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

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51 views

Replacing Probabilities with Densities

I recently read through the calculation of the probability that one independent exponential RV is less than another and have been left with a nagging question. Thank you very much for any answers.
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152 views

Integrating a special skew normal — the CDF of a convolution of a normal with a truncated normal

I am having a little trouble trying to compute an integral. In short, I wish to solve the following: $$F(x) = \int_{-\infty}^x \phi(au-b)\,\Phi(au+b)\,du $$ My intuition is that this might be ...
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88 views

independent chi squares mean independent non central chi square?

Let $Y$ be a multivariate normal random vector with covariance $\Sigma$. Let $A_0,A_1$ be matrices such that $$A_0\Sigma A_1=0.$$ It is known that in this case $Y'A_0Y$ and $Y'A_1Y$ are independent ...
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105 views

Expected values in a sequence

We draw 2 numbers from a normal (gauss) distribution with mean $\mu$ and variance $\sigma$ and we add them to find the first value $a_1$ of a sequence. The second value $a_2$ of this sequence is the ...
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43 views

Got Stuck with these probability problems

I tried my best to solve 'em , but after waiting a few sheets of paper , I got nothing on me . A litle help from you guys might do the trick , Thanks ! ...
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146 views

Mean and variance of geometric function using binomial distribution

Can anyone help solving this question please? I tried but not sure of the steps to reach the conclusion.
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130 views

poisson random variable distribution using probability

Let $X$ be the number of emails that a company receives in a day. Assume that $X$ is a Poisson random variable with parameter $\lambda$. The company classifies each email as spam or not spam. The ...
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1answer
69 views

Covariance combined with normal distribution

We have $N_1$ and $N_2$, normal distributed random variables with averages $µ_i=E[N_i]$ and variances $σ_i^2=Var[N_i]$ and $c = Cov(N_1, N_2)$. We want to compute $E[e^{N_1} I(N_2>0)]$, where I is ...
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49 views

Proof this form

I am trying to proof this form: Let $g(.)$ be a function, for $y_n$ is a a nonnegative random variable, $\varepsilon>0$, $g(x)>0$ is increasing function for $x>0$, and $E[g(x)]>0$, then ...
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1answer
141 views

Conditions for positive dependence

Consider two random variables $X$ and $Y$ with joint distribution $F_{X,Y}$ and strictly positive density function $f_{X,Y}$. Additionally, let $x^*$ be the value of $x$ that solves: $$ \Pr[Y\leq ...
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1answer
2k views

Continuous uniform distribution over a circle with radius R

I started to do this problem with the standard integration techniques, but I cant help but think that there has got to be something I am not seeing. Since it is a uniform distribution, even though x ...
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150 views

How to calculate probability using multinomial distribution?

So according to the multinomial distribution, the probability function $\Pr(X_1 = x_1, X_2 = x_2, \dots, X_k = x_k)$ is equal to $\dfrac{n!}{x_1! x_2! \cdots x_k!} \cdot p_1^{x_1}\cdot p_2^{x_2} ...
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73 views

Problem of basic probability

We have 6 machines; the lifetime of each is ~ $\exp(-\ln(.7)=0.35)$ if one machine fail it is repair the next day .What is the probability that some particular day NO machine works?? So far I got : ...
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1answer
65 views

Calculating the probabilities of different lengths of repetitions of numbers of length 4

I'm trying to calculate the probabilities of different lengths of repetitions of X length number however I know I'm doing it incorrectly since when I add all the probabilities together they don't ...
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1answer
553 views

Approximating a Poisson distribution to a Normal distribution

I have the following problem I'm trying to solve: I know that the quantity of complains in a call center is a Poisson variable with $\lambda=18 $ costumers/hour, and that the probability of being ...
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3answers
108 views

How I can find the expected value of $G$?

Suppose two teams play a series of games, each producing a winner and a loser, until one team has won two more games than the other. Let $G$ be the total number of games played. Assuming each ...
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1answer
133 views

Conditional Probability Proof

Suppose that X and Y are independent discrete random variables. Let h(x,y) be a bounded two-variable function. Show that: E [h(X,Y)|X = x] = E [h(x,Y )] Explain why this is usually not true if X and ...
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2k views

Distribution of a difference of two Uniform random variables?

