1
vote
0answers
38 views

The probability that exactly / at-least $k$ numbers are in the correct position [duplicate]

Given a sequence of $[1,\dots,n]$ in random order: Let $P_k$ be the probability that exactly $k$ numbers are in the correct position Let $Q_k$ be the probability that at least $k$ numbers are in the ...
1
vote
1answer
20 views

Probability with Exp distribution, CDF, and multiple variables

You have a list of chores to do at home, but are expecting family to arrive shortly. The amount of time until their arrival (measured in hours) can be modeled as an Exp(2) random variable. Your list ...
1
vote
2answers
27 views

Probability with Uniform Distribution with Multiple Variables

Every time you go to a beach for vacation, you take home a little sand to keep as a souvenir. Over your lifetime, you have done this exactly 100 times. On each visit, the weight of sand you take home ...
2
votes
1answer
71 views

Probability Question: Who's right, me or the book?

I'll be giving some classes on probability theory later this year, and so I've been going through the textbook to check that I'm up to speed. I came across the following question: The discrete random ...
1
vote
3answers
53 views

Transformation of two independent uniform random variables

Suppose $X,Y \sim \text{Uniform} \left(0,1 \right)$ are independent. Then I need to find the PDF for $W=X/Y$. By the CDF technique this is seen to be : $$F_W( w)=\int_{0}^1 \int_{0}^{wy} ...
2
votes
0answers
10 views

Poisson-Binomial distribution approximated by binomial distribution

I am looking for strategies how to approximate poisson-binomial distribution (PB) via the binomial (B) distribution. I have seen a few papers [Ehm91,Roos01,LeCam59] on them. The papers uses total ...
3
votes
1answer
53 views

Example of non continuous random variable with continuous CDF

Can someone provide an example of $X$ being a non-continuous random variable with continuous cumulative distribution function? For instance: $X$ is discrete if it takes (at most) a countable number ...
-2
votes
0answers
32 views

Birthday Problem [on hold]

This is an extension of birthday problem, please help In a class of 85 students, let X be the number of students who share a birthday with at least two other members of the class. a) ...
-1
votes
2answers
26 views

Geometric distribution related probability questions [on hold]

I am learning Probability, and I have this problem. Suppose $X\sim {\cal Geom}(p_1)$ on $\{1,2,3,...\}$, $Y\sim {\cal Geom}(p_2)$ on $\{1,2,3,...\}$, and $X, Y$ are independent. Let $S=X+Y$. ...
0
votes
1answer
45 views

Transformation theorem, Cauchy distribution

I have derived the density for the ratio of two independent random variables,via the transformation formula. In this way: $V = X/Y $ and $ U = X $ inversion yields: $Y = U/V$ och $X =U$ , the ...
0
votes
0answers
40 views

What did I do wrong when using Jacobian transformation

A device containing two key components fails when, and only when, both components fail. The lifetimes, $T_1$ and $T_2$, of these components are independent with common density function $f (t) = ...
0
votes
2answers
32 views

probability of a flipped coin

A fair coin is flipped three times. Let $A$ be the event that a head occurs in the first flip and $B$ be the event that exactly one head occurs. a) Find $p(A/B)$ b) Are $A$ and $B$ independent? ...
-1
votes
2answers
31 views

If pages in a book have an iid Poisson number of errors, in 10 pages what is the probability that exactly 3 pages have exactly 1 error?

Suppose the number of spelling error on any given page in particular book can be modeled by a Poisson distribution with $\lambda=2$, and assume that the number of errors on different pages is ...
-1
votes
0answers
39 views

Choosing random marbles until one is divisible by $X$ [on hold]

A box contains twelve marbles on which they are numbered by $1,2,3,...,12$. Now let $X$ represent the number of marbles you must choose with replacement until you obtain one with a number that is ...
1
vote
3answers
39 views

Determine the law of $F^{-1}(U)$, $U$ uniformly distributed on $[0,1]$

i'm trying to understand the following problem Let $X$ be a real random variable, its distribution function is $F(t):\Bbb{P}(X\le t), \forall t\in \Bbb{R}$. Define the right-continuous inverse by ...
2
votes
2answers
56 views

conditional probability about sum and product rule

I am reading Bishop's Pattern Recognition and Machine Learning. In page 73, chapter 2.1. I can't understand the formula 2.19 : $$p(x=1|\mathcal{D})=\int_0^1 p(x=1|\mu)p(\mu|\mathcal{D})\text{d}\mu ...
0
votes
0answers
26 views

What is the optimal prize for a prize ticket in a raffle [on hold]

What, if any is the optimal price for a prize ticket given the value of a prize? For example if you were to raffle a TV and wanted to cover the cost of the prize? Let say the people were aware of how ...
3
votes
1answer
65 views

Sum of normally distributed independent random variables, where one has a different (exponential) unit

$$X \sim \mathcal{N}(\mu_X,\,\sigma_X^2)$$ $$Y \sim \mathcal{N}(\mu_Y,\,\sigma_Y^2)$$ $\mu_X$ and $\sigma_X$ have unit decibel watt ($\text{dBW}$); $\mu_Y$ and $\sigma_Y$ have unit watt ($\text{W}$). ...
0
votes
1answer
22 views

Polynomial joint pdf $f(x,y)$ such that of $f(x) \neq f(y)$

How can I build a polynomial joint pdf $f(x,y)$ for $x \in [x_1, x_2]$ and $y \in [y_1, y_2]$ such that of $f(x) \neq f(y)$ or equivalently, $x$ and $y$ are depended on each other?
5
votes
4answers
131 views

What is the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$ and some of the properties of $E[X \mid Y]$?

