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

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0
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1answer
35 views

Discrete Uniform Probability: isn't my textbook just wrong?

My textbook is showing me examples of discrete probability distributions, one of them is in the picture: I learned in Calculus that the summation of the series $1/n$ where $n\to \infty$ is ...
0
votes
1answer
27 views

What type of distribution is this?

$$p(x) = \frac{1 }{\theta} . \frac{(ln\theta)^x}{x!} $$ I have no idea what type of distribution is this? And what will be its first and second moment?
1
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0answers
35 views

Let $X$ be a random variable with cumulative distribution function $F_X$, find $F_Y$ for $Y = \sin(X)$ and $Y = \cos(X)$

Let $X$ be a random variable with continuous cumulative distribution function $F_X$, find $F_Y$ for $Y = \sin(X)$ and $Y = \cos(X)$ My approach: I would say that $F_Y(y) = \mathbb{P}(Y \leq y) = 1$...
2
votes
0answers
27 views

'Finding' a normally distributed random variable

Let a random variable $Z$ have a standard normal distribution. Suppose that we start at $0$. We 'walk' right, along the number line, till we reach $a$. We then turn around, walk back, past $0$, till ...
-1
votes
1answer
34 views

How to find value of RV for this condtion [closed]

The probability density function of the time to failure of an electronic component in a copier(in hours) is $$f(x)=\exp(-x/1000)/1000$$ for $x>0$ and $f(x)=0$ for $x\leq 0$. How determine the ...
0
votes
1answer
28 views

Extension of Simple Random Sample without Replacement

First, I will show an extension of simple random sample without replacement, and then put forward the question. (1) From Wiki, a simple random sample is a subset of individuals (a sample) chosen ...
0
votes
1answer
19 views

Most likely value of negative binomial random variable

If $X$ is a negative binomial random variable let's say with $p =0.2$ and $r = 4$ then how can we calculate most likely value of $X$? I thought it is expected value but that is $20$ and I guess most ...
2
votes
1answer
69 views

Limit of a multiple integral [closed]

$$\displaystyle\lim\limits_{n\to \infty} \int_0^1 \int_0^1...\int_0^1 \cos^2\left(\frac{\pi}{2n}(x_1+x_2+...x_n)\right)dx_1 dx_2...dx_n$$ I don't know how to begin.
0
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0answers
27 views

How to define a uniform probability distribution over a convex polytope / polyhedra and add them?

Let $P$ be a convex 3d polyhedra / 2d polytope constrained by a set of linear inequalities $Ax<= b$. 1.How to define a uniform probability distribution over a polytope/polyhedra? Let us say we ...
0
votes
1answer
38 views

Loss probability and VaR

I would like to estimate Value-at-Risk analytically and through delta-gamma aproximation. I don't know if my idea is ok, but i would like to build a portfolio of European option. Suppose that in this ...
0
votes
1answer
34 views

generating distribution density function for a system of events with exponential distribution

We have a system in which events happen after each other. Events are i.i.d. An event, shown by random variable $X$, follows exponential distribution with $E(X)=\frac{1}{\lambda}$. We suppose the ...
0
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0answers
15 views

The distribution described by the inverse of the sum of two power law distributions

I'm trying to fit real-world data, and have been trying a whole lot of different distributions, some of which I've been making up on the fly. One distribution that fits the data very well is described ...
0
votes
1answer
28 views

Discrete probability distribution where (max - min) $\not=$ average [closed]

Let $X$ be a discrete random variable with $N$ possible values and some distribution. What are some well-known distributions for $X$ such that $\mathbb{P}(x_{max}) + \mathbb{P}(x_{min}) \not= \...
1
vote
2answers
25 views

Show that $Z^2 \sim Gamma(\frac{1}{2},\frac{1}{2})$ where $Z \sim N(0,1)$ using change of variables method

