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Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions.

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Proving formula for math expectation

In a book I met a formula for math. expectation of a random variable $\xi$ with distribution function $F(x)$: $$M{\xi}=-\int_{-\infty}^{0}F(x)dx+\int_{0}^{\infty}(1-F(x))dx$$ I wonder how do I prove ...
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Correct notation for probability

Suppose I have a sequence of $n$ iid random variables (RV) $ X_1,X_2,...,X_n$ I want to denote the probability of at least one of those RV being lower than a threshold $x$. Is this notation correct? $...
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conditional expectation $E[{X_1}^2+{X_2}^2|X_1+X_2=t] $ of normal distributed variables

to find the conditional expectation $E[{X_1}^2+{X_2}^2|X_1+X_2=t] $ if $X_i$'s are independent and both are std. normal distributed. My attempt: as given is $X_1+X_2=t$ , take $X_2= t-X_1$ and ...
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Joint density of $(R,X)$ when $(X,Y)$ is uniform on the unit circle and $R^2=X^2+Y^2$

The question is like this: A point $(X,Y)$ is picked at random uniformly in the unit circle. Find the joint density of $R$ and $X$, where $R^2 = X^2 + Y^2$. The TA drew a diagram like this and ...
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Which curve to select for finding the CDF of a function of a continuous joint distribution?

I came across a question which required to find the CDF of a function of a continuous joint distribution: $$W=X*Y$$ The following is the joint PDF: The PDF of the joint distribution Therefore, $40 \...
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Find the conditional density

I have a sequence of iid random variables $X_1, \ldots, X_n$ following the pdf: $$ f_\theta (x) = \theta x^{\theta-1} $$ for $\theta >0$ and $0 <x<1$. I have prove that T= $$\sum_{i=1}^n \...
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What is the mathematically appropriate and concise way of writing a constant added to a random number?

Let $x$ be a constant in $\mathbb{R}$. Let $y$ be a random number that is generated according to a certain probability distribution. I want to make the sum $$x + y$$ However, I am not sure how to ...
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Implementing Gibbs Sampler on joint distribution $X$ and $N$ where $X$ is continuous and $N$ is discrete

Q Random variables X and N have joint distribution, defined up to a constant of proportionality, $$f(x,n) \propto \frac{e^{-3x} x^n}{n!} ~,\quad n=0,1,2, \ldots , x>0$$ Implement a Gibbs sampler ...
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Probability distribution on a transformed variable problem

I am not sure if my process to solve this particular problem is correct and not looking for particular solutions. The Question: Given $X^3 $~ $ N( \mu , \sigma^2), $ x>0$ $ and $Y=(\frac{X^2}{4})$...
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To what PDFs does the Chebychev Inequality Theorem apply? [on hold]

Does the Chebychev Inequality Theorem apply to any & all probability distribution functions? Thank you. SDH
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Find the underlying normal distribution of a folded normal distribution

This might be a very basic question, but I haven't found a clear answer anywhere, so I hope someone here can help. How does one find the mean and standard deviation of a normal distribution that ...
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1answer
29 views

Given an $X_1,…,X_n \sim Unif[0,1]$ i.i.d. sample, after ordering them, one of $X_k^*=x$, give a maximum likelihood estimation for $k$ using $x$.

The problem: The title sums up the problem pretty well, we have a sample from an independent idential (uniform) distribution (i.i.d.): $X_1, ..., X_n \thicksim Unif[0,1]$, and we only know one ...
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Mathematically bad behavioral distributions

I'm studying asymptotic theory and have found that most of the distributions shown in textbooks have "good" properties like differentiability and integrability. Edgeworth expansion, for example, ...
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$X_1, X_2, …, X_n \sim Exp(\lambda)$, what's the joint distribution of $X_1, X_1+X_2, …, X_1+X_2+…X_n$ and is it a uniform ordered distribution?

To elaborate on the title, here is the entire problem: Let $X_1, X_2, ..., X_n \thicksim Exp(\lambda)$ be an independent sample. What's the joint distribution of the sequence of $X_1, X_1 + X_2, ...,...
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Binomial Probability Number of Trials Formula and Code

This is an interview question I've had. Your number space is between 0 to N. You can only draw M random samples from 0 to N (N >= M). How many times (T) would you need to draw from your number space ...
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1answer
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Does anything like expectation of joint distribution exist?

I know how to find the expectation of a function of a Random Variables, I was just wondering that does expectation of a joint distribution exists? I think, since expectation is an average which by ...
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Identically distributed?