I was trying to understand both intuitively and rigorously what the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$. Let me tell you first the things that do make sense to me. $E[X\mid Y=y]$ makes ...
1
vote
1answer
25 views

Probability of multiple variables, geometric distribution?

You are on a basketball team, and at the end of every practice, you shoot half-court shots until you make one. Once you make a shot, you go home. Each half-court shot, independent of all other shots, ...
0
votes
1answer
24 views

How do you get the probability distribution of the sum of random variables by using the inverse of the transform?

I read the following statement: If X and Y are independent random variables, the distribution of their sum W = X + Y can be obtained by computing and then inverting the transform $M_W (s) = ...
-1
votes
0answers
16 views

what are the joint distribution functions and copula? [closed]

Let $U$ and $V$ be two independent uniform (0,1) random variables and let \begin{eqnarray*} R &=&\sqrt{\frac{U^{2}+V^{2}}{2}},\\ A &=&\frac{U+V}{2}, \\ G &=&\sqrt{UV}, \\ H ...
0
votes
1answer
36 views

Show that $Pr[X \gg Y]\approx 1$

Can one show (and how) that $$Pr[X \gg Y]\approx 1$$ for $$X:=\sum_{i=1}^k Bin\left(n\left(\frac{1}{2}\right)^i,i\right)$$ and $$Y:=\sum_{i=k+1}^{\infty} ...
0
votes
0answers
61 views

Prove $Pr[X + Y \geq x] \sim Pr[X \geq x]$

We have two independent random variables $X_n$ and $Y_n$, where $$X_n=\sum_{i=0}^n x_i$$ and $$Y_n=\sum_{j=0}^n y_j,$$ where $x_i$,$y_j$ are (non-identically) Bernoulli distributed and independent. ...
1
vote
2answers
37 views

Does $E[X]\gg E[Y]$ for independent RV imply that $Pr[X+Y \geq x] \sim Pr[ X \geq x]$?

We have two independent random variables $X$ and $Y$, where we know that $E[X]\gg E[Y]$, thus $\frac{E[Y]}{E[X]}\rightarrow 0$. I am now interested in $Pr[X+Y \geq x]$ and would like to show that ...
0
votes
1answer
20 views

Finding the boundaries of integration when calculating P(X + Y > a) or P(X + Y < b) (Jointly Distributed Continuous Random Variables)

I have a problem on setting the boundaries of integration when I'm trying to find probabilities like $P(X + Y > a)$ or $P(X + Y < b)$. For example, when I have $f(x,y) = \frac {x} {5}\ +\frac ...
1
vote
1answer
39 views

Poisson approximation of $X$ by $Poisson(E[X])$

I've tried to find something, but couldn't find anything about the following question. Is it possible to approximate any random variable $X$ with $E[X]=o(1)$ by a Poisson random variable ...
1
vote
2answers
73 views

Partial sum of binomial

I 'm trying to figure out a closed form solution for the following summation: $\sum_{j=0}^{\omega} j{n \choose j}p^{j}(1-p)^{n-j}$ where $\omega < n$ Is there any closed form solution?
0
votes
1answer
33 views

Probability of Random Variable Minus Random Variable

$X_1 , X_4$ ~ $ Binomial(18000,1/6)$. So $X_1+X_4$ ~ $Binomial(18000,1/3)$. I am asked to find $P(X_1-X_4)\leq 80)=?$. The solution is to find $Var(X_1-X_4)=6000$, $E[X_1-X_4]=0$ and then do the ...
0
votes
0answers
16 views

Maximum-Likelihood estimator [closed]

Given are $N$ Elements, labeled $1, ..., N$. The Elements are randomly shuffeled and then $n$ ($n < N$) Elements are choosen ($x_1, ..., x_n$). How do I found the Maximum-Likelihood estimator ...
0
votes
1answer
47 views

Geometric Distribution - How to show that a certain event is unplausible?