So I found a couple of solutions that use integration but I would like to solve this using random variable transformations. I found a similar solution but was hoping to get some more clarity. Here is ...
0
votes
0answers
11 views

Help with finding a particular joint distribution of a Bayesian Network

Consider a Bayesian Network defined by the following matrix: $$\left[\begin{array}{ccccccc} 0&1&1&0&0&0&0 \\ 0&0&0&1&1&0&...
1
vote
1answer
25 views

Finding the distribution of a random vector in a conditional probability problem [closed]

Players A and B are playing a game of drawing coins from two boxes without returning/replacing them. Box1 has three coins with values 0, 1 and 2 and Box2 has two coins with values 1 and 2. In the game,...
1
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2answers
25 views

Conditional Survival Function

Let $X$ and $Y$ be random variables with joint distribution $F(x,y)$. Let $F(y|x)=P(Y\leq y|X=x)$ denote the conditional distribution of $Y$ given $X=x$. Let $S(y|x)=P(Y>y|X=x)$ the conditional ...
1
vote
1answer
27 views

Problem with the density of the compound distribution

My problem is to calculate $E[\max(S-5000, 0)]$ where $$S = \sum_{i=1}^{N} X_i,$$ $N$ is a random variable with geometric distribution, parametrized as follows: $$P(N=n) = \frac{\beta^n}{(1+\beta)^{...
-1
votes
1answer
16 views

what is the relation between variance and mean for two Laplace distribution variables? [closed]

if X and Y are two Laplace distribution variables. assume that $sx^2$ and mx are the variance and mean for variable X. And $sy^2$ and my are the variance and mean for variable Y. What is the relation ...
1
vote
1answer
31 views

Is this a Multinomial distribution?

I find it hard to notice when do I have a Multinomial distribution and if its possible to "transform" problems into a Multinomial distribution problems. For example I have the following exercise: ...
0
votes
1answer
21 views

CDF probability problem [closed]

For random variable $X$ and $Y$ defined on the same sample space, let $U = \min\{X,Y\}$ and $V = \max\{X,Y\}$. a) Determine the CDF of $V$ in terms of the joint CDF of $X$ and $Y$ b) ...
0
votes
1answer
89 views

Average number of terms required in a sum of exponential variables to reach a specific limit

I have a sum $Y=\sum_{i=1}^{\infty}(X_i-t)u(X_i-t)$ where all $X_i's$ are i.i.d exponentially distributed random variables with parameter $\lambda$ and $t$ is a constant. I want to know how many term ...
2
votes
3answers
45 views

Expected value of a geometric distribution with first step analysis.

I am trying to understand the "story proof" found in this lecture. I am a bit confused as how the expected value of a random variable differs from the the random variable itself when considering ...
0
votes
2answers
38 views

Expected value of $Y=\prod_{j=1}^{N} X_{j}$, where $N\sim\operatorname{Poisson}(\lambda)$.

Let $$Y=\prod_{j=1}^{N} X_{j},$$ $X_{j}$, for $j=1,2, ...$, are identically and independently distributed with mean $\lambda$ and variance $\sigma^{2}$ and $N\sim \operatorname{Poisson}(\lambda).$ ...
3
votes
3answers
183 views

Uniform bounded of Riemann-like sum and improper integral

For any $h>0$, suppose $\{(y_i,y_{i+1}]\mid i\in \mathbb{Z}\}$ be a uniform partition of $\mathbb{R}$ with mesh size $h$. I am considering under what condition for a continuous transition density ...
1
vote
2answers
33 views

Finding the probably density function of $Z=\sqrt{X^2+Y^2}$ where Y~N(0,1) and X~N(0,1)

X and Y are normal random variables that are independant. Finding the probably density function of $Z=\sqrt{X^2+Y^2}$ where Y~N(0,1) and X~N(0,1). Attempt: Let $z \in R$. If $z \lt 0$ then $P(Z\le ...
1
vote
2answers
43 views

