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

learn more… | top users | synonyms

0
votes
0answers
11 views

Survival and Cumulative distribution of Binomial

Could someone please point me to a document that presents the Survival and Cumulative distribution of the Binomial distribution? Even better, could you provide the equations directly?
0
votes
0answers
9 views

How to do non-analog sampling from a pdf

I have a distribution given by $$ f(x)=\frac{1}{2}\sinh(\sqrt{2x})e^{-x}. $$ Due to the shape of this distribution, most of my samples will be in the range $x\in(0,2]$. However, I am interested ...
0
votes
1answer
2k views

Joint cdf and pdf of the max and min of independent exponential RVs

Let $X$ and $Y$ be independent random variables. Each has an exponential distribution with parameter $\lambda$. Define two new random variables by $W = \min({X,Y}) $ $Z = \max({X,Y})$ Find the ...
1
vote
1answer
29 views

Order Statistics with two Groups of Draws

Let $X_{1},X_{2},\ldots,X_{m},\ldots,X_{n}$ be independently drawn from a distribution $F$ and let $Y_{k}^{(n)}$ be the $k$-th order statistic (Convention: ...
1
vote
0answers
23 views

Expected maximum of maxima

Let $F(x)$ denote some CDF, and $\{f_i\}_{i=0}^m$ be a set of random variables independently drawn from that distribution. I would like to compute $$ E\bigg[ \max\bigg\{ \max\bigg\{\{f_i\}_{i=0}^m ...
6
votes
2answers
2k views

Derivation of the maximum entropy distribution

I am reading a book and having trouble following something. The problem is to try to maximize the differential entropy $-\int_{0}^{\infty}p(r)\log p(r)$ under the constraints that ...
1
vote
0answers
5 views

Distribution of an autoregressive process

Say that we are given a AR process. Also, lets assume that the residuals of the process come form a distribution $P_R$ which, while known to us, is not necessarily normal. Can I derive the ...
0
votes
1answer
34 views

Absolute value of a random variable

I have never encountered this concept before. Is this equation valid for $y>0$? $$\mathbb{P}(|X|>y) = \mathbb{P}(-|X|<y<|X|)$$ What about this? $$\mathbb{P}(|X|>y) = ...
0
votes
1answer
36 views

Differences of heads and tails in a fair coin

I'm very new to this so I would appreciate a detailed explanation. I wrote a very simple Matlab program that "flips a coin" (randi([1 2])) $n$ times. Every time I ...
2
votes
2answers
20 views

Linear Transformation of Poisson Point Process

Suppose we have a random variable that follows a Poisson Point Process: $ X \sim poisson(\lambda t) $ and a function $f(x) = ax + b $ where $a,b \in \mathbb{R}$. What is the pdf of $Y = aX + b$? I ...
1
vote
1answer
40 views

If $X = X_1+\dotsb+X_N$, and $N\sim\operatorname{Pois}(\lambda)$, then what is the distribution of $X$ given $N$?

I have a question that I'm really struggling with (below): It's hard to understand exactly what is the question actually states. does this mean the number of trials itself is a distribution with a ...
0
votes
0answers
6 views

Which probability distribution(s) $f(x)$ allow for a closed form solution to $\int\left(x-a\right)^{-\gamma}f\left(x\right)dx$?

I'm trying to find if there is a specific probability distribution $f\left(x\right)$ (or many) such that the following integral $$\int\left(x-a\right)^{-\gamma}f\left(x\right)dx$$ has a closed form ...
0
votes
0answers
11 views

How to approach deriving a folder distribution's pdf from original pdf?

Let's say I have an Erlang-distributed random variable $x$, and now I'm only taking the samples of $x$ for which it holds that $x>T$, where $T$ is some constant. The probability $P[x>T]$ can be ...
1
vote
1answer
64 views

Sum of probabilities is infinite

I'm stucked solving this problem: Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d. random variables with exponential distribution and $\lambda=1$. Show that ...
1
vote
1answer
4k views

Understanding the difference between normal distribution and lognormal distribution

I'm having trouble understanding the difference between a normal distribution and lognormal distribution. Here's what I've done so far. Definitions of lognormal curves: "A continuous distribution in ...
0
votes
1answer
95 views

What is an Hypergeometric distribution where the last event must be a success?

I'm trying to find out the name of a distribution that is like negative binomial, only for finite population and without replacement. Or like Hypergeometric distribution where the last event has to be ...
0
votes
1answer
56 views

I am not given figures to answer this question. Whats the right approach?

