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

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1answer
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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1answer
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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2answers
500 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
163 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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2answers
1k 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
103 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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0answers
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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3answers
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
165 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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1answer
77 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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2answers
434 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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2answers
282 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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2answers
293 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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2answers
741 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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0answers
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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2answers
245 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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2answers
2k 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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0answers
374 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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1answer
103 views

Bernoulli Distribution

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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1answer
168 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
80 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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2answers
334 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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1answer
532 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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2answers
456 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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29 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 ...
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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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1answer
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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1answer
23 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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1answer
59 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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1answer
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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1answer
43 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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1answer
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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1answer
37 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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1answer
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 ,& ...
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150 views

Show that $ \hat{\theta}_2 = Y_{(n)} - \frac{n}{n+1}$ is unbiased estimators of $θ$.

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the uniform distribution on the interval $(θ, θ + 1)$. Let $$ \hat{\theta}_2 = Y_{(n)} - \frac{n}{n+1}$$ Show that $\hat{\theta}_2$ is ...
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1answer
36 views

multivariate probability generating function

Suppose I have three random variables $X_1$, $X_2$ and $X_3$, with probability generating functions $g_1(z)$, $g_2(z)$ and $g_3(z)$. Now I have a joint-distribution $P(X_1-X_2,X_1-X_3)$, whose ...
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Is $(\ln l(y))^2 l(y)^x f_0(y)$ integrable over the real numbers?

Is $(\ln l(y))^2 l(y)^x f_0(y)$ integrable over $\mathbb{R}$ for any continuous pair of densities $f_0$, $f_1$ and $l=f_1/f_0$ with some known constant $0\leq x\leq 1$? It seems that $(\ln ...
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1answer
140 views

questions on bias of estimator

a) Let $X_{1},...,X_{n}$ be i.i.d Uniform$[0,\theta]$. Show that estimator $\beta(X)=max(X_{1},..,X_{n})$ is a biased estimator for $\theta$.Find an unbiased estimator, based on $\theta$. My attempt: ...
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1answer
52 views

How to multiply a standard normal RV times a uniform{-1.1} RV?

I'm not really sure how to even start this question. I know how to get the pdfs of both, but don't get what it equals when you multiply them together, especially since they have different ranges.
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1answer
203 views

Suppose that Z has a standard normal distribution. Find the density function of $U = Z^2$.

Suppose that Z has a standard normal distribution. Find the density function of $U = Z^2$. First of all, do I need to use probability function for normal distribution to find the cdf for U. If so, ...
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0answers
24 views

Tranform residual to match expected moments

Let's say we have the a set $(X_i,Y_i)$, $i\in I$, $I$ is an arbitray finite set of indexes, and the model $$ Y = g(X\beta)$$ Using some method, we obtain the individual first four moments of the ...
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1answer
59 views

How to obtain a pdf of a random variable defined as a function of many variables?

Given $N$ independent random variables ($X_1$,$X_2$,...,$X_N$) with individual pdfs $f_1$,...,$f_N$: How to determine the pdf of a random variable $Y=G(X_1,...,X_N)$?
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100 views

Proof of the affine property of normal distribution for a landscape matrix

The widely used/mentioned/assumed affine property of multivariate normal distributions says that: Given a random vector $x \in R^N$ with a multivariate normal distribution -- $x \sim N_x(\mu_x, ...
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1answer
209 views

Joint probability density function using the same random variable

My original problem statement: Let $x$ be a random variable with pdf $f_x(x)$ and let y be $y=x$. (a) Find the joint pdf $f_{x,y}(x,y)$. (b) Find the conditional pdfs $f_{x|y}(x|y), f_{y|x}(y|x)$ ...
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1answer
85 views

Finding integration bounds for density of sum of two independent random variables

Let $X, Y$ be independent random variables, both uniformly distributed over the interval $(0,1)$. That is, $$f_{X}(a)=f_{Y}(a) = \begin{cases} 1 & \text{if $0 < a < 1$} \\ 0 & ...
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1answer
797 views

Finding joint cdf and pdf of independent random variables

Let X and Y be independent random varaibles. Each has an exponential distribution with parameter λ. Define two new random variables by W = min{X,Y} Z = max{X,Y} Find the joint cdf of W and Z, ...