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

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0answers
10 views

Moments of censored exponential distribution

I have a question as to whether my calculation of moments of censored exponential distribution is correct. I have two random variables $T_A=\min(\tau,t_1)$ and $T_B=\min(\tau,t_2)$, where $t_1<t_2$ ...
0
votes
1answer
106 views

Distribution in Polya's Urn / existence of mgf / Stolz–Cesàro alternative / dominated convergence theorem

I know this has been asked elsewhere, but I think the values or random variables are different or something. From Williams' Probability with Martingales: I proved that $M_n$ is a $\sigma(B_1, ...
1
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0answers
8 views

maximum likelihood estimators of a shifted gamma distribution?

i had this question in my exam but didn't know how to solve this apart from constructing the likelihood function and differentiating .but got stuck in the middle of nowhere.please help .
0
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1answer
13 views

Expectation of scaled sum of squares of iid random variables

Let $X_1, \dots, X_n$ be iid standard normal random variables. Consider the vector $X = (X_1, \dots, X_n)$ and the vector $Y = \frac{1}{\|X\|}(X_1, \dots, X_k)$ for $k < n$. What is ...
1
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2answers
120 views

How to I find the distribution of $\log p(X)$ when $X$ is distributed under $p$?

I have a feeling there's no general solution to this problem, but I'll ask anyway. I have a multivariate PDF $p$ and, given a random vector $X\sim p$, I'd like to find the the PDF of $\log p(X)$. ...
2
votes
1answer
44 views

Find a continuous PDF on $[0,6]$ for given probabilities

Find a continuous probability density function $f$ on $[0,6]$, such that $\mathbb{P}([0,2]) = 0.6$, $\mathbb{P}([1,4]) = 0.5$ and $\mathbb{P}([3,5]) = 0.2$. After some calculations I came up with ...
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0answers
7 views

Invariant distributions: Applications in the real World

I'm studying about invariant distributions for Markov processes; say in the context of dynamics of Random Neural Networks (biological Networks). I can't fully understand what does an invariant ...
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0answers
7 views

Derivation of variance of a linearly transformed vector

I am trying to derive the variance of a linearly transformed vector. A result was given here. $$ \mathbf{y} = X \, \mathbf{b} $$ $$ \mathbf{b} \sim \mathcal{N}( \mathbf{0}, \sigma^2 I) $$ If we say ...
0
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1answer
9 views

Pseudo-inverse of the Cumulative Distribution Function of X

The goal of these calculations is to write a Python function that generates pseudo-random values with the distribution described below. This isn't relevant to the question (or even to this ...
2
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0answers
28 views

Showing a relation between binomial and negative binomial analytically

If $X$ is binomial random variable $B(n,p)$ and Y is negative binomial $(r,p)$, How can I show that $F_X(r-1) = 1- F_Y(n-r)$. While it is possible to show that using the definition of binomial and ...
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0answers
36 views

Joint probability distribution $X, Y$.

$f(x,y)= \frac{3}{2}(x²+y²)$, $\:\:0 \leq x,\: y \leq 1$ $0$, elsewhere Determine whether or not $X$ and $Y$ are independent. Independent characteristic when $f(x,y)=f(x)f(y)$ To find f(x) and ...
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0answers
27 views

Proportional probability of payouts with defined expected value.

Assume we have a lottery with payouts $(2,3,5)$. So if you buy a ticket you can win a pot which will payout your ticket price multiplied by one of those numbers. The organizer expects a margin profit ...
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0answers
19 views

Probability and Statistics [on hold]

How can I check if a Moment Generating Function is valid or not? I tried using the definition for that but it didn't help me at all.
1
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0answers
39 views

Distribution and expectation value of ceiling function of Poisson

There is Poisson random variable $X$ $$P(X=x)=\frac{\lambda^{x}}{x!}e^{-\lambda}$$ And define random variable $Z=\lceil \beta X \rceil$ ( $\beta$ is rational number which is less than 1). How can I ...
2
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0answers
97 views

Probability mass function of the sum of the function of the sum of iid random variables

How can I get an expression of the probability mass function of: \begin{equation} Y_i=\sum_{k=1}^i f\left(\sum_{n=1}^{k} X_n\right) \end{equation} being $x_n, n=1,2,...$ iid random variables and ...
0
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1answer
23 views

Calculate the characteristic function $\varphi_W$ of W

$p(x)=xe^{-x}$ for $x\geq 0$ or $0$ otherwise. I tried to substitute $e^{-x}$ but then i found there is still a $x$ in front.
1
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3answers
47 views

Find the distribution of $Y = -\log (1-X)$ given that $X\sim U(0,1)$.