Show that $\frac{X}{X+Y}\sim Beta(\alpha,\beta)$

Let IG denote Inverse-Gamma distribution Inverse-Gamma. If $X\sim IG(\alpha,1)$ and $Y\sim IG(\beta,1)$. Show that $\frac{X}{X+Y}\sim Beta(\alpha,\beta)$ I tried with jacobian transformation ...
2
votes
1answer
26 views

Joint distribution weird result

We have 3x3 board Each board cell has $0.5$ chance to be white (there is no dependency between different cell colors) Let X = number of white rows (a row with only white cells on it) Let ...
0
votes
2answers
36 views

Expected value of hyper geometric distribution

Question: Say, I have to calculate the expected value of the number of aces from a deck. I pick cards without replacement. Thus, the distribution of the number of the cards is hypergeometric. ...
0
votes
1answer
23 views

What is the closest apporoximation for pdf of log-normal distribution?

I am unable to compute a complex integral which uses the pdf of log-normal distribution. Hence, I want to replace the pdf of log-normal distribution with an alternate function(s) (piece-wise ...
0
votes
0answers
18 views

Contradiction in the CDF derivation from two different strategies

I have a sum $Y=X_1u(X_1-t)+\cdots X_Nu(X_N-t)$ where all $X_i's$ are i.i.d with exponential distribution with parameter $1$ and $u(x)$ is the unit step function. As can be seen from the expression of ...
1
vote
2answers
28 views

Right limit of integral

Following is the joint PDF of RV $X,Y$ and $Z$ $$f(x, y, z) =\begin{cases} kxy^2z;& 0 < x,y < 1, 0 < z < 2,\\ 0,& \text{elsewhere}. \end{cases}.$$ To find value of $k$ I tried ...
0
votes
0answers
39 views

Integral of non-Gaussian distribution, random walk?

I would like to evaluate $$ F = \frac{\mathbb{E} \left\{\left(\int_0^T x^3(t) dt \right)^2\right\}}{\mathbb{E} \left\{\left(\int_0^T x(t) dt \right)^2 \right\} } \approx \frac{\mathbb{E} \left\{\left(...
0
votes
1answer
15 views

Solving for a Conjugate Prior in search of MAP estimator

I am trying to prove that if a given random variable $X \sim Exp(\lambda)$ and $\lambda \sim Gamma(\alpha,\beta)$ hen $\lambda | X \sim Gamma(\alpha^{*},\beta^{*})$ for some parameters $\alpha^{*}$ ...
0
votes
2answers
50 views

CDF of sum of N exponentially distributed random variables with condition

I have $Y=X_1u(X_1-x_{th})+X_2u(X_2-x_{th})+\cdots+X_Nu(X_N-x_{th})$, with all the $X_i\sim\lambda e^{-\lambda}$, $u(t)$ is the unit step function and $x_{th}$ being the threshold which means that any ...
-1
votes
0answers
44 views

Three random variables with exponential distributions

Having $X$, $Y$ and $Z$ as three independent identical random variables all having exponential distribution $E(X)=E(Y)=E(Z)=\frac{1}{\lambda}$, What is the answer of the following probability: $P(X+Y&...
1
vote
1answer
34 views

Why aren't CDFs left-continuous?

Let $F$ be a cumulative density function on $\mathbb{R}$. From an argument in a textbook, it is shown that $F$ must be right-continuous: Let $x$ be a real number and let $y_1$, $y_2$, $\ldots$ be ...
0
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0answers
27 views

Why does the following equality hold in proving Meyer's inequality?

I have a question in proving Meyer's inequality. The proof I read is taken from the book "Malliavin Calculus and related topics" by Nualart. I just have one equality which I am not sure, I will ...
4
votes
1answer
148 views

Limit theorem for changed time

This post seems long, but its almost everything proofed in this post. Only one step seems to be left, for the desired proof. I would be very gratefull for any help. The setup Given a Levy-Process $U_{...
0
votes
1answer
29 views

Let $X$ have a Poisson distribution with parameter $\lambda$.