We have given a geometric distribution with parameter $p$ as well as some result $r$, which we doubt is an outcome of the given distribution. What is the best way to show that $r$ is indeed not a ...
2
votes
1answer
40 views

Expected Payment under limited policy

The unlimited severity distribution for claim amounts under an auto liability insurance policy is given by the cumulative distribution: $$ F(x) = 1 - 0.8e^{-0.02x}-0.2e^{-0.001x} , x \geq 0$$ ...
1
vote
1answer
56 views

The pdf of $X+Y$

$X,Y$ are independent. $X\sim U(0,1)$ and $$f_Y(y)=\cases{2y,\;0<y<1\\ 0,\;Else.}$$ What is the pdf of $X+Y$? (i.e. $f_{X+Y}$) I know that $$f_X(x)=\cases{1,\;0<x<1\\ 0,\;Else.}$$ But ...
4
votes
1answer
59 views

Expected maximum of a sequence of i.i.d. Poissons

Let $X_i \sim \mathrm{Pois}(1)$ be a sequence of $n$ i.i.d. random variables (with Poisson distribution with parameter 1). I'm interested in the asymptotic behavior of $$\mathbb E[\max_{i \in ...
0
votes
1answer
35 views

Question about exp. distribution

We know that $X\sim \exp(1),Y\sim \exp(2)$ and they are independent. What is $P(Y>X)$? exp=Exponential... Thank you!
1
vote
2answers
29 views

What is Cumulative Binomial probabilities?

I am new to this so don't know if I am asking the right question as I just read about its usage but didn't know what exactly a Cumulative Binomial probability is. So my question is, What is ...
0
votes
0answers
42 views

Is the following probability distribution stationary/constant

For a conservative system, we know that angular momentum, $l$, and total energy, $E$, are constant, i.e. $\dot{l}=\frac{dl}{dt} = 0$ and $\dot{E}=\frac{dE}{dt} = 0$, where $t$ indicates time. Let ...
1
vote
0answers
23 views

How to find the conditional probability [closed]

I need to calculate the joint probability $p\left(x,y,z,\dot{x},\dot{y},\dot{z}\right)$ using the following relation: \begin{equation} p\left(x,y,z,\dot{x},\dot{y},\dot{z}\right) = ...
0
votes
1answer
36 views

Understand step in computing marginal distribution of restricted Boltzmann Distribution

Proof taken from http://image.diku.dk/igel/paper/AItRBM-proof.pdf (page 24) I understand everything up to and including: (1) $$p(\textbf{v}) = \frac{1}{Z}e^{\sum_{j=1}^mb_jv_j} \prod_{i=1}^n\sum ...
0
votes
1answer
31 views

Box-Muller method for correlated normals

The standard Box-Muller method produces two independent normal variables given two uniform ones. Is it possible to extend the method such that given a correlation coefficient $\rho\in[-1, 1]$ and two ...
1
vote
1answer
39 views

Find Limiting Distribution of $|X_n|$

Let $Z_1,Z_2,...,Z_n,...$ be a sequence of independent standard normal random variables. Let $X_n=\sum^n_{k=1}\frac{Z_k}{\sqrt{k}}$. Does the limiting distribution of $|X_n|$ exists? If yes, find it; ...
0
votes
1answer
60 views

Operations on Random Variables

It is known that the equivalent resistance of a parallel combination of two resistors is equal to \begin{align*} R = \frac{R_1R_2}{R_1+R_2} \end{align*} which could be also written as ...
-1
votes
2answers
29 views

We are making a Bernoulli experiment…

We are making series of independent Bernoulli experiment with $\frac13$ chance to success. What is the probability that we got success at the first experiment, if we know that we get two successes at ...
0
votes
1answer
33 views

Method of moments for Beta $(\alpha_1,\alpha_2)$ distribution

I am trying to solve for the first two moments of a Beta$(\alpha_1,\alpha_2)$ distribution. We know that the first moment is equal to: $\mu_1 = \frac{\alpha_1}{\alpha_1+\alpha_2}$ and the second ...
3
votes
1answer
50 views

Prove Number of Arrivals $N(s)$ up to time $s$ follows $\mathrm{Poisson}(\lambda s)$ Distribution

This comes from my self-study of Durrett's "Essentials of Stochastic Processes" book, page 97. Definition Let $\tau_1,\tau_2,\ldots$ be independent $\mathrm{exponential}(\lambda)$ random variables. ...
0
votes
1answer
41 views

What are random variables and its connection with functionals?

Here is an image of the conversation which I had with my Prof. (He's the one in violet and myself in orange) The topic was random variables and other probability related definitions. I tried to ...
0
votes
0answers
12 views

Combining independent Gaussian probabilities

I am using three Gaussian distributions with which I generate random numbers to represent many candidate xyz points. I use some selection criteria (details not particularly relevant) to decide on ...
-1
votes
1answer
23 views

Asymoptotic distribution of identically distributed random variables [closed]

$Y_1, Y_2, ..., Y_N$ are independent and identically distributed random variables with the distribution function $F := F_{Y_1}$ and $F'_n(y) = \frac{1}{n}\sum_{i=1}^{n}\mathbf{1}_{\{Y_i \leq x\}}$ as ...
0
votes
1answer
24 views

Distribution of the sum of two independent normal variables [duplicate]

Given the two variables $A\sim \text{N}(\mu, \phi^2)$ and $B\sim \text{N}(\xi, \omega^2)$ with $\mu, \xi \in R$ and $\phi^2, \omega^2 > 0$ how do I prove that $C := A + B\sim \text{N}(\mu + \xi, ...