Given a set of independent and identically distributed random variables $(\xi_i)_{i\in I}$ i know that $(X_i)_{i\in I}$ and $(Y_i)_{i\in I}$ ar conditionally independent and identically distributed. ...
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Estimation of a geometric distribution's parameter by the reciprocal of the sample mean

I came accross this exercise when studying statistics, but I can't get to what's the solution. The exercise simply asks to show whether the reciprocal of the sample mean is an unbiased estimation for ...
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2answers
44 views

Name of the distribution $p^x(1-p)^{1-x}$?

We are given a random variable $X$ with PDF: \begin{align*} f(x ; p) &= p^x(1-p)^{1-x} \ , \\ \end{align*} where $0 \leq p \leq 1$ is the parameter and the support is $x \in \{0,1\}$. Anyone ...
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30 views

for poisson distribution, show that $P(0<X<2(\lambda +1)) \ge \frac \lambda{\lambda +1}$

If $X$ is poisson distributed with mean $\lambda \ge 0$, (integer). Show that $P(0<X<2(\lambda +1)) \ge \frac \lambda{\lambda +1}$ I applied markov inequality but I can't match the answer ...
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Finding the pmf of a r.v. S ~ Poisson's multinomial distribution.

Notation # categories $= c$. # trials $= t$. Side $i$ = $si$. Random vector $= S = \left[S_1\;S_2\;\ldots\;S_c\right]^T$. # occurrences of $si$ vector $= s = \left[s1\;s2\;\ldots\;sc\right]^T$. ...
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a problem in Durrett third edition exercise3.7 Remark [on hold]

how to understand if $\phi(t) \to 0, as\ t \to \infty, \mu$ has no point mass. $\phi$ is characteristic function.
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1answer
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Show that $n · Y_n$ converges in distribution as n $\rightarrow 1$, and find the limit distribution.

Let $X_1, X_2, . . .$ be a sample from the distribution whose density is: $f(x) = \begin{cases} \frac{1}{2}(1+x)e^{-x}, & \text{for $x>0$} \\ 0, & \text{otherwise} \end{cases}$ ...
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Poisson Process Distribution of waiting time$W_{X(t)+2}$ in poisson process

Let's say $N(t)$ is a poisson process with parameter $\lambda$. Let $W_n$ to represent the waiting time of $n$th arrival. What is $P(W_{N(t)+2}≤t+s)$? I solved this question by the following way:$$P(...
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1answer
29 views

Relation between probabilities such as $P(AB>a) > P(Ab>a)$

Let $X=AB$, $A$ and $B$ are random variables which are NOT independent and I know that $A>0$, $B\geq b >0$ with $b$ is a deterministic constant. Then for any constant $a$, probability $P(X>...
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Finding pmf for 4 variations of an urn problem.

I would like to test my understanding of distributions by verifying my answers to 4 deliberately related questions. Urn R has 5 R marbles. Urn G has 11 G marbles. Urn B has 4 B marbles. Random vector ...
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Finding the probability that each apple in a randomly selected bag has a mass less than 105 grams.

I am practising normal distribution exam type questions but I am stuck at this one: The masses of individual apples sold in a food store are normally distributed. The supplier who provides the ...
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Product of two normal distribution $Z = XY$; where $X$ is $N_1(\mu1,\mathrm{std}_1)$ and $Y$ is $N_2(\mu_2,\mathrm{std}_2)$

If we multiply directly, then the new mean is $\mu = (\mu_1 \cdot \mathrm{std}_2^2 + \mu_2 \cdot \mathrm{std}_1^2)/(\mathrm{std}_1^2 + \mathrm{std}_2^2)$ and $\mathrm{std}^2$ is $(\mathrm{std}1^2 + \...
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1answer
23 views

Conditional probability of two dependent continuous random variables

I have two continuous random variables $V_1$ and $V_2$ defined as $$\begin{aligned}V_1 &:= a_1 \cdot W_1 + a_2 \cdot W_2 + a_3 \cdot W_3 + a_4 \cdot W_4 + a_5 \cdot W_5 \\ V_2 &:= b_1 \cdot Y ...
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Finite Moments of Vector in Exponential Family

I am studying some notes on exponential families and there is a section on the computation of moments. The exponential family has the form $$\exp(\sum_{j = 1}^k \phi_j B_j(x) + C(x) - D(\phi))$$ I ...
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1answer
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How can we plot an equation in 3 variables on a 2D plane in context with functions of pair of Random Variables?

I came across the following question: The Question I tried solving it, the following is my attempt: $$ P[W\le w] = P[XY\le w] = P[Y\le w/X] $$ And then I simply double integrated keeping the ...
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Sample Median and a single observation asymptotically independent? [on hold]

Given a sample of i.i.d. random variables $X_1,...,X_n$, can it be shown, that $X_j$ and $med\{X_1,...,X_n\}$ are asymptotically independent for any $j$? (The impact of one random variable on the ...
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1answer
28 views

Why a normal distribution would not give a good approximation to a specific discrete distribution?