If $X \sim U (0,1)$ then if we define a new random variable $Y=-\log (1-X)$ then what will be distribution of $Y$. Please explain.
6
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4answers
17k views

Combining two probability distributions

I have a variable $X$. In a measurement $A$, $X$ follows the normal distribution $N_1$ with mean $m_1$ and standard deviation $\sigma_1$. In a similar measurement $B$, $X$ follows another normal ...
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0answers
25 views

Appropriate distribution for set of probabilities $p_1 ,…, p_n$

I am doing some evaluation of a system, that has set of probabilities $p_i$ $i= \in \{1,...,N\}$, I need to model them as random variables such that : $$ \sum_i p_i \leq 1$$ and $$ 0 \leq p_i \leq 1 ...
2
votes
1answer
41 views

problem finding marginal distribution for a PDF [on hold]

I need to find the marginal distribution function $f_y$ for $$f_{xy}(u,v)= \begin{cases} 1\over u, & \text{$u\ge 1, 0\le v \le {1 \over u}$} \\ 0, & \text{else} \\ \end{cases}$$ my problem ...
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1answer
43 views

How do I renormalize these probability distributions?

So I have two random variables, $X_1$ and $X_2$, both uniformly distributed on $[0, 1]$. If $Y = (X_1 + X_2) / 2$, it will also be distributed between 0 and 1, but it won't be uniformly distributed ...
0
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0answers
11 views

Failure boundary for simple routing problem

As an absolute beginner concerning probability theory I am currently trying to solve the following problem: Given a grid that has $x$ columns (here $x = 4$) and $y$ rows (here $y = 5$), we insert a ...
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0answers
16 views

Continuity of random variable as function of a random variable

Suppose, we are given a continuos random variable $X$ and a continuous and nondecreasing function $f$. Can it be shown that a second random variable $Y=f(X)$ is continuos on the support of $X$? What ...
3
votes
3answers
54 views

Upper bound for difference of Poisson random variables

Let $X, Y$ be random variables with Poisson$(\lambda)$ and Poisson$(2\lambda)$ distributions, respectively.Then (i) If we assume that $X, Y$ are independent, $$\mathbb{P}(X \geq Y) \leq ...
1
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2answers
45 views

Methods for calculating the mean and variance of a distribution created from the addition of two normally distributed quantities

I'm trying to understand how to interpret the following which refers to determination of the mean and variance of a distribution that's the result of adding two normally distributed random variables. ...
4
votes
1answer
22 views

Ehrenfest Chain: stationary distribution

In the Ehrenfest Chain model: There are M balls which are divided between urn A and urn B. At each stage, if a ball is chosen, then it would be moved into a different urn. Let $X_n$ be the # of ...
0
votes
2answers
21 views

Find PDF on $[0,6]$ such that $P([1,3]) = 0.5$

Find a probability density function $f$ on $[0,6] \subset \mathbb{R}$, such that $\mathbb{P}([1,3]) = 0.5$ That is we need to find an $f$, such that $\int_{[0,6]} f(x)dx = 1$ and $\int_{1}^{3} ...
0
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0answers
11 views

Expected values of Hermite polynomials under Gaussian distribution

On Wikipedia there's a nice result stating that $$E[He_n(X)]=\mu^n,$$ where $He_n$ is the (probabilists') Hermite polynomial of order $n$ and $X$ is a $N(\mu, 1)$ random variable. I'm interested in ...
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2answers
41 views

Distribution of the sum of $N$ loaded dice rolls

I would like to calculate the probability distribution of the sum of all the faces of $N$ dice rolls. The face probabilities ${p_i}$ are know, but are not $1 \over 6$. I have found answers for the ...
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0answers
11 views

Continuity of Monte-Carlo simulations with uniformly distributed input parameters

Suppose a continuous and monotone function $f:\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ to be given. So, in the general case, if I slightly change parameters $a$ and $b$, the function ...
0
votes
1answer
62 views

Distribution of a function of a random variable

Suppose we have continuous random variable $X$ with distribution $f_X$. That is $$ P\left(a \le X \le b \right) = \int_{a}^{b} f_X(x) \ dx $$ Now suppose I have a function $\phi: \Bbb{R} ...
1
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1answer
17 views

Method for determining distributions of sum of Normal distribution unknown mean and variance

I've been trying to complete this question but have been struggling to see how to approach it. Any help would be greatly appreciated. Is there a standard way of approaching and answering ...
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0answers
23 views

What is expected value of only positive numbers [duplicate]

Normal distribution with mean zero and standard deviation 1. What is the expected value of just the positive variables
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0answers
13 views

Position error probability distribution when distance and angle error distributions are zero mean Gaussian

In one problem we are estimating the position of an object from the measurement of its distance $\mathbf{r}$ from a point as well as its angle $\mathbf{\theta}$ from the reference direction. The error ...
0
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2answers
16 views

Estimate on Probability of a standard normal variable

In the book written by Karatzas & Shreve, at the page - 111; the authors have mentioned about a result: If $Z_{v}$ be a standard normal variable; then for $\epsilon \gt 0$ ; $\mathbb P ...
0
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1answer
16 views

Show that Uniform$(1,5)$ is neither singular nor absolutely continuous with respect to Uniform$(0,3)$.

