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

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2
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2answers
27 views

sum of two consecutive bernoulli rv

$\newcommand{\Var}{\operatorname{Var}}$I have $X_1,X_2,\ldots,X_{n+1}$ i.i.d. rv, each $X_i$ is a Bernoulli rv with parameter $p$, i.e. $X_i \in \{0,1\}$, $P(X_i=0)=1-p$ and $P(X_i=1)=p$ with $0 \leq ...
1
vote
1answer
252 views

joint probability distribution of one discrete, one continuous random variable

This is a problem on the joint distribution of a discrete and a continuous random variable. Kitty Oil Co. has decided to drill for oil in 10 different locations; the cost of drilling at each ...
0
votes
0answers
9 views

An example of $k$-independent distributions.

I'm trying to better understand the idea of $k$-independence in distributions. The idea is that a distribution $\mu$ over $\{0,1\}^n$ is $k$-independent if any restriction of $\mu$ to $k$ variables ...
0
votes
0answers
5 views

distribution of a quadratic form

Let $A$ be a symmetric positive definite matrix, and $x$ a random vector. Suppose we know the distribution of $x^\top A x$. What can we say about the distribution of $x^\top x$?
1
vote
2answers
20 views

characteristic function of $\sum_i^N X_i$, $N$ is a Poisson distribution

I have a series of $X_i$ random variables, identically and independent distributed. $S_n=\sum_i^N X_i$, with $N$ which has a Poisson distribution and is independent from $X_i$. I have to compute the ...
1
vote
1answer
20 views

Conditional probability with a normal distribution

Given that Y and L are normally distributed, the expectation of L given Y is $\mu (Y)$ and the variance of L given Y is $\sigma ^2 (Y)$, why is the conditional probability $P(L > x| Y) = \Phi ...
0
votes
2answers
28 views

Conditional probability for random variables with different distributions

Random variables $X$ and $Y$ are independent, where $X$ is exponentially distributed with parameter $1$ and $Y$ has uniform distribution on $[-1,1]$ interval. Find $\mathbb{P}(Y>0|X+Y>1)$. My ...
2
votes
2answers
32 views

Distribution of numbers in everyday life

If you were to read tomorrow's newspaper it is intuitively more likely that the whole number 1 would appear more times than 643689443. Is there an expected distribution of numbers used in general? ...
-1
votes
1answer
62 views

Transforming distributions

There is an economy, populated by a large number of agents. A first order condition common to all agents, is the following: $$E[\exp^{(1-\theta)\eta_i}(r-R+\eta_i)]=0$$ the index $i$ indicates the ...
3
votes
1answer
44 views

Uniform sampling with replacement item frequency

Suppose we are sampling from $N$ distinct items uniformly with replacement $M$ times. What can be said about the distribution of frequencies of items drawn? For example, if I sort all the frequencies ...
0
votes
1answer
14 views

expected value product dependent random variables

My question is strictly operative, if I have, for instance, two random variables $X$ and $Y$, $X$ is a $\mathcal{N}(m,\sigma^2)$ and $Y=e^{h(X-m)-1/2(h^2\sigma^2)}$. $E[Ye^X]$ is $\int y e^x p(x) ...
0
votes
0answers
25 views

Normal approximation with dependent variables

I have a sequence of $N$ dependent random variables $$y_i = \frac{x_i}{||\vec x||_2} \quad \mathrm{for} \quad \vec x \sim \mathcal N(0,\mathbb{1}_N),$$ where the $x_i$ are the iid elements of $\vec ...
1
vote
1answer
42 views

Poisson, Gamma distribution example.

Can someone explain me answer for these questions? Suppose customers arrive at a store as a Poisson process with λ = 10 customers per hour. The Poisson process of X ∼ Poisson(λ) the time until k ...
0
votes
0answers
23 views

Testing for the independence of random variables

In probability theory, $X$ and $Y$ are independent if: $f_{X|Y}(x|y)=f_X(x)f_Y(y)$ If I have sample $Y_1,...,Y_n$ and I would like to test if $Y_i$ is independent from the rest of the sample, I ...
0
votes
1answer
14 views

normality of data

Does the qqplot below suggest that the data is normally distributed? The fact that it's nearly perfectly linear is to me an indication of normality. However, the Anderson-Darling test for some reason ...
-1
votes
0answers
29 views

Random walk probability [on hold]

A particle executes a simple unrestricted random walk on a straight path, a step to the right of length 1 occurring with probability 1/3 and a step to the left of length 1 occurring with probability ...
2
votes
0answers
19 views

