p$ liters of an unmixable fluids mixture: $p_i$ liters of fluid A and with $p-p_i$ liters of fluid B. What is the probability that a spill of $m$ liters of liquid, at least ...
1
vote
1answer
37 views

order statistics

Suppose $X_1,..., X_n$ are i.i.d. continuous r.v. with distribution function $F(x)$, and density function $f(x)$. $X_{(1)}<\cdots<X_{(n)}$ are the order statistics. I've already showed that ...
0
votes
0answers
11 views

Kullback-Leibler or Jensen-Shannon divergence between two distributions.

i would like to understand well what Kullback-Leibler or Jensen-Shannon divergence between two distributions will tels us about two distribution,for instance let us consider following code ...
2
votes
2answers
24 views

Select an element without uniform distribution with a uniform random without iterations

I have N elements (numbered from 0 to N-1) and I must choose one but without the same probability. For example, I need that the 0 must happen 50% of times, 1 with a 25%, 2 with a 12.5%, etc. I don't ...
0
votes
1answer
31 views

Measurability and knowledge

there seems to be a subtle relationship between knowledge and measurability. If I have a stochastic process $(X_n)_n$, then for example a stopping time ( other examples would be martingales, ...
1
vote
3answers
70 views

Finding expected value??

In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the ...
0
votes
1answer
20 views

Finding the $75$th percentile of a distribution.

So, I came across a couple of homework problems on finding percentiles. The first was: pdf of $X$ is $f(x)=\frac{10}{x^2}$ for $x\gt 10$, and $0$ otherwise. Finding the $75$th percentile here was ...
1
vote
0answers
29 views

Probability: NEED HELP to Understand with the follow [duplicate]

I need help to understand the probability derviation of a paper. Please help me. For the following, please only treat $|h_{R,B}|^2$ and $|h_{A,R}|^2$ as random variables (other parameters can be ...
1
vote
1answer
40 views

Non-uniform sampling of N-sphere

Suppose I have a unit $N$-sphere from which I want to draw points at random. To obtain uniformly distributed points I do the usual technique of drawing $N$ random variables $x_i$ from a Gaussian ...
2
votes
1answer
18 views

Convergence in total variation

There are the very basic convergence types in probability theory: almost sure, in $L^p$-norm, in measure and in distribution. Besides that there is the concept of convergence in total variation norm. ...
1
vote
1answer
26 views

Product of 2 random variables:domain of integration

I am trying to compute the PDF of the product of two ind. random variables: $Z=XY$, where $0\leq x \leq d$ and $ 0\leq y \leq 1 $. ($0<d<1$) I found this formula : $ f_Z(z) = ...
1
vote
1answer
61 views

Exponential Distribution question

I'm having some trouble understanding the mechanics of how to solve with this distribution. The question: The number of years that a washing machine functions is a random variable whose hazard rate ...
0
votes
1answer
37 views

An exercise in probability

I've read this exercise in the internet: "The PIRON Software Company currently develops marketing software for primarily service-based organizations. They are considering expanding their operations ...
1
vote
0answers
24 views

What is the variance of an arbitrary “good” function of several independent normally distributed random variables

During my studies years ago I came over a formula that states something like if $x_i$ are independent normally distributed variables with variances $\sigma^2_i$ and $f(x_i)$ is differentiable (and ...
1
vote
1answer
53 views

Question about transformations and sums on uniformly distributed random variables.

I'm looking into a few problems as a hobby of mine, and found myself with the following problem: let $X$ be a random variable uniformly distributed on $[0,1]$. What is the probability that after $N$ ...
0
votes
2answers
28 views

Joint Probability Density Function, marginal probability density, joint cumulative distribution, probability [on hold]

Let $X$ and $Y$ have a joint probability density function given by $$ f_{X,Y} (x, y) = \begin{cases} \dfrac{12}{5} xy (1 + y) & \text{if } 0 \le x \le 1, 0 \le y \le 1 \\ 0 & ...
0
votes
0answers
27 views

Count number of repetitive customer visiting a restaurant [closed]

Suppose a restaurant at the end of year, come to know that he served 10800 Customers. What number or percentage of customers would be repetitive customers. Because all 10800 Can not be new customers. ...
0
votes
1answer
15 views

Constructing Confidence Interval given data [closed]

I have some data: 2216, 2225, 2318, 2237, 2301, 2255, 2249, 2281, 2275, 2204, 2263, 2295 Now the question says to construct a 95% confidence Interval on mean. Then it says to construct 99% confidence ...
0
votes
1answer
33 views

How to prove that convergence in MGF implies Convergence in Distribution?

I know that if the moment generating function of two distribution converges to the same function then the two distribution converges in CDF. But how can we prove this thing explicitly ?
0
votes
1answer
32 views

Distribution of marbles on number line

I have a set of marbles and a number line from 0 to infinity. Every step I either put a new marble on the number 0 or I move one existing marble (chosen uniformly) to the next number. The ratio ...
0
votes
1answer
29 views

A moment's question.

Let G be a (absolutely) continuous distribution such that $$\displaystyle{\int_{-\infty}^{\infty}{x^{2}dG(x)}}<\infty$$ or $$\displaystyle{\int_{0}^{1}{\left[G^{-}(t)\right]^{2}dt}}<\infty.$$ ...
15
votes
0answers
181 views
+250

Zombie outbreak on a $k$-regular graph

Suppose we have a zombie outbreak on a connected $k$-regular graph of order $n$. There are $n_0$ initially infected zombie nodes, and each turn, each zombie infects its neighbors with probability ...
1
vote
1answer
44 views

What's the distribution of $\Phi(X), X \in N(0,1)$?