Let $X$ and $Y$ both be distributed between $[1,2]$, what is the distribution of $Z=X-Y$?
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95 views

Does $0$ correlation imply independence for marginally normal distributions?

Assume $X \sim \mathcal N(\mu_1, \sigma_1^2)$ and $Y \sim \mathcal N(\mu_2, \sigma_2^2)$. If $\rho_{X,Y} = 0$ then $X \bot Y$. Can someone give a hint why this is true ?
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530 views

The correlation between two normal distribution

Let $X$ have the $N(0,1)$ distribution and let $a>0$, show that the random variable $Y$ given by $$Y=\begin{cases} X & \text{if }|X|<a\\[5pt] -X &\text{if }|X|\geq a\; \end{cases}$$ has ...
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1answer
168 views

Lower bounds of laplace transform of characteristic functions

I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function of some undetermined probability ...
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2k views

Poisson process. Time between two events.

Suppose that people immigrate to a territory according to a Poisson process with a $\lambda =$ rate of 1 per day. What is the probability that the time between the tenth and eleventh exceeds two ...
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1answer
106 views

Help with understanding the $\chi^2$-distribution

I'm studying statistics and there's one part in my book I can't understand. I tried to make as good translation as I can of the problematic part...here goes: Chi squared $\chi^2$ distribution Let ...
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133 views

Probability, Poisson Process, My solution is correct?

Jobs are submitted for a computer according to a Poisson process with a rate λ (jobs / hour). Determine the probability of at least two jobs are submitted within the 'a' first few minutes. I tried ...
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29 views

Coin tossing - Two tosses, one is a head, probability other is a tail? [duplicate]

A friend of mine tossed a fair coin twice. Suppose instead that I happen to see the result of one of his tosses, and it is a head. What is the probability that the other toss is tail?
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63 views

Geometric distribution, tossing a die [duplicate]

Possible Duplicate: geometric distribution throwing a die Yesterday I posted a question which was answered but I disagree with the answer so I'd like to ask again so we can discuss it ...
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170 views

Probability distribution function of the length of an interval taken from a uniform probabilty distribution.

This is a consequence of my suggested solution to this question. Consider the probability distribution function that is uniform over the interval $[-a,a]$: $$F(x)=\begin{cases} 0 & x \leq -a\\ ...
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78 views

Joint Distribution Function

Hi Guys, Just need help understanding how to go about doing this question. I know how to convert single distributions but I'm unsure about how to do the joint ones. I usually draw a diagram ...
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487 views

Question on uniform distribution

Two people agree to meet each other on a particular day, between 5 and 6 PM, They arrive independently on a uniform time between 5 and 6 and wait for 15 mintues. What is the probability that they meet ...
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288 views

Proof of $\sum_{k=0}^n k \text{Pr}(X=k) = \sum^{n-1}_{k=0} \text{Pr}(X>k) -n \text{Pr}(X>n)$

$X$ is a random variable defined in $\mathbb N$. How can I prove that for all $n\in \mathbb N$? $ \text E(X) =\sum_{k=0}^n k \text{Pr}(X=k) = \sum^{n-1}_{k=0} \text{Pr}(X>k) -n ...
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308 views

What is the density of the sum $Z = X+Y$?

Find the density of the sum $Z = X+Y$ when $X$ and $Y$ are independent, standard uniform random variables. $$f_X(x) = 1\quad\mathrm{if}\quad 0\le x \le 1$$ $$f_Y(y) = 1\quad\mathrm{if}\quad 0\le y\le ...
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1answer
71 views

Transforming a Continuous Function

My math is quite limited so please bear with me. I will get to the point: Is there a way to transform a continuous function into a bounded one? In essence I have a normalized Gaussian distribution ...
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766 views

a problem of trinomial distribution

$X_1$, $X_2$, $X_3$ are distributed according to the trinomial distribution with $n$($=X_1+X_2+X_3$) and $p_1$, $p_2$, $p_3$ ($p_1+p_2+p_3=1$). What is a correlation of $X_i$ and $X_j$? Is the ...
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116 views

Windowed Linear Correlation

$\DeclareMathOperator \Cov {Cov}$ $\DeclareMathOperator \Var {Var}$ $\DeclareMathOperator \E {E}$ Consider the following experiment: For $N\geq1$, consider $N$ black balls. Let us paint each black ...
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251 views

