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

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8 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) ...
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0answers
18 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 ...
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1answer
31 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 ...
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0answers
24 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
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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 ...
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1answer
13 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 ...
3
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1answer
31 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 ...
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2answers
32 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 ...
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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 ...
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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 ...
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0answers
16 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 ...
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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 ...
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1answer
36 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 ...
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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 ...
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0answers
44 views

Transforming distributions

I have a random variable $\eta \sim N (0, \sigma^2)$ and I need to find the following expected value. The random variable is also assumed to be $IID$ (independently and identically distributed across ...
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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?
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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, ...
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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 ...
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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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2answers
56 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 ...
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0answers
37 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
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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 ...
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0answers
20 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 ...
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1answer
23 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
19 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) = ...
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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?
2
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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 ...
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1answer
22 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 ...
2
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1answer
40 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 ...
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0answers
14 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 ...
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2answers
17 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 ...
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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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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 ...
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2answers
27 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 ...
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2answers
52 views

Expectation of maximum of iid random variables

Let $X_1, X_2, \ldots, X_n$ be independent random variables having the common density function $f(x)$. We have $$f(x) = \begin{cases} 1 & \text{for } 0 < x < 1, \\ 0 & \text{otherwise} ...
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1answer
19 views

Bivariate Normal probability question

I have this homework question Suppose $(X,Y)\sim BN(u_x=0,u_y=0,w_x^2=1,w_y^2=1,p=-0.6)$. Find: a) $c$ such that $8X+10Y$ and $cX+5Y$ are independent b) $P(X<0,Y>0)$ My thoughts are (a) ...
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1answer
46 views

Need an example/counterexample of continuous and increasing function.

If $\mu$ is a finite measure on the measurable space $\big( X, \mathscr{F} \big)$, $f : X\to [ 0, +\infty)$ is measurable. Then $\textbf{does it exist a continuous function $g : [ 0, +\infty)\to [ ...
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1answer
11 views

Bounds on joint distribution

$X$ and $Y$ are distributed according to the joint PDF $$ f_{X,Y} (x,y) = \{ \begin{array}{lr} \frac{3}{7}x & : 1 \leq x \leq 2, 0 \leq y \leq x\\ 0 & : otherwise ...
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0answers
13 views

Limiting Poisson distribution

Do you know a paper where the limiting joint Poisson Distribution to a Multinomial Distribution is stated? I read that for a multinomial Distribution $Mul(n, (p_i)_{i=1}^k)$, where $p_i \rightarrow ...
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0answers
36 views

Sum of independent discrete random variable

Here is my attempt of deriving the sum of independent random variable in the discrete case : $\underline{\textbf{Sum of independent random variables}}$ Let $\mathcal{C_1}, \mathcal{C_2}$ be ...
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0answers
35 views

Do you know this theorem?

I have read the following Theorem: Suppose that $$(X_n(t_1), ..., X_n(t_k)) \rightarrow (X(t_1),...,X(t_k))$$ holds, whenever $t_1,...,t_k$ all lie in $T_P$, that $$P\{X(1) \neq X(1-)\}=0$$ and that ...
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1answer
24 views

(Multidimensional) Standard Brownian Motion: Convergence

Relating to this question, I have a further one, and hope, someone can help me. I know that $$\left(X_j - X_{j-1}\right)_{j=1}^t \xrightarrow{d} \left(Y_j\right)_{j=1}^t.$$ Further, we know that ...
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1answer
22 views

convergence to standard brownian motion

Could you help me with the following: I have that $$T(x):=\frac{X(nx)-E[X(nx)]}{\sqrt{n}} \xrightarrow{d} N(0, \frac{x^k}{k})$$ for each fixed $x>0$, where we also have that $\frac{X(nx)}{t}$ is ...
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40 views

Vector distribution after Girsanov transform

Let $X$ be a gaussian vector under $P$ and $U$ a variable such that the vector $(X,U)$ is gaussian. $dQ = Y dP$ with $Y = e^{(U −E_p(U) − 1/2 var_p[U])}$. I have to show that $X$ is gaussian under ...
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1answer
23 views

Finding $P(X < Y)$, where $X$ and $Y$ have two different Beta distributions

I have two random variables, $X$ and $Y$, that have the distributions $$X \sim \beta(r_1 = 1, s_1 = 5) \qquad \text{and} \qquad Y \sim \beta(r_2 = 3, s_2 = 1)$$ I need to find $P(X < Y)$. I ...
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1answer
42 views

P.d.f of $X_{(1)}/X_{(n)}$

Let $X_{i} \sim U(0, \theta) $ and $X=(X_1,\dots,X_n)$. Find the pdf of $$ \frac{X_{(1)}}{X_{(n)}}$$ I coulxnt find a way of doing it that looks convenient. Any idea? P.s: $X_{(i)} $ are the order ...