Let $X$ have a Poisson distribution with parameter $\lambda$. (a) Show that the moment-generating function of $$Y = \dfrac{(X − \lambda)}{\sqrt{\lambda}}$$ is given by $$M_Y(t)=exp(\lambda e^{\frac{t}...
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votes
1answer
32 views

$P(T ≤ 5 | T ≥ 2)$ from CDF [closed]

If for discrete random variable T the CDF is defined as $$F(t) = \begin{cases} 0, & \text{t<1}\\ 1/4, & \text{1≤t<3}\\ 1/2, & \text{3≤t<5}\\ 3/4, & \text{5≤t<7}\\ 1, &...
0
votes
2answers
28 views

Chebyshev's inequality to find probability of interval

Here is how I solved the problem: $$ X\sim N(\mu=.13, \sigma^2=.005^2)\\ .12\le x\le .14 \\ \mu-2\sigma\le x \le \mu+2\sigma\\ $$ Using Tchebychev's inequality, I get $$ P(|x-\mu|\le 2\sigma)=1-\frac ...
1
vote
1answer
28 views

Can I run a regression when both independent and dependent variables are all dichotomous?

I have conducted a survey where all my questions are asked in a dichotomous manner (Yes/No). Eg IV:"Are you a smoker?", "Are you obese", "Is your gender male/Female" etc. DV: "Have you ever had a ...
0
votes
1answer
14 views

How the value of denominator calculated here?

I found this example in a book and it has to find probability distribution as stated below: If a car agency sells 50% of its inventory of a certain foreign car equipped with side airbags, find a ...
0
votes
4answers
41 views

Convolution: Give a proof that $f_T(t)=\int_{-\infty}^{\infty}f_X(x)f_Y(t-x)dx$ where $f_T(t)$ is the PDF of random variable T

Here is the question: Let $X$ and $Y$ be independent, continuous r.v.s with PDFs $f_X$ and $f_Y$ respectively, and let $T=X+Y$. Find the join PDF of $T$ and $X$, and use this to give a proof that $...
2
votes
2answers
24 views

Getting the marginal distribution from the joint pdf

To bein with, I did the following calculations: $$ Y\sim Uniform(0,x)\\ f_x(x)=\{\frac{1}{x^2},x\ge1\}\\ f_{y|x}(y)=\{\frac{1}{x},0\le y \le x\}\\ f(x,y)=f_x(x)f_{y|x}(y)=\frac{1}{x^3},x\ge 1,0\le y\...
1
vote
1answer
26 views

Joint probability distribution.

I am trying to calculate P(Y|Z) given the following distribution $\ P(X,Y,Z) = P(X)P(Z)P(Y|X,Z)$ Now, initially I did the following calculation. $$P(Y|Z)=\sum_{x}^{}P(X,Y|Z)=\sum_{x}^{}P(X,Y,Z)/P(Z)=\...
0
votes
0answers
23 views

bloom filter: how to estimate probability and tune the filter

My goal is to tune bloom filter in such a way so that I'd get best possible results. I have a dictionary of N=100000 strings, and I have distinct sets of strings S0, S1, S2. For each string from ...
2
votes
0answers
32 views

joint-probability of Langevin equation

I am working on Langevin equations: $\frac{dx}{dt}=u$ $m\frac{du}{dt}= -\gamma u + \theta(t)$ where $\theta(t)$ is delta-correlated in time Gauss-distributed noise with zero-mean $\langle \theta (...
1
vote
1answer
37 views

Change of Uniform Continuous Variable

Let $X$ be a $U(-1, 1)$ random variable, we define $Y = X^4$. Calculate the correlation coefficient between both variables. Are they uncorrelated? PS. I don't know how to use MatJax equations, I'm ...