Product of gamma distributed independent random variables

Let $X_1,\ldots,X_n$ be identical and independent gamma distributed random variables with density function $f(x)=\frac1{\Gamma(\alpha)}x^{\alpha-1} e^{-x}$. I am interested by the product $X_1\cdots ...
0
votes
2answers
36 views

Soccer and probability distributions

The USA soccer team is going to play a championship with 7 other tems. The 8 teams, are going to be divided in two groups of 4 each one. From the participants, Brazil is considered the strongest team ...
1
vote
1answer
523 views

Correlation between Beta distributions

I have a Computer Science background and not very knowledgeable in Probability and Statistics. So excuse me if my question,notation, or language is flawed. Anyways, the problems is that we have two ...
1
vote
1answer
291 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 ...
0
votes
1answer
23 views

Finding the conditional probability from a conditional distribution function

I'm taking a probability theory class and I'm having troubles with multivariate distributions. In particular, I don't really understand how to find conditional probabilities. Here's a question I'm ...
0
votes
0answers
15 views

Combination of exponential distribution and geometric distribution

I am trying to figure out the distribution times for dark times for the following process. An atom is prepared in state 1 (dark) and decays to state 2 with characteristic time scale T. From state 2 ...
0
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0answers
17 views

When is Complex Normal Distribution equal to Normal distribution for real numbers

Let $Z = X+ iY$ be a complex random vector with real and imaginary part equal to $X$ and $Y$ respectively. Assuming that $Z$ has complex Normal distribution, can we say that making $Y=0$, the ...
1
vote
1answer
37 views

Question about the Bayesian Inference of a parameter

In order to understand the difference between the Frequentist and Bayesian inference, I was reading the presentation at: http://www.stat.ufl.edu/archived/casella/Talks/BayesRefresher.pdf . In order to ...
1
vote
1answer
22 views

Probability: Gamma Function vs Gamma Distribution

Could someone help me with setting up the function of this question. I've been setting it up with the gamma distribution function but kept getting the wrong answer. What I did was I used the Gamma ...
1
vote
2answers
28 views

Identifying the distribution which represents a negative binomial distribution as a compound poisson distribution

Suppose that the random variable $X$, which has a negative binomial distribution with probability $p$ and parameter $r$, can be represented as the summation of $N$ iid random variables $Y_1, Y_2, ...
0
votes
3answers
44 views

Find $P(X+Y\le 0)$ given the joint probability function of $X$ and $Y$

I am struggling with part c of this question. Could someone please tell me how to approach and solve this type of questions?
1
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2answers
60 views

Proof of, and requirements for, the reverse of Jensen's Inequality for concave functions

As I understand it, Jensen's Inequality states $$\int_{U}f_{V}\left(h(u)g(u)\right)du\geq f_{V}\left(\int_{U}h(u)g(u)du\right)$$ For a convex function $f_{V}$, a probability distribution $g(u)$ on ...
0
votes
1answer
819 views

Finding joint cdf and pdf of independent random variables

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
8 views

Finding the percentile of a normally distributed variable

I'm taking a probability theory class and I'm stuck on a question. Here's the question: A manufacturing plant utilizes 3000 electric light bulbs whose length of life is normal distributed with mean ...
1
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0answers
29 views

Probability distribution of k consecutive successes with n maximum trials

Let $X$ be a random variable that represents the number of trials of a given experiment. The outcome of a single trial is a Bernoulli random variable, with probability of success $p$, and trials are ...
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votes
0answers
38 views

Characteristic functions

Here $E(Y)$ means the expected value of $Y$. 1) Could any one explain for me how to get from (2.7) to (2.8) ? 2) Why does the author know to define $\phi_1(u)$ and $\phi_2(u)$ in such a way? ...
2
votes
2answers
42 views

Probability Distribution, where $E(X^2) = 2E(X)$

May I please get help with this question? What is the answer and how do I get to it? [Within the context of discrete random variables]. Consider a probability distribution where $E(X^2) = 2E(X)$. In ...
0
votes
0answers
23 views

Prove Logarithmic function is part of exponential family

The aim is to prove that the logarithmic distribution with parameter $p (0<p<1)$ is part of the exponential family and hence, give its canonical parameter. To prove a distribution is part of ...
0
votes
1answer
23 views

Total law of probability in continuous space

I am finding little difficulty in the following definition of total probability specified in a NLP related paper. Say $q^i$ is a partition of my continuous sample space. The authors have defined the ...
0
votes
1answer
25 views

For which function $f$ is $1 \ll \sum_{i=1}^{n} i \cdot i^{-f(n)} \ll n$?

