Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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2
votes
1answer
32 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
18 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
1answer
16 views

Distribution combinations

How many ways can 25 identical pencils be distributed between two people?.Each all pencils must be shared out. A) Each person must have at least 5 pencils? B) Each person must have at least 7 ...
1
vote
0answers
8 views

elementary proof for discrete Kantorovich-Rubinstein theorem?

For the Kantorovich-Rubinstein theorem, please see the wikipedia page http://en.wikipedia.org/wiki/Wasserstein_metric (which does not contain a reference for the proof). I am only interested in the ...
0
votes
0answers
8 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
31 views

Law and Brownian Bridge

Let $Z_{t}= W_{t}-tW_{1}$ and $Y_{1}=\sup_{0\leq t\leq 1}Z_{t}$, $(W_t, t \geq 0)$ standard Brownian motion Find the law of $Y_{1}$ I know that $\textbf{P}(\sup_{0\leq t\leq 1}W_{t}\geq x , ...
3
votes
2answers
46 views

What is the reason for the one-half in the normal pdf's gaussian (i.e. : why $\exp(-x^{2}/2)$ instead of $\exp(-x^{2})$ )

It doesn't seem to relate to normalization, as the normalizing constant adapts to every possible "upstairs formulation", and in the standard case is $\displaystyle\frac{1}{\sqrt{2\pi}}$. Does it ...
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 ...
0
votes
1answer
43 views

Martingale based on normal PDF evaluated at normalized i.i.d. sums

I have the following problem. $(X_n)_{n\geq0}, n\in\mathrm{R}$, is a family of iid r.v., normally distributed $\mathcal{N}(0,1)$ $\mathcal{F_n} := \sigma((X_i)_{1\leq i\leq n})$ $x\in\mathrm{R}, ...
-3
votes
0answers
9 views

Proof of “Normal approximation to the log-normal distribution” [on hold]

I saw the post about the normal approximation to lognormal (Normal approximation to the log-normal distribution). The proof is shown as well. Yet as I'm looking for the proof in a journal article form ...
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
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
56 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} ...
1
vote
2answers
33 views

P/1 Actuary Exam Question

I was doing problems and came across this one and was wondering why the P[1<=x<=2] is F(2) - lim (x->1-) F(x) rather than F(2)-F(1)? Could someone please explain this for me?
2
votes
0answers
12 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
34 views

$\sum_{k\ge 0} e^{-an} \frac{(an)^k}{k!}f(\frac{k}{n}) = \Bbb{E}\left(f\left(\dfrac{X_1+\cdots + X_n}{n}\right)\right)$

Hello everybody i need to show following equality $$\sum_{k\ge 0} e^{-an} \frac{(an)^k}{k!}f(\frac{k}{n}) = \Bbb{E}\left(f\left(\dfrac{X_1+\cdots + X_n}{n}\right)\right)$$ Where $(X_i)_i$ are ...
1
vote
1answer
46 views

Exponential of Squared Brownian Motion

Long time lurker, first time posting! Have a problem, that looks familiar but I can't put my finger on it. Need to calculate $\mathbb{E} [\exp(aW_T^2)|F_t]$ where $W_t$ is an $F_t$ adapted standard ...
2
votes
1answer
56 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 ...
1
vote
2answers
24 views

Joint density distribution and Variance

I was wondering if there is a way to calculate the joint distribution of two fully correlated variables, both with known distributions, expected value and variance, without knowing the conditional ...
0
votes
0answers
49 views

A city wants to encourage downtown

could you please help me with this ( part d ) A city wants to encourage downtown employees to use public transportation. Below is the time in minutes to get to work on one morning according to ...
0
votes
1answer
50 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 ...
2
votes
1answer
18 views

Continuous Random Variables: Uniform

Problem: A person drives to work via a road with a single traffic signal. The light cycles, green for 45 seconds, red for 15 seconds – ignore yellow. Assume the person approaches the signal at a ...
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) = ...
2
votes
1answer
25 views

Column space of stochastic matrix.

Consider an arbitrary matrix $M \in \mathbb{R}^{n \times m}$. Denote the column space of $M$ as $\mathcal{C}(M)$. Is it always possible to construct a right stochastic matrix $S$ such that ...
1
vote
1answer
35 views

Question about a change of variable used to compute $E(X)$ from the CDF of $X$

I was studying a proof where the author shows that if the range of x is $\mathbb R_+$ and $F$ is the cumulative distribution function then: $$E[X] = \int_{0}^\infty (1-F(x))dx $$ However on one ...
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 ...
0
votes
0answers
7 views

finding the variance of a complex scenario

construct a sequence of cubes with one vertex at the origin and having lengths 1/2^x, x=1,2,3,... Select a point randomly from the unit cube with one vertex at the origin and sides of length 1. Let ...
1
vote
3answers
40 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 [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 ...
0
votes
0answers
12 views

Sum of a truncated normal random variable and normal random variable (correlated) [duplicate]

I'm wondering if there is a closed form of pdf of sum of a "correlated" normal random variable and a truncated normal random variable. I found a paper providing the pdf for "uncorrelated case" but I ...
0
votes
2answers
16 views

What is the probability density function of $g(S) =S/2$ for a triangle pdf

Say we have the following "triangle" probability density function: $ p_{S}(s) = \left\{ \begin{array}{lr} s & : s \in[0,1]\\ 2-s & : s \in [1,2]\\ 0 & o.w. ...
2
votes
2answers
78 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 ($\text{W}$). ...
1
vote
0answers
17 views

Integrating with indicator functions

I want to evaluate $$\int_{-\infty}^{\infty}(A_1e^{-\beta_1(b-x-y)}+B_1e^{-\beta_2(b-x-y)})(pn_1e^{-n_1y}1_{\{y\geq0\}}+qn_2e^{n_2y}1_{\{y<0\}})dy,$$ $b>x, \beta_1<n<\beta_2$. I am trying ...
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
133 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
0answers
26 views

How to estimate the covariance matrix if the unnormalized pdf is known but integral is intractable?

Assume a $d$-dimensional random vector $x$, whose unnormalized pdf is known as the product of N multivariate t-distribution: $$Pr(x)\propto\prod_{i=1}^nt_{\nu_i,\mu_i,\Sigma_i}(x)$$ Is there any ...
0
votes
0answers
21 views

How to estimate the covariance matrix if the unnormalized pdf is known but integral is intractable? [duplicate]

Assume a $d$-dimensional random vector $x$, whose unnormalized pdf is known as the product of N multivariate t-distribution: $$Pr(x)\propto\prod_{i=1}^nt_{\nu_i,\mu_i,\Sigma_i}(x)$$ Is there any ...
-1
votes
0answers
18 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
0answers
21 views

Convergence of Beta Distribution to Bernoulli Distribution [closed]

How will I show that the $$\beta\left(\frac{a}{n} , \frac{b}{n} \right)$$ distribution converges to the $$\operatorname{Bernoulli}\left( \frac{a}{a+b} \right)$$ distribution?
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 ...
1
vote
1answer
18 views

Singular vector of random Gaussian matrix

Suppose $\Omega$ is a Gaussian matrix with entries distributed i.d.d. according to normal distribution $\mathcal{N}(0,1)$. Let $U \Sigma V^{\mathsf T}$ be its singular value decomposition. What would ...
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 ...