Actually, I'm just studying singular continuity, absolute continuity.I know the definitions.And have solved few very basic sums. Now, in this problem, I'm not understanding what does this 'with ...
0
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0answers
8 views

Statistical distance between a multiplicative mask and a random number

Given $x \in \{1,\ldots,2^n\}$ and a uniform random $r \in \{1,\ldots,2^{n+k}\}$, then the statistical distance $\Delta(x + r\bmod q; r) < 2^{-k}$, for a $q > 2^{n+k+1}$. With addition this is ...
1
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2answers
71 views

How to integrate $\int\limits_0^\infty e^{-a x^2}\cos(b x) dx$ where $a>0$

How to integrate $$\int\limits_0^\infty e^{-a x^2}\cos(b x) dx$$ where $a>0$ The real problem is this integral $$\lim\limits_{\alpha\rightarrow 2}\int\limits_0^\infty e^{-a x^\alpha}\cos(b x) ...
8
votes
3answers
302 views

Maximum of a sum of random variables

Let $X_1, \dots, X_n$ be independent and identically distributed random variables with $E(X_i) = 0$ and $$S_k = \sum_{i \leq k} X_i$$ What is the probability distribution of $M_2 = \max \{ X_1, ...
1
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2answers
29 views

Convergence of Sum of Random variable to another - Cantor function

Let $(X_{n})_{n\geq1}$ be i.i.d. Ber$\left(\frac{1}{2}\right)$. I want to show that $$\sum_{{n\geq1}}\frac{2X_{n}}{3^{n}}$$ converges almost surely to a random variable $X$, without saying that this ...
1
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1answer
10 views

Expectation of the minimum of two $\mathcal U(0, 1)$ r.v.'s conditional on it being greater than or equal to some value

Let $X_1, X_2$ be i.i.d. $\mathcal U(0, 1)$ (continuous) r.v.'s, and let $0 \le R \le 1$ be some number. What is $\mathbb E[\min(X_1, X_2) \mid \min(X_1, X_2) \ge R]$? My attempt: Let $Y = ...
1
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1answer
11 views

How to calculate the transition density for a multivariate jump process

I have the following stochastic process: $dX = (A-I)XdN$, where $X$ is a $2\times1$ vector of random variables, $A$ is a constant, real, symmetric, $2\times2$ matrix, $I$ is the identity matrix and ...
7
votes
1answer
2k views

A function of two cumulative probability distributions with same first 2 moments

Let $\Phi_1$ and $\Phi_2$ be cumulative probability distribution functions with domain $[L, \infty)$, $L\geq 0$, both distributions having the same expectation $\mu$ and the same second moment (hence ...
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0answers
9 views

Sampling Distributions. Statistics [on hold]

I'm stuck in this problem: Problem Picture I did the literal a and b, but the rest of them I don´t understand, the reason why.
1
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2answers
45 views

What is the probability that at least $2$ out of $4$ digits in a code is the same?

If I choose $4$ digits for a code randomly out of the digits $0$ to $9$. What is the probability that at least $2$ of these digits are the same? By at least I mean that you have to count with the ...
0
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0answers
28 views

Calculate the discrete density of the variables of a Markov chain

$X$ and $Y$ are independent random variables of Bernoulli with parameter $\frac{2}{3}$. $Z=X+Y$ $\{X_n\}_{n \in \mathbb{N}}$ with values in {0,1,2} having $Z$ such as initial law and the transition ...
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0answers
32 views

On the probability distribution of iterated permutations

I have this little problem that has been nagging me for a couple of months now. It occurred to me when considering the fairness of card shuffling methods. Here's my best attempt at formalizing it: ...
6
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2answers
63 views

Estimate Grade Distribution Based on Performance of Each Question

As the title states, I would like to be able to estimate the grade distribution of an exam based on the mark distribution of each individual question. To give a quick example of what I mean, suppose ...
1
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1answer
160 views

Polya's urn model - limit distribution

Let an urn contain w white and b black balls. Draw a ball randomly from the urn and return it together with another ball of the same color. Let $b_n$ be the number of black balls and $w_n$ the number ...
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2answers
32 views

What probability mass, density or distribution function might corresponds to this moment generating function? [duplicate]

I have somehow come up with a random variable $X$ with moment generating function (assuming it exists) $$M_{X}(t) = -t (1 - e^t)$$ What is the probability mass, density or distribution function? It ...