Z is a random variable defined as the sum of N independent Bernoulli trials where the probability of each Bernoulli taking the value 1 is given by p. The number of Bernoulli trials N is itself a ...
15
votes
1answer
643 views

Integrate this monster

Can you please help me? I've been trying for some time now to integrate this: $$\int_0^\infty g^{-(a+1)} \; \exp\left\{-\left(\frac{b}{g} + \frac{1}{2} \sum_{i=1}^{n} ...
0
votes
0answers
4 views

Existence of first passage time density for time-inhomogeneous diffusion

Let $X$ be a time-inhomogeneous diffusion process in $\mathbb{R}^d$: $$dX_t=b(t,X_t)dt+\Sigma(t,X_t)dB_t,$$ where $\Sigma_{d\times d}$ is uniformly elliptic, and coefficients are such that the above ...
0
votes
2answers
39 views

Probability problem with combination of poisson and binomial distributions

Exercise The number of clients that enter to a bank is a Poisson process of parameter $\lambda>0$ persons per hour. Each client has probability $p$ of being a man and $1-p$ of being a woman. After ...
0
votes
1answer
479 views

Joint probability of sum of two random variables and one of its terms

Let $X$ and $Y$ be two independent random variables (weibull distributed) and $Z=X+Y$. I am trying to find $\mathbb P\big(Z\geq z ~\cap~ X\leq x\big)$. I know that $$ \mathbb P(Z\geq z \cap X\leq x) = ...
2
votes
0answers
23 views

Distributions on the simplex with beta marginals

Consider a random vector $(X_1,X_2,\ldots,X_n)$ such that 1) $\; X_i\sim\text{Beta}(a_i,\sum_{j\neq i}a_j)\qquad i=1,2,\ldots,n$, 2) $\; X_1+X_2+\ldots+X_n=1$. Can we conclude that ...
0
votes
0answers
11 views

Is output from softmax function continuous probability distribution?

I want to ask: Is a output from softmax function a continuous probability distribution?
0
votes
0answers
13 views

measure-theoretic probability, (sets of) null events and non-zero probability

Assuming a well-defined probability space $ (\Bbb{R},\mathscr{B},\Bbb{P}_X) $, where $\mathscr{B}$ is the Borel $\sigma$-field, and for a random variable $X$ having a continuous probability density ...
1
vote
1answer
168 views

Definition and statistics of the Negative-Hypergeometric distribution

The Encyclopedia of Mathematics defines the Negative Hypergeometric distribution (NHG) in the following way: There are $N$ elements, of which $M$ are marked and the rest are unmarked. Elements are ...
0
votes
1answer
47 views

Infinite probability density?

I've read that for a "[..]random variable strongly "localized" around a single value", the probability density function (PDF) could be: $p(x)=\frac {1}{2\epsilon}$, with $\epsilon \to 0$, and ...
1
vote
1answer
1k views

Uniformly Most Powerful Test and Rejection Region of Poisson Distribution

Let $X_1, \dots,X_n$ be a random sample from a Poisson$(\lambda)$ distribution where $\lambda > 0$. (1) Find the Uniformly Most Powerful (UMP) level $\alpha$ test for the following set of ...
0
votes
1answer
27 views

square-root rule of time

I tried to test the square-root-rule of time for quantiles of a normal distribution. So i created with the statiscal programming language R two variables a<-rnorm(100,mean=2,sd=1) ...
0
votes
0answers
25 views

Waiting Time Distribution

Let X be a random variable which denotes the amount of time spent in a state(say state 'I') before changing state. As X is a random variable it must have a Probability space/sample space and a sigma ...
9
votes
2answers
707 views

Bingo Probability Problem

A Bingo card has 25 squares with numbers on 24 of them, the center being a free square. The integers that are placed on the Bingo card are selected randomly and without replacement from 1 to 75, ...
0
votes
0answers
17 views

Change of variable formula for density function

We all are aware of the change of variable formula whereby if $$[A, B] = g(X, Y) $$ and g is invertible, then the joint density function of A, B is given by $$f_{ab} (A, B) =1/|J| f_{XY} (g^{-1}(a, ...
1
vote
1answer
41 views

Expected value of exponential function

Suppose two identical component are connected in a piece of factory equipment. The two lifetimes X1 and X2 are independent each having exponential distribution with beta =2. The value of the equipment ...
1
vote
3answers
899 views

Derivation of mean and variance of Hypergeometric Distribution

I need clarified and detailed derivation of mean and variance of a hyper-geometric distribution. If a box contains $N$ balls, $a$ of them are black and $N-a$ are white, and $n$ number of balls are ...
0
votes
0answers
10 views

Security of $(k, 2k)$-bit generator for small seeds

Here is the problem I am working on for context. I have $\epsilon \le 1 - 2^{-k}$ and $\epsilon$ approaches 1 as $k \to \infty$ but I'm stuck on part c). The $f$ is secure iff there does not exist ...
2
votes
1answer
25 views

probability problem with Poisson distribution

Problem A retailer knows that the demand of boxes is a random variable with Poisson distribution of parameter $\lambda=2$ boxes per week. The retailer completes his stock on monday so as to have four ...
1
vote
1answer
25 views

