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

learn more… | top users | synonyms

0
votes
1answer
77 views

Realisation of a Poisson process

Is anyone able to explain the section highlighted in green?
0
votes
1answer
36 views

number of ones with neighbours in a random binary string

Consider a sequence of i.i.d. random variables $(\xi_i)_{1 \leq i \leq L}$ such that $\xi_1 \in \{0,1\}$ and $P(\xi=1)=p$. Introduce the function $N : \{0,1\}^{L} \rightarrow \mathbb{N}$ which counts ...
1
vote
2answers
21 views

Question about independence

First of all is true that given $X,Y$ two random variables indenpendent; $(X,Y)\in D\subset \mathbb{R}^2$ then $\text{Cov}(X,Y)=0$? I tried to prove it and this is my solution: If $D=[a,b]\times ...
0
votes
1answer
32 views

Find $k$ if $f(x)=k(1/2)^x$ for $x=1,2,3,$ and $0$ elsewhere.

Find $k$ if $f(x)=k(1/2)^x$ for $x=1,2,3,$ and $0$ elsewhere. Here is what I did: $$\int_1^3 k\left(\frac12\right)^{x}dx=1$$ After integrating, I found it to be: ...
2
votes
0answers
56 views

$X,Y\sim U(0,1)$ are independent, $W=\max (X,Y)$.

$X,Y\sim U(0,1)$ are independent, $W=\max (X,Y)$. How do I find the PDF of $W$? How do I find the expectation of $W$ at two ways: 1. with the PDF of $W$ and without the PDF of $W$. I ask this ...
1
vote
1answer
101 views

Can someone please help to understand the following probability

I was reading something on communication, then I came across the following equation: $Power_{rx}=Power_{tx}*|R|^2/(1+d^2)$ where $Power_{tx}$ and $d$ can be assume to be constant, and R is the ...
1
vote
2answers
64 views

Probability Generating Functions with Three Dice

Three identical dice are thrown. The dice are fair, that is, for all three dice the probability of turning up face $j$ is $1/6$, $1 \le j \le 6$. Let $X_1,\ X_2,\ X_3$ be the independent random ...
1
vote
2answers
88 views

Approximate distribution for the sample mean?

A random variable $X$ is said to follow a discrete uniform distribution if its probability function is given by $$p_X(x) = \left\{ \begin{array}{ll}\frac{1}{\theta}, & x = 1, 2, \ldots, ...
1
vote
2answers
112 views

Uncorrelated, Non Independent Random variables

I don't understand the parts highlighted in green. I understand that the supports imply that X and Y are not independent but not how the graph shows this graph. I'm a bit confused by all aspects of ...
1
vote
1answer
115 views

The Weibull as the limiting distribution of the Burr distribution

I often deal with "payout patterns" which are vectors of the cumulative percentage of a loss that has been paid over time. For example, for $t \in [0, 1, 2, 3, 4, 5]$ I may have $p_t = (5\%, 15\%, ...
0
votes
1answer
52 views

How to draw a distribution of the function?

I have $f(x)= \frac{1}{2}$ and $x\in[0,2]$. How i can draw a distribution for it ?
0
votes
1answer
28 views

Getting the pmf from probability generating function?

I uploaded a picture so that my question may be accurate. It is question 3a that I am struggling with. I've learnt that the P(X=r)is the rth derivative(w.r.t. t) of the pgf at t=0 . Divided by r ...
0
votes
1answer
46 views

Bivariate Transformation

Why can I not let $V=X$ in this transformation as opposed to $V=Y$? I have tried it with $V=X$ and i get a different joint pdf.
1
vote
2answers
278 views

Sum of two truncated gaussian

What is the CDF and the PDF (or approximation) of the sum of two truncated gaussian $X = TN_x(\mu_x,\sigma_x;a_x,b_x)$ and $Y = TN_y(\mu_y,\sigma_y;a_y,b_y)$ ? where $TN(\mu,\sigma;a,b)$ is a ...
0
votes
0answers
38 views

Expectation and Variation of dependent RVs

This is a really nice question, and while I can think of a solution to both parts, I wonder if there's a more elegant one to the latter: A fair 6-faced dice is tossed once. In a box there are 6 ...
0
votes
2answers
42 views

Prove the existence of exactly two maxima for a positive $L^1$ function

I have a function $f:\mathbb R \rightarrow \mathbb R^+$ with the following properties it is $L^1$ and $C^2$ it has one single extremum (maximum) at $x=0$ it is symmetric: $f(x)=f(-x)$. it is ...
0
votes
1answer
81 views

Finding expected value of mean of an estimator

We have a set of unidimensional data, $X_1, . . . , X_n$. : The data are drawn from a uniform distribution on the interval $[a, b]$. This model has two positive real parameters, a and b, such that $0 ...
2
votes
1answer
82 views

probability in roulette!