I am interested in the expected value of a power-law Distribution. I would like to let the Parameter $f(n)$ depend on $n$ for $n \rightarrow \infty$. And now I would like to determine $f(n)$ such ...
2
votes
1answer
46 views

Relation between uniformly distributed random variable and i.i.d Bernoulli sequence (Cantor space)

(Uniform RV <==> i.i.d Bernoulli sequence) (1) Let $(X_n)_n$ be a sequence of i.i.d. Bernoulli random variables($P(X_n=0)=P(X_n=1)=\frac 12$) on a probability space. Then show that $\xi:= \sum_n ...
0
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0answers
28 views

Probability Distributions and Random Discrete Variables

How do you read this? For (a) do we let $X= 1/6, 1/2, 1/5$ and $2/15$ and sub into the equation, $$ Y=X^2-2X. $$ How do we go about solving this?
1
vote
1answer
34 views

Closed-form term for this expression

I have a normal Distribution $X \sim N(\mu, \sigma)$. Is there an easy way to give an asymptotic estimate with small error (I would prefer with relative error $\rightarrow 0$) for $P[X \geq k]$? We ...
0
votes
1answer
24 views

Applying Markov's inequality to a sequence of random variables

Does the Markov inequality also work for infinite $a$ or only for constant $a$? More precisely: If $X(n)$ is a sequence of random variables and $f(n)$ is some sequence of numbers,is it allowed to ...
0
votes
2answers
12 views

$Y_{(n)} = X_{(n)}/\mu$?

If $ X_1, ...,X_n$ are iid random variables such that $ X_i \sim U(0, \mu)$, is that true that if $Y_i = X_i/\mu$, then $Y_{(n)} = X_{(n)}/\mu?$ I am sorry if the question looks so simple and I am nt ...
1
vote
1answer
21 views

Show that statistic is (not) sufficient

I need to verify ifthe statistic $|X|$ is or npt sufficient for $\mu$, if $ X \sim N(\mu, 1)$ Using the definition, I've obtained the pdf of X given $ T(X)=|X|:$ $$f_{X|T}(x|t) = ...
2
votes
1answer
31 views

Difference between $\lim P[…]$ and $P[ \lim ]$

In a Galton-Watson branching process the extinction probability is sometimes given by $$\lim_{t \rightarrow \infty} P[X(t)=0]$$ and sometimes as $$ P[\lim_{t \rightarrow \infty}X(t)=0]$$ Is there a ...
0
votes
1answer
23 views

Conditional Probability using a Matrix

I understand how to find P1: that is simply: P(D1|D0)=0.8 P(W1|D0)=0.2 P(D1|W1)=0.4 P(W1|W0)=0.6 I do not however, understand how to find P2 using the matrix. Normally I would solve it as ...
0
votes
0answers
15 views

How to characterize $F(x \mid Y = y)$ satisfies $\exists ! G(\cdot)( G(x) = \int F( x \mid Y = y) dG(y))$?

$F(x \mid Y = y)$ is the conditional distibution function $P(X \leq x \mid Y =y)$. $G(x)$,$G(y)$ represent $P(X \leq x)$ and $P(Y \leq y)$. Is there a charaterization of $F(x \mid Y = y)$ such that ...
1
vote
2answers
18 views

Proving the Probability of an Event Through Bayes Theorem.

The question goes as such: An event A can occur if only one of the mutually exclusive events B1, B2, or B3 occur. Show that P(A) = P(B1)P(A|B1)+P(B2)(A|B2)+P(B3)*(A|B3) my working out: P[A|(B1 U B2 ...
1
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1answer
28 views

Expected Value of a mixed distribution

I have a question from my practice actuary exam... I understand one method of arriving at the answer, however the alternative method is giving me a bit of confusion! I have the lifetime of seismic ...
0
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0answers
34 views

Sampling and averaging in Monte Carlo Simulation

(First of all, I apologize for the vague title. Couldn't think of rather proper one.) Let's say that we have 10 items where each item has probability distribution of one's own, say Lognormal ...
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2answers
28 views

Finding the mean and variance of an exponential probability distribution

I'm taking a probability theory course, and I'm struggling a bit with gamma and exponential distributions. Here's a question that I'm stuck on: The length of time Y necessary to complete a key ...
0
votes
1answer
13 views

Find the probability generating function of a GW process

Consider a Galton-Watson process with offspring distribution $\mathrm{Poisson}(1)$. That is, $\textbf{p}(k) = \frac{e^{-1}}{k!}$. Given this information, and that $P(z) = ...