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

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2
votes
1answer
41 views

Intuition for probability density function as a Radon-Nikodym derivative

If someone asked me what it meant for $X$ to be standard normally distributed, I would tell them it means $X$ has probability density function $f(x) = \frac{1}{\sqrt{2\pi}}\mathrm e^{-x^2/2}$ for all ...
-2
votes
1answer
19 views

The probability that two randomly selected $2$ year old male feral cats will live to be $ 3$ years old is? [on hold]

The probability that a randomly selected $2$ year old male feral will live to be $3$ years old is $0.82666$. (a) what is the probability that two randomly selected $2$ year old male feral cats will ...
0
votes
1answer
23 views

Suppose that $E$ and $F$ are two events? [on hold]

Suppose that $E$ and $F$ are two events and that $P(E\cap F)= 0.4$ and $P(E)= 0.8$. What is $P(F\mid E)$ ?
2
votes
2answers
24 views

Probability Distribution

I'm thinking about a set of n users on Facebook. Between each of the $\binom{n}{2}$ pairs of distinct friends, lets say an edge (indicating that the two people are friends) is independently present ...
2
votes
1answer
16 views

The distributions of incomes in two cities follow the two Pareto type pdfs. Find P(X<Y)

The distributions of incomes in two cities follow the two Pareto type pdfs $$f(x)= \frac{2}{x^3}, 1 < x < \infty.$$ $$g(y) = \frac{3}{y^4}, 1<y<\infty.$$ Here one unit represents ...
1
vote
0answers
36 views

Of strings and substrings: A problem of probability

Problem statement Let $\Sigma$ be an alphabet. Let $\Sigma^*$ be the Kleene star of $\Sigma$, or the set of all strings of $A$. Let $\Sigma^+$ be the Kleene plus of $\Sigma$, or the set of all ...
2
votes
0answers
49 views
+50

Find a probability density

I am going through a paper trying to understand all the single steps, but I got stuck. I need to calculate $$p(x+\delta t) \mid x(t), t)= \int p(x(t+\delta t) \mid \mu , x(t), t)p(\mu\mid x(t), t) ...
0
votes
0answers
12 views

Reconstruct multivariate binary distribution from marginals

I'm have a random vector $\bf a$ with binary entries, $a_i \in \{0,1\}$. The probability distribution $P({\bf a})$ is not fully specified, but I have the marginals $p_i$, which are the probabilities ...
3
votes
2answers
503 views

Poisson Distribution when only given using mean

I'm doing the following homework problem and am unsure of whether or not my answers are correct. This is my first time working with Poisson distribution and I want to make sure I am doing it ...
3
votes
1answer
63 views

Distribution of $\sin(X) *\cos(Y)$ where $X,Y$ are iid r.v., uniformly distributed on $[0, 2 \pi]$

What is the probability density of $R = \sin(X) * \cos(Y)$ where $X,Y$ are independent random variables, uniformly distributed on $[0, 2 \pi]$? I am stuck with complicated integrals, not sure if ...
0
votes
1answer
413 views

Product of two exponentially distributed random variables

I am trying to find the close form expression of probability distribution of $Z$ such as $Z=X_1X_2$ where $X_1$ and $X_2$ are two independent exponentially distributed variables with PDF ...
0
votes
2answers
24 views

how to estimate parameters of a triangular distribution? [on hold]

I have a set of observations, and they come from a triangular distribution. Now I want to estimate its parameters, but how?
-1
votes
0answers
24 views

Distribution Problem based on unknown function [on hold]

I got struck at this problems as Function is not given. Any help will be appreciated
1
vote
1answer
19 views

What is the “Cumulative Distribution of the magnitude of the N-dimensional standard gaussian”

I am confused by this line from a paper: "Let $F_1(x)$ be the cumulative distribution of the magnitude of an $n$−dimensional standard Gaussian random variable and $F_2(x)$ be the cumulative ...
0
votes
1answer
14 views

Geometric distribution with given probability value.