I have a discrete distribution $X$ with mean $56.87500$ and standard deviation $70.725$. I also have that the support of $X$ is contained in $[0, 8750]$, the maximum value of $X$ is $8750$ and $p(X=0)=...
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1answer
34 views

Density function of a sum of 75 discrete random variables

I need help with find the density function of $S$ if $$S=\sum_1^{75}X_j,$$ where $X_j=I_jB_j$, and $I_1,\ldots, I_{75},B_1,\ldots, B_{75} $ are independent, $$Pr(I_j=1)=0.01 , \ Pr(I_j=0)=0.99 \ \ ...
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superposition of infinitely many poisson processes

I know that the superposition of two Poisson process with rates $\lambda_1$ and $\lambda_2$ is again a Poisson process with rate $\lambda_1+\lambda_2$. Thus this process has interarrival times ...
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2answers
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Find the probability of a Normal Distribution random variable

The excercise is given as it follows: Let the Temperature $T$ during a month of a year has a normal distribution with mean $68°$ and a standard deviation of $6°$. Find the probability $p$ that the ...
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4answers
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Intution behind the fact that $X,Y$ i.i.d $\not \implies \mathbb E[X|A] = \mathbb E[Y]$

Let $A=\{X=Y\}$ be the event that $X$ and $Y$ take the same value. If $X$ and $Y$ arediscrete, independent and identically distributed, then is it true that $E[X|A] =E[Y]$? I know that it is false (...
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Understanding examples of Poisson Distribution

Sheldon Ross describes the following as examples of random variables that generally obey the Poisson probability law: $1.$ The number of customers entering a post office on a given day. $2.$ The ...
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How do I include the fact that Poisson λ follows a gamma distribution? [on hold]

A marine biology research group is looking for a kind of precious aquatic plant in a particular sea area. Let X be the number of the plants per cubic kilometer in the sea area and assume X has the ...
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Distribution function of X-Y for normally distributed random variables [duplicate]

I have two independent normally distributed random variables $X,Y\sim \cal N(\mu,\sigma^2)$ and want to calculate the distribution of $X-Y$. I tried with $F(z)=P(X-Y \leq z)$ but failed. Does anyone ...
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1answer
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CDF from PDF of a function

I have this problem where I need to graph the CDF, for that I need to find the constant $c$. The formula below is a PDF: $f(x) = c(x^2+1)\space\space\space if\space X \in [0,1];\space$ otherwise $0$...
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1answer
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How do I choose a set of numbers from a PMF with a specified total?

So basically I'm choosing a set of numbers from a probability mass function, (say binomial or scale-free). By which I mean I'm performing a weighted choose operation using the PMF as weights. However ...
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1answer
27 views

Statistics and Confidence Intervals

Given the following set of values: 10,11,14,95,73,30,29,9,97,94,70 How do I calculate a 99% confidence interval for the sample mean? I am assuming that the variance is 10 Well, the idea I have is ...
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Existence of a joint distribution given the conditional and marginal distribution

Can anyone point me a book where it has a proof of Theorem 1.7 (ii) of Jun Shao's book - Mathematical Statistics? I need this to show that given a distribution on one space and a collection of ...
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How to assign weight while approximating multiplication

I have two summations: $y = \sum_{i=1}^{s} w_{i} x_{i}$, and $y' = \sum_{i=1}^{s'} w'_{i} x'_{i}$. $s$ and $s'$ are the number of terms in each summation, and both are big numbers. $w_i$ and $w'_{i}...
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Is there a generative network model for arbitrary distributions that guarantees an edge count?

So what I am trying to do is rewire a random directed graph (specifically a boolean network) so the out-degree distribution is scale-free. However I need a generative model that will allow me to ...
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1answer
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Product of independent random variables following different distributions

I need to find the CDF of the product of two independent random variables $Z=XY$. $X$ is defined in $\left ( -\infty,0 \right )$ and $\left ( 0, \infty \right)$. Y is defined in [$0,A_o^{2}$], whith $...
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Probability, expectations [on hold]

i need help with this exercise. (I google translated from norwegian so sorry if the english is bad) We want to find the proportion of the working population that is unemployed at some point. A ...
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1answer
33 views

Confused about how to show independent random variable $Y$ has the Poisson distribution with parameter $t\lambda$

Assume there are N independent exponential random variable $(X_1, X_2,..., X_N)$ with parameter $\lambda$. Fix a real number $t > 0$. Let Y be the largest $N$ so that $X_1 + X_2 + \ldots + X_N \...
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1answer
28 views

Sucker Bet - Coin Flipping Stochastic Process

Having a lot of trouble working out this exercise. I have tried constructing the 8x8 matrix with all possible combinations of three flips of the coin {HHH, HHT, HTH, ... , TTT} and then calculating an ...