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

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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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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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148 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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72 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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63 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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529 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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131 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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513 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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166 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
104 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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62 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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2answers
166 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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461 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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283 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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301 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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751 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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115 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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246 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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261 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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385 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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106 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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1answer
81 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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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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339 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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546 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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467 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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40 views

Two random variables and finding the expectations

I have a question similar to this Link. The main difference is that now there exist two random variables, which are both normally distributed and independent to each other. There are two equations ...
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74 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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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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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 ...
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63 views

Prove $Pr[X + Y \geq x] \sim Pr[X \geq x]$

We have two independent random variables $X_n$ and $Y_n$, where $$X_n=\sum_{i=0}^n x_i$$ and $$Y_n=\sum_{j=0}^n y_j,$$ where $x_i$,$y_j$ are (non-identically) Bernoulli distributed and independent. ...
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39 views

Compute a conditional probability of normal random variable

Suppose $X, T$ are continuous random variables, and $X \sim \mathcal{N}(0, 1)$, $T$ have density function $f_T$. (But $X,T$ do not have joint density) Is there any way to compute the following ...
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24 views

monotonicity of binomial coefficient

I am interested in $$f(x):={k-1 \choose x-1} p^{x} (1-p)^{k-x}.$$ How do I find out in which Domain this function is monotonically increasing, in which it is monotonically decreasing? For which $x$ ...
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66 views

Quick Question Integration with Joint PDF

Let $X_1, X_2, \ldots, X_n$ by independent and identically distributed random variables with probability density function (pdf) $$f_X(x) = \left\{\begin{array}{ll}1, & 0 < x < 1\\ 0, ...
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28 views

distribution min

I have random values $X,Y$ and $f(x,y) = 2x+4y$ , where $0\le x\le 1$ , $0 \le y \le 1-x$ How to calculate distribution of $Z = \min(X,Y)$ ? As I calculated, $X$ and $Y$ are dependent, because ...
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44 views

Joint cumulative probability with dependent interval

Edit 2: Tried to make the question clearer by adding a graph Edit: Please note: This is not a duplicate of another question I asked earlier (Finding joint density of dependent variables). The ...
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29 views

Solving this random variable problem

This is an earlier problem Proving this random variable problem but generalised, maybe you want to take a look at that one first? $X_1,X_2,X_3,\ldots$ are IID random variable taking values in ...
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46 views

Formula needed for calculating probability of recurring events

I'd like to find an answer for calculating the following recurring events: You have X opportunities of picking a ball from a sack. Every time after a ball is picked, the ball is returned to the sack. ...
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50 views

Average minimum distance between two random vectors

Let $\mathbf{y_1} =\begin{bmatrix}g_1x_1 & g_2x_1 & \dots & g_Nx_1 \end{bmatrix}$ and $\mathbf{y_2} = \begin{bmatrix} f_1x_2 & f_2x_2 & \dots & f_Nx_2\end{bmatrix}$. All the ...
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50 views

$$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& elsewhere.\end{cases}$$ Find the MLE for $θ$.

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...