The probability of a man hitting a target is $2/3$. If he doesn't stop shooting until he hits the target for the first time, a) What is the probability of taking 5 shots to hit the target? b) Which is ...
0
votes
0answers
33 views

Binomial-like distribution

Starting with $1$, for $n$ trials multiply by either $1+p$ or $1-p$, with $0 \le p< \le 1$. Does this distribution have a name? What are its properties, such as density (PDF)? It is like a skewed ...
1
vote
1answer
560 views

Solving Probability Density Function for continuous random variable

The probability density of a random variable $x$ is $$f(x)=a\ \cdotp x^2\ \cdotp \mathrm{e}^{−kx}\ (k>0,\ 0\leq x\leq \infty)$$ Then, the coefficient $a$ equals $$(i)\frac{k^3}{2}\ \ \ \ (ii)\ k^3 ...
1
vote
1answer
19 views

Find the required Chi-square score for an arbitrarily low p-value (2 degrees of freedom)

I'm trying to use the Chi-Square test to find the significance of data that suffers from the multiple testing problem. Because I have this multiple testing problem, the required p-value to view a test ...
2
votes
1answer
16 views

Properties of unimodal functions

A probability density function $f$ is said to be unimodal if the value at which the maximum value of the function is attained is unique. I am reading some papers on econometrics that appear to use ...
0
votes
1answer
30 views

Infrequent fail of the popular parameter estimators, having several beta-distributed random variables to be estimated

I have a project in which there exist $N$ Beta-distributed Random variables each of which should be estimated, having a sample for each of them. The sample domain is $\{0.1,0.3,0.5,0.7,0.9\}$ and the ...
1
vote
2answers
33 views
-1
votes
1answer
31 views

Does these inequalities hold in General for probability distribution? [on hold]

Let $Q(y)$ be a probability density of $y \in [-1,1]$. Then for $t> 0$, the inequalities are $\displaystyle \int_{0 \leq y <t} y^2 Q(y) \, dy \leq t^2 \int_{0 \leq y <t} Q(y) \, dy $. ...
0
votes
0answers
27 views

How do I calculate conditional PDF?

Obtain $$P(2 < Y < 3 | X = 1)$$ where the joint pdf of X and Y is $$f_{X,Y}(x,y) = (6-x-y)/8$$ where $$0 < x < 2$$ and $$2 < y < 4$$? so first, I did $$f_Y|X=1(y) = ...
0
votes
1answer
27 views

Slow convergence simulating log-normal sample from the normal

I am trying to simulate a log-normal random variable $Y$ with mean $m = \mathbb{E}[Y] = 0.001$ and standard deviation $s = 0.094$ by simulating a normal sample instead, and then exponentiating it. ...
0
votes
0answers
27 views

Interpretation of integral as ratio of joint and conditional densities?

A common exercise in Bayesian statistics is specifying a prior $p(\theta)$ on some parameter $\theta$. We then observe a collection of data $D=(X_1,\dots,X_N)$, the distribution of which is ...
-3
votes
1answer
20 views

Uniform distribution and real values [on hold]

If the random variable $k$ is uniformly distributed in $(0,5)$, What is the probability that the roots of the equation $4x^2+4xk + k + 2 = 0$ are real?
3
votes
2answers
89 views

Given a variable $X$ with a PDF, what is the PDF of $\sqrt{X}$

I feel this is simple and I'm overlooking something really basic. Let's say a have a variable $x$ which obeys the exponential distribution. So if collect 100000 occurrences of $x$ and plot its ...
-2
votes
1answer
22 views

Probability of a user references in a network [on hold]

I am trying to figure out no of possible referrals of a user in a network. Where the size of a network is not fixed but we can set an assumption of 1000 persons. Edit: A user knows few users in a ...
-1
votes
3answers
165 views

Union of three independent events

Let $ A_1, A_2, A_3 $ be independent events with probabilities $ \frac 12,\frac13,\frac14,$ respectively. How to compute for $P(A_1\cup A_2 \cup A_3).$ My solution starts from using the probability ...
3
votes
1answer
17 views

CDF of the difference of two Gaussian mixtures

I have two Gaussian mixtures, $X_D$ and $X_{\overline{D}}$: $$ f(X_D) = \sum_{c=1}^m f(X_D\mid C=c)P(C=c) = \sum_{c=1}^m \phi(x-\mu-g(c))P(C=c), $$ $$ f(X_\overline{D}) = \sum_{c=1}^m ...
1
vote
1answer
14 views

How to calculate a posterior probability with a given Gaussian Mixture Model?

I'm building a GMM-based classifier in speech processing and I'm using GMM as a probabilistic scoring mechanism (therefore I don't intrinsically care about the underlying mixture components). For ...
0
votes
1answer
23 views

Recovering density parameters from distribution function

Let $X$ be a random variable with probability density function $g(x;\theta_1,\theta_2)$, where $g$ is parameterized by two real numbers $\theta_1$ and $\theta_2$. I'd like to specify that $$ P(a \leq ...
0
votes
1answer
17 views

computing weight from distance metric

I have a distance between two points in meters. I want to convert this distance into weight such that as distance increases the weight decreases. What are some good weighting function that can ...
2
votes
1answer
22 views

Probability in knockout games.