So I have read how to play roulette...still a little confused, and now I'm faced with a probability question about it which makes the problem a little harder. Please help me reason this where ...
0
votes
2answers
30 views

A joint PDF question: $\displaystyle f(x,y)=1-\frac x3-\frac y3$

Really stuck on this problem: If $\displaystyle f(x,y)=1-\frac x3-\frac y3$ for $0 \le x \le 2$ and $0 \le y \le h,$ the find $h$. I know I need to integrate but confused how to set it up. ...
1
vote
1answer
82 views

Finding the pdf of an estimator

We have a set of unidimensional data, $X_1, \ldots , X_n$ drawn from the positive reals. We define a model for its distribution: The data are drawn from a uniform distribution on the interval $[0, ...
1
vote
1answer
74 views

Probability of event in normal distribution

Let $X$ be a random variable that is normally distributed and $X_1,\ldots,X_n$ be (independet) copies of $X$, then we can estimate this probability by using a simple Monte-Carlo estimator: $p := P (X ...
1
vote
2answers
56 views

Weibull Distribution, what is $R^2$?

Given a Weibull Distribution $f_R$, how do I transform $R\to R^2$, and what is the distribution for $R^2$? Attempt: Since $f_R$ is distributed with parameter $k$ and $h$ as a function of $x$, so . ...
1
vote
1answer
93 views

How does a pdf of the difference of two random variables relate to the pdf of each random variable

Let $T_1$ and $T_2$ be non-negative continuous random variables (rv) denoted in the form $T_i = \mu_i + \sigma_i X_i$ for $i=1,2$ where \begin{eqnarray*} T_{1} &=&\mu _{1}+\sigma _{1}X_{1} \\ ...
1
vote
0answers
167 views

Finding the nth moment of the geometric distribution: Why does interchanging the derivative and summation operators not work after n=1?

I am trying (and failing) to find a recursive formula for the $nth$ moment of a geometric distribution. I have arrived at bogus results, and I think it has something to do with the convergence of the ...
1
vote
1answer
32 views

How to get a Gaussian curve fitting a given range of values?

I was trying to find a way to make a gaussian function out of a range of values: $1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9\ 10\ 11\ 12\ 13\ 14\ 15\ 16$ I want the mean to be the most probable value, $8$ and the ...
0
votes
1answer
20 views

Random variable with pdf proportional to Normal

I don't understand the step highlighted in green. I know $f_Z(z)= \frac{k}{\sqrt{2\pi}}$ $ e^{-\frac{z^2}{2}}$ when $z>-\frac{\mu}{\sigma}$ and $0$ elsewhere; but i'm stuck at this point.
1
vote
0answers
46 views

pdf of area of a circle

$X,Y$ are random variables with standard normal distribution (they are independent). $W$ is the area of the circle that has center at $(0,0)$ and passes through $(X,Y)$. What is the pdf of $W$? I ...
0
votes
1answer
117 views

Quick Question Integration with Joint PDF

Let $X_1, X_2, \ldots, X_n$ by independent and identically distributed random variables with probability density function (pdf) $$f_X(x) = \left\{\begin{array}{ll}1, & 0 < x < 1\\ 0, ...
0
votes
1answer
34 views

Find marginal and conditional distributions [closed]

Consider the probabiility density function $f_{X_1, X_2}(x_1, x_2) = \left\{\begin{matrix}\frac{1}{8x_2} \exp\left\{ -\left( \frac{x_1}{2x_2} + \frac{x_2}{4}\right)\right\}, & x_1 > 0, ...
0
votes
1answer
17 views

Expectation of minimum set of i.i.d random stopping times with the same distribution

What is the expectation of the minimum set of n i.i.d random stopping times? is it \frac{T}{n}
1
vote
1answer
81 views

Marginal PDF with dependent variables

I don't understand how to work out the limits of integration in b).
0
votes
1answer
35 views

Is this a Markov chain property

For $A,B$ measurable sets and $(X_n)_n$ a Markov chain. Do any of the following properties hold? $$P(X_2 \in B | X_1=x_1,X_0 \in A) = P(X_2 \in B|X_1=x_1)$$ or $$P(X_2 \in B|X_1 \in A,X_0=x_0) = ...
1
vote
0answers
79 views

Moment-generating function of a generalised normal random variable

Let $X$ be a random variable that follows the "version 1" generalised normal distribution described here, with p.d.f. ...
2
votes
1answer
79 views

Urn with balls, distribution of random variable

From an urn containing $6$ balls numerated $1,\ldots,6$ we randomly choose one, then again and stop only when we picked the ball with number $1$ on it. Let $X$ be the greatest number that appeared on ...
0
votes
2answers
35 views

Gamma and Exponential distribution question?

