0
votes
0answers
16 views

Infinite fourth moment and maximum entropy

Alright, I expect this is a silly question, but I don't actually know, so. Suppose there is some random variable that's distributed on the reals, and all I know about the distribution is its mean ...
1
vote
1answer
20 views

Relation between the distribution functions of random variables $Y$ and $-Y$

I'm having trouble understanding a certain property of CDFs for negative random variables. Let $Y$ be an exponential random variable and let $f_y, F_Y$ denote the PDF and CDF respectively. My book ...
0
votes
1answer
25 views

Proving Continuity and equivalence

I have posted ths on the Quant Finance page as it is part of a QF problem but realised I may get a swifter response here! Iam working on a problem where I have successfully reduced a version of Black ...
0
votes
1answer
17 views

Combining biased coin flips

Suppose one has a biased coin $C_1$ with probability $p$ of landing heads and $(1-p)$ prob. of landing tails. If one wants to sample a coin $C_2$ with $p^2$ probability, one can flip the coin $C_1$ ...
0
votes
0answers
36 views

Compound Gaussian distribution

Let $\mathbf{a},\mathbf{b}\sim \mathcal{N}(\mathbf{0},\sigma^2\mathbb{I})$ and let $A$ be the circulant matrix defined to have $\mathbf{a}$ as its first column. I'm trying to study the behaviour of ...
3
votes
0answers
23 views

Probability and sums of prime factors

Of the first N natural numbers, we select two different numbers at random. We'll call the greater one A and the lesser one B. What is the probability (P) that the sum of A's prime factors is LESS than ...
2
votes
1answer
31 views

$X$ ~ $\Gamma(s,\lambda)$. Using $M_X(t)$ find the following…

a) $E(X) =$ ? b) $E(X^2) = $ ? c) $Var(X) = $ ? My thoughts: I know that moment-generating function for $\Gamma(k,\theta) = ( 1 - t\theta)^{-k}$ for $t < \frac{1}{\theta}$. I also know that ...
0
votes
1answer
37 views

Mixing continuous and discrete distributions

I'm wondering how, if it is at all possible, to write the p.d.f. for the following random variable. Given RVs $X_1$ and $X_2$ distributed according to some joint distribution having known density ...
2
votes
1answer
32 views

Deriving a lower bound for a probability involving a random variable $X$ with the Gamma distribution.

Question Let $X$ have the $Gamma(\alpha, \beta)$ density. I.e. $$f_X(x) = \frac{1}{\gamma(\alpha)\beta^\alpha}x^{\alpha-1}e^{-\frac{x}{\beta}}$$ when $x >0$ and $0$ elsewhere. The moment ...
0
votes
2answers
29 views

Probability density use for biased outcome

I am not a mathematics pro so do not mind if this is dumb let us suppose I have a method for generating random real values between 0 and 1 . All the values between 0 and 1 are equally likely to be ...
0
votes
0answers
27 views

Prove Joint distribution of estimators

Let $X_1,...,X_n$ iid r.v. with distribution F, with mean $\mu$ and median $\theta$.Assume that $Var(X_i)=\sigma^2$ and $F'(\theta)>0$. If $\hat{\mu}_n$ is the sample mean, and $\hat{\theta}_n$ the ...
1
vote
1answer
28 views

Probability density function of two uniformly distributed stochastic variables

I'm currently stuck on an exercise involving two independent stochastic variables X and Y. Both X and Y ~ U(0,1) (uniform distribution) The goal of the exercise is to calculate the probability ...
0
votes
1answer
37 views

Simple Expected value of MLE

Let $X_1,..., X_n$ be iid $Exp(\lambda)$. The MLE for $\lambda$ is $\hat{\lambda}=\frac{1}{\bar{X}}$, where $\bar{X}=1/n \sum^n_{i=1}X_i$ How can I conclude that $E(\hat{\lambda}) = n\lambda/(n-1)$? ...
0
votes
1answer
21 views

Distributions of local times of a single excursion of 1D random walk

Consider Simple Random Walk in one dimensions, starting from $x \in \mathbb{Z}^+$. The walker jumps to the right with probability $p$ and to the left with probability $1-p$. Assume $p \leq ...
1
vote
1answer
14 views

Histogram with different sample probabilities

Assume we are given a list of samples $L_1,L_2,\ldots,L_n$ of some random variable $L$. By classing them into bins we can easily create a standard histogram. But now suppose that we associate a ...
1
vote
0answers
20 views

Working with the sum of two independent random variables, and estimating a parameter

A network source sends a sequence of zeros and ones, $X_1, X_2, ...$ with $X_i$(iid) Bernoulli with $p = P(X_i = 1), 0 < p < 1$. Due to disturbances the received sequence is $Y_1, Y_2, ...$ ...
1
vote
1answer
42 views

$E(X_i \cdot I(X_i>\theta)$ expected value of when X is greater than the median.

Let $X_1, ..., X_n$ be iid with a distribution F. Let $\theta$ be the median of F. What is the value of $E(X_i \cdot I(X_j>\theta))$? If $i\neq j$, then $E(X_i \cdot I(X_j>\theta))= 1/2 \cdot ...
2
votes
1answer
39 views

Finding conditional distribution

Let $X$ and $Y$ be independent $Exp(1)$-distributed random variables. Find the conditional distribution of $X$ given that $X + Y = c$ ($c$ is a positive constant). this is my idea: $$f_{X \mid ...
0
votes
2answers
22 views

Laplace transform of noncentral chi-square distribution

I'm interested in non central chi-square distribution. More specifically, i want to derive the laplace transform of noncentral chi-sqruae disribution or density function. Let me know whether it ...
0
votes
0answers
9 views

Generate quadrature points from a distribution

Is there any method to generate quadrature points from any arbitrary probability distribution, $p_{X}\left(x\right)$? We already know about Gauss Hermite rule for Normal distribution, Gauss-Laguerre ...
1
vote
0answers
41 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
21 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
74 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
58 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
15 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 ...
2
votes
1answer
59 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 ...
0
votes
1answer
51 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
33 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 $X$ [closed]

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 ...
2
votes
3answers
42 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
57 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 [closed]

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 ...
2
votes
2answers
84 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
136 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
25 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) = ...
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
63 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
40 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
76 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
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 ...