Suppose in a knockout tournament 32 players p1 , p2 .....p32 participate. In each round players are divided into pairs at random and winner goes to the next round. If p5 reaches semifinal what is ...
0
votes
0answers
11 views

characteristic function problem 4

which of the following is not a characteristic function? a) 1 b) $e^{it} $ , $t \in R$ c) $\frac{1}{1-it} $, $t\in R$ d)$e^{|-t|}$, $t \in R$
7
votes
1answer
2k views

Singular jacobian matrix?

I have a series of questions, in various degrees of befuddled muddledness (and they are related to my previous questions: this and this) First question: how do I do a change of variable if the ...
-1
votes
0answers
16 views

What are interventions / intervention distribution?

I don't understand what interventions are: Definition 2.2.1 [Intervention Distribution] Consider a distribution $\mathbb{P}^\mathbf{X}$ that has been generated from an SEM $\mathcal{S} := ...
0
votes
1answer
386 views

Mixture Gaussian distribution quantiles

Let $f_1(x), \dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, \dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = \sum_i w_i f_i(x)$ is also a ...
3
votes
1answer
57 views

sign of the conditional expectation

I'm working on the following problem: Let $X$ be a random variable defined on $(\Omega,F,P)$ and $G$ a $\sigma$-algebra contained in $F$. Show that, if $E(|X|)<\infty$ and $E(X\mid G)$ has the ...
2
votes
3answers
4k views
0
votes
1answer
29 views

Lottery probability with payout system

Assume we have a lottery which has following payouts 1,2,5,6,9,10,16. The organizer expects 4% profit from the lottery. I wrote ...
0
votes
1answer
27 views

How to find median from a probability distribution?

Having trouble on something that should be really, really easy. I need to find the median of the following probability distribution...but according to the website I linked below...I'm doing it ...
-1
votes
1answer
21 views

Continuous probability function [on hold]

The probability density function of the random variable X is given by $$f(x) = \begin{cases} \frac{c}{\sqrt x}, & \texttt{for } 0<x<4 \\0, &\texttt{otherwise} \end{cases}$$ a) Find ...
1
vote
4answers
86 views

Difference between $E[X^2]$ and $E[X^3]$

Hope to ask a dumb question. $Y = aX$,with $a \in N_+$. Here, we know the correlation coefficient is 1. Now, suppose $X \sim N(0,1)$. Here, we know $X, Y$ are not independent. Cov($X,Y$) = ...
1
vote
0answers
72 views

Alternative ways to prove $\{f:f(0)=\sum_k f(\frac{k}{\sqrt{n}})g_n (k)\}$ is dense in $\{f\in C^2 (\mathbb{R}) : f(0)=\int_{\mathbb{R}} f(u)g(u)du\}$

I want to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{\sqrt{n}}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) ...
1
vote
1answer
18 views

How to derive formula for marginal probability of choosing nest in nested logit model?

I am trying to understand all the details of the nested logit and what confuses me is the formula for marginal probability of choosing the nest. In more details: the joint probability of individual n ...
0
votes
1answer
39 views

What is the expected value of the random variable with the following pdf

Let $X$ be a random variable with pdf $$f(x \mid \sigma) = \dfrac{1}{2\sigma}\exp\left(-\dfrac{|x|}{\sigma}\right)\text{, } x \in (-\infty, \infty)\text{, }\sigma > 0\text{.}$$ Here are my steps: ...
2
votes
1answer
38 views

Conditional distribution of $X$ exponential given $U\leq e^{-X}$, with $U$ uniform on $(0,1)$

Let $X$ be exponentially distributed with mean $1$ and $U$ be a $U(0,1)$ random variable independent of $X$. Define $$I= \begin{cases}1,&U \leq e^{-X}\\ 0,&\text{ ...
0
votes
2answers
17 views

Question about finding a distribution without taking into account previous events

We have 8 prisoners, each has a probability of escaping (independently) each day of $0.4$, what is the distribution of the amount of escaping prisoners on the third day? This is the answer: the ...
0
votes
0answers
19 views

Probability distribution to failure [on hold]

I am going to do a simulation for a manufactruing system, i must consider a scenario as: a $20\%$ probability of failures occurring in $M1$. Q: What is the probability distributions the time to ...