Finding density of a function of i.i.d. R.V.s

I'm finding it difficult to understand the explanation for my problem's solution, I would like to post it here so maybe one of you could enlight my understanding. Just to clarify, I'm studying for a ...
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3k views

Conditional Covariance of Functions of Random Variables

$\newcommand{\Cov}{\operatorname{Cov}} \newcommand{\E}{\mathbb{E}}$ I realize that $\Cov(X,Y) = \E[(X-\mu_X)(Y-\mu_Y)] = \E[\Cov(X,Y|A)] + \Cov(\E[X|A], \E[Y|A])$. But I am not sure how this is ...
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267 views

maximum of two non central chi squared random variable

Let $$s_i \sim \chi(k_i, \lambda_i), i\in \{ 1, 2\}$$ be two non-central chi-squared random variables with $k_i$ degrees of freedom and $\lambda_i$ parameter of non-centrality I am wondering if ...
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398 views

Density of compound Poisson process

Is the probability density function (pdf) of the Compound Poisson $X(t)=\sum_{i=1}^{N(t)}Y$ known? Where $N(t)$ is a Poisson process and $Y$ is normally distributed with mean $\mu$ and variance ...
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108 views

Bernoulli Distribution with support different from $\{0,1\}$

Suppose the support of a distribution is $\{12 , 13 \}$ with $P(X = 12) = p$ and $P(X = 13) = 1-p$. Is this still a Bernoulli distribution even if the support is not $\{1, 0 \}$?
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169 views

Help with a short paper - cumulative binomial probability estimates

I was hoping someone could help me with a brief statement I can't understand in a book. The problem I have is with the final line of the following section of Lemma 2.2 (on the second page): Since ...
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82 views

Asymptotic equivalent of the law of lotto minimal value

This question is inspired by this one, where the law of the minimum $X$ of $m$ elements sampled without replacement from $\{1, \dots, n\}$ was investigated. In this question we wrote that the number ...
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3answers
2k views

The expectation of the half-normal distribution

For the density function below, I need to find $E(X)$ and $E(X^2)$. For $E(X)$, I did the following steps and got the answer of $-2/\sqrt{2\pi}$. However, this is incorrect as the correct answer is ...
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346 views

Definite integral of cdf of the form $\Phi(\alpha+\sqrt{d^2-\frac{x^2}{2\sigma^2}})$

Any solution for the following definite integaral? Here $\Phi(x)$ represents the cumulative distributive function of standard normal distribution ...
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554 views

PDF of $f(x)=1/\sin(x)$?

What is the probability density function (PDF) of $f(x)=1/\sin(x)$ when $x$ is uniformly distributed in $(0,90)$? $f(x)=\sin(x)$ has a known PDF, which has the form $2(\pi\sqrt{1-\sin(x)^2})^{-1}$, ...
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475 views

Are these transformations of the $\beta^\prime$ distribution from $\beta$ and to $F$ correct?

Motivation I have a prior on a random variable $X\sim \beta(\alpha,\beta)$ but I need to transform the variable to $Y=\frac{X}{1-X}$, for use in an analysis and I would like to know the distribution ...
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Conditions for the solution to a matrix differential equation (with exponentials) to be a valid probability distribution

Define $F(z) : \mathbb{R} \to [0,1]^2$, $C \in \mathbb{R}^{2 \times 2}$ and $D \in \mathbb{R}^{2}$. The scalar $z \in [0, \infty)$ A few further assumptions: $C$ is negative definite and ...
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31 views

Trouble deriving sum of squared normals is Exponential with mean $2$

Box-Muller method hinges on the fact that $R = Z_1^2 + Z_2^2$ is Exponential with mean 2, where $Z_1, Z_2$ are independent standard normals. I want to derive this fact but am getting stuck. I proceed ...
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79 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 ...
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23 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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32 views

Prove Joint distribution of estimators

Let $X_1,...,X_n$ iid r.v. with distribution F, with mean $\mu$ and median $\theta$.Assume that $Var(X_i)=\sigma^2$ and $F'(\theta)>0$. If $\hat{\mu}_n$ is the sample mean, and $\hat{\theta}_n$ the ...