Finding distribution of random variable

During my exam there was the following question which I could not answer: Let $X_1, X_2$ be real valued random variables. Assume that $X_1$ is exponentially distributed. Given that ...
0
votes
0answers
22 views

show that $U_n$ converges to $0$ in $L^1$ and almost surely.

let $(X_n)_{n\geq1}$ be a sequence of independent random variables. Suppose that the density function of $X_n$ is: $$ f(x)=\dfrac{1}{2}.e^{|x|} \quad x \in \mathbb{R} \quad \forall n \quad ...
1
vote
0answers
28 views

Finding the marginal density

The joint probability density function of $X$ and $Y$ is given by $$f(x, y) = 1/y^2 , 0< x< 1, y\geq 1 $$ $[I]$ - Find the joint density function of $U = XY$ and $V = X/Y$ $[II]$ - What are ...
-3
votes
0answers
18 views

Function of two continuous random variables. find CDF [on hold]

[\begin{array}{l}{\rm{Let X be a continuous random variable with uniform distribution on }}\left[ {0,1} \right].{\rm{ }}\\{\rm{Let Y be a continuous random variable with uniform distribution on ...
-1
votes
1answer
39 views

Let X be a continuous random variable with pdf… [on hold]

a.) Let X be a continuous random variable with pdf $f_x(t) = \exp[-t-e^{-t}]$ for all t in the reals. Find $F_X(x)$ My solution is $$F_X(x)= P(X \le x) = ...
1
vote
3answers
45 views

Computing mean of probability density function without integration

Is there any method of determining the mean of a random variable $X$ without integration? Its probability density function is given by: \begin{align} f_X(x)= \begin{cases} 1 − x & \text{if } 0 ≤ ...
1
vote
1answer
25 views

Correct notation for compound conditions of random variables

Let $X$ be a discrete random variable, and $P(X = x)$ is its probability mass function of a binomial experiment. If I want the probability of obtaining between 5 and 10 successes, I write ...
0
votes
1answer
16 views

Show that $\sum\limits_{i=1}^{\inf} p_i \prod\limits_{j=1}^{i - 1}(1 - p_j) + \prod_{i=1}^{\inf}(1 - p_i) = 1 $

The question comes from Hoff's "A First Course in Probability" book. Let $p_i$, $i = 1, 2, ...$ be probabilities (so that $0 \leq p_i \leq 1$, and show the that the equation in the title holds, ...
0
votes
0answers
13 views

Properties of Kernel Integral inner Product of Gaussian Process

Can anyone give any reference / suggest how to get the rigorous mathematical properties of the following : $$ Y=\int_{a}^{b} K_{X} (t) \ f(t) \ dt $$ where $$f \sim GP (\mu(\cdot), R ...
0
votes
0answers
10 views

Connect the MGF of the Survivor, Cumulative and Mass disttributions

Assume that $X$ has a known distribution $P_X$, with a generating function $\hat P_X$. What relationship links $\hat P_X$ with the MGF of X's CDF ($\hat C_X$) and SDF ($\hat S_X$). Would that ...
1
vote
2answers
36 views

Probability of winning a 5 game series by winning 3 games where probability of winning each game is given

Two teams play a series of baseball games, team A and team B. The team that wins 3 of 5 games wins the series. The first game takes place in the stadium of the team A, the second in the stage of team ...
0
votes
1answer
19 views

Probability of inequality of Random variables

Assume that $X_1,X_2,X_3$ are independent random variables with known distributions. How can I calculate the distribution of $P(X_1<X_2<X_3)$.
0
votes
0answers
26 views

What can be said about the joint and conditional probability distributions given some property of the marginals?

Let $Y_1$ and $Y_2$ be two random variables where $Y_i$'s take values in $\{0, 1, 2\}$ for $i \in \{1, 2\}$ with the property that $\sqrt{P_{Y_i}(0)} \ + \ \sqrt{P_{Y_i}(2)} \le 1$. Assume also that ...
0
votes
0answers
29 views

Dependence/Independence Problem in probability [on hold]

Suppose a student is taught by $N$ teachers. The pdf of the marks that the student gets from $i$-th teacher is $p_{m_i}(x)$ and we assume that all $p_{m_i}(x)$'s are i.i.d i.e. ...
1
vote
0answers
13 views

PMF of Bernoulli trials needed to produce at least one success and at least one failure

Let X be the number of Bernoulli(p) trials until we get at least one success and at least one failure. This means we need a string of successes followed by a failure or a string of failures followed ...