In a course on statistics, this set of non-compulsory exercises were supplied (in Swedish). I'm stuck on 8.10. My translation of the exercise: The stochastic variable X has a $N(0,1)$ ...
0
votes
0answers
18 views

Repetitive sampling from the uniform and an unknown distribution

I am trying to model an experiment. We have n ''players''. Each one picks independently a sample from a continuous uniform distribution in the [0,$2^{64}$], let's call it $u_i$. He also picks a sample ...
0
votes
0answers
15 views

Distribution maximum with small sample related to large sample

Suppose the random variables $X_i$, $i=1,\cdots,n$ and $Y_j$, $j=1,\cdots,m$ all have distribution $F(x)$, with order statistics denoted by $X_{(i)}$ and $Y_{(j)}$. Assuming $n<m$ (e.g. $n=m/100$), ...
3
votes
0answers
21 views

Gamma distribution Norming constant for extreme minima

the norming constants for extreme maxima of Gamma distribution is known and is give in link.springer.com/article/10.1007/s10687-010-0125-3. I would like to know is there reference or paper that states ...
0
votes
0answers
33 views

Mathematical Probability and Statistics( all the math need)

I would like some suggestions about mathematical techniques and knowledge are required to understand and master 2nd year undergraduate probability and statistics. I am mature student with some ...
1
vote
0answers
29 views

Showing $\lambda_V(x)\leq \min\{\lambda_1(x),\cdots, \lambda_n(x)\}$.

Suppose $X_1, \cdots, X_n$ are independent, nonnegative continuous functions, each $X_i$ has hazard function $\lambda_i(x)$. If $V=\max\{X_1, \cdots, X_n\}$, I need to show that ...
0
votes
1answer
19 views

Negative Binomial distribution as a Gamma mixture distribution

Let $f(x;\theta)$ be the poisson frequency function with mean $\lambda$. and $p(\lambda)$ the Gamma distribution with mean $\mu$, and variance $\mu^2/\alpha$. I have to show that ...
3
votes
1answer
57 views

What is the joint probability distribution of number of balls after $n$ draws?

The following problem came into my mind when I am studying the Polya Urn Model. At the beginning, from a bin containing $c_1$ balls labeled $1$, $c_2$ balls labeled $2$, … , $c_m$ balls labeled $m$, ...
0
votes
1answer
32 views

Normalizing constants for Extreme value distributions

I have a question regarding the normalizing constants $\mu$ and $\sigma$ that appear in the following problem. Let the random variable $Y_n$ be $Y_n=max(a_1,a_{2},\cdots, a_n)$ and $X_{n}$ be ...
1
vote
2answers
46 views

If $X$ is distributed normally with mean $0$, is it correct to say $X$ and $-X$ “have the same distribution”?

Q: If $X$ is distributed normally with mean $0$, is it correct to say $X$ and $-X$ have the same distribution? In a way, this seems correct: both $X$ and $-X$ have the same probability density ...
0
votes
1answer
25 views

Using the Weibull Distribution, derive $E(X^k)$

If $X$~WEI$(\theta,\beta)$, derive $E(X^k)$ assuming $k\gt-\beta$. Note that $X$~WEI$(\theta,\beta)=\frac{\beta}{\theta^{\beta}}x^{\beta -1}e^{-({x}/{\theta})^{\beta}}$ I am having a very difficult ...
1
vote
0answers
20 views

Deriving joint distribution from expectation

Given two random variables $X$ and $Y$ and let $K$ be a constant value. Assume the expectation $\mathbb{E}[X(Y-K)^{+}]$ is given for all possible values of $K\geq 0$. Is there a way to derive the ...
4
votes
2answers
53 views

Homework problem - Ways to test if a density function is cumulative density function

I have a problem that states: Let $F : \mathbb R \to R$ be defined by $$F(x) =\begin{cases}e^{\frac{-1}{x}} &\text{if } x > 0\\ 0&\text{if } x \leq 0\end{cases}$$ Is $F$ a ...
0
votes
1answer
27 views

Conditional probability and distribution

Let Y ∼ Exp(1/5). Find P(Y ≤ 18|Y > 13). Could anyone give me any hints?
0
votes
1answer
30 views

Joint distribution of independent random variables

Say I have two independent random variables $X$ and $Y$ both having the exponential distribution. I.e. $f_X(x) = \lambda_1 e^{-\lambda_1 x}, \ x \ge 0, 0$ elsewhere $f_Y(y) = \lambda_2 e^{-\lambda_2 ...
-1
votes
0answers
29 views

Homework help - Random Variable min - can't understand what teacher wants me to do with problem

The problem is: Let X(1), . . . ,X n be independent random variables, with X(i) having an exponential with parameter λ(i) distribution, for any i. Then the distribution of the random variable X = ...
1
vote
6answers
168 views

Producing a CDF from a given PDF

So I have this PDF: $$ f(x)= \begin{cases} x + 3 & \text{ for } -3 \leq x < -2\\ 3 - x & \text{ for } 2 \leq x < 3\\ 0 & \text{ otherwise} \end{cases} $$ To make this a CDF, I ...
2
votes
2answers
59 views

Probability Distributions and Probability

Suppose $X \sim N(3, 4)$, and let $Y = X^2$. Find $\Pr(Y ≥ 12)$. What does $Y$ mean?
0
votes
2answers
31 views

Probability of CDF and PDF [closed]

Suppose continuous random variable $X$ has a cumulative distribution function $FX$ satisfying $FX(x) = 2x^2 − x^4$ for $0 \leq x \leq 1$. (a) Compute $\displaystyle P\left(\frac{1}{4}\leq X \leq ...