The working time of one bank has an exponential distribution with a parameter λ=0.1 (in minutes). You came in the bank, but there were already 35 people before you. What's the probability that all of ...
0
votes
1answer
59 views

Distribution of minimum of two uniforms given the maximum

Let $X_1$ and $X_2$ be two random variables uniformly distributed on $(0, 1)$. It is easy to calculate the distribution of minimum and maximum of these two numbers: $$ P[\max(X_1, X_2)<x] = x^2 $$ ...
0
votes
1answer
97 views

Reability of any CDF in Excel based on the binomial one as Cumfreq does.

I'm trying to get my own excel sheet to calculate the confidence limits or belts. I'm interesting in apply it to the Two Components Extreme Values (TCEV) Distribution for Flood Frequency Analysis and ...
0
votes
2answers
87 views

What is the pdf of $X,Y$?

We know that the common pdf of $X,Y$ is constant function, on the triangle $(0,0),(0,1),(2,0)$ (and out of this range the value of the function is zero). What is $f_X(x)$ and $f_Y(y)$? My solution: ...
0
votes
1answer
63 views

Probability Estimator

Hi I was going through the MIT 2005 Machine Learning homework assignments and I was having trouble understanding a few concepts in probability theory. I would be obliged if anyone could validate my ...
0
votes
0answers
54 views

How to calculate total probability from independent events

Assume Y is caused by two independent events A & B, upon investigating a data set carrying 1000 entries we see. $$\begin{align}\text{Number of occurrence of events } A = 497 \text{ and } ...
1
vote
4answers
54 views

Verification of this summation [closed]

How do I check or evaluate this summation $$\sum_{k\ge 0} \left(\frac12\right)^{k+1}k=1$$
1
vote
1answer
44 views

Check or Evaluate this Summation

How do I check or evaluate this summation$$\sum_{k=0}^n \frac{2(k+1)}{(n+1)(n+2)}=1$$ for $0\le k\le n$
0
votes
0answers
31 views

Solution to a certain moment problem

I'm looking for a function $f$ that satisfies $f(x)\geq0$ $\int f(x) \mathrm{d}x=1$ $\int xf(x) \mathrm{d}x=0$ $\int x^2f(x)\mathrm{d}x=1$ $\int x^4f(x)\mathrm{d}x=\delta$ $\int ...
1
vote
2answers
54 views

Binomial/Negative Binomial Distribution? Why not Poisson here?

When I looked at the below problem, I thought of Poisson immediately. I converted the rate to making 9/10 shots. However the answer told me to use the binomial/negative binomial distribution for parts ...
1
vote
2answers
42 views

Joint distribution: show the components of the joint distribution are independent.

Very odd question I think... Show that if $(X,Y)$ is a random vector in $\mathbb{R}^{2}$ with density $f_{(X,Y)}(x,y) = f(x)g(y)$ for a pair of non-negative functions $f$ and $g$, then $X$ has ...
1
vote
1answer
85 views

Conjugated priors (Pareto and Beta): Does this distribution have a name?

$$F_X(x)=\begin{cases} \quad\dfrac{\alpha}{\alpha+\theta}\left(\dfrac x\omega \right)^\theta &\text{ if } x<\omega \\ \\ 1-\dfrac{\theta}{\alpha+\theta}\left(\dfrac\omega x\right)^{\alpha} ...
1
vote
1answer
23 views

Distribution functions of a probability measure on a probability space $(\mathbb{R},\mathcal{B})$

Let $F$ denote a distribution function of a probability measure $P$ on a probability space $(\mathbb{R},\mathcal{B})$, where $\mathcal{B}$ denotes the Borel $\sigma$-algebra on $\mathbb{R}$. Given ...
4
votes
1answer
206 views

Derivation of the negative hypergeometric distribution

Suppose we've given an urn which contains $R$ red and $W$ white balls. These balls are drawn randomly from the urn and are not placed back. Let $X:=$ number of attempts, before we've drawn at least ...
0
votes
1answer
20 views

Is there a function to tell if two probability vectors' max values are in the same dimension?

Is there a method or function to tell two probability vectors' max values are in the same dimension? Or Is there a bound for the angle of two normalized probability vector which their max values are ...
1
vote
3answers
41 views

Determine the Cumulative Distributive Distribution(CDF) of a truncated value?

It is the last part(part h) that I am having problems with. I know you use integration and then split it into 2 parts. But how exactly do you do it ? A detailed answer would be very helpful ! ...