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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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Finding a continuous function that minimizes an integral expression

I am trying to find a continuous function $x(t)$ defined over positive real numbers that minimizes the expression below: $$\frac{\nu y_1 + (1-\nu)y_2}{y_0}$$ where $\displaystyle \nu = \int_{0}^{\...
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e raised to the power of negative Kulback-Leibler divergence

When looking at the similarity between two distributions, I found the Bhattacharyya coefficient to be pretty intuitive to understand since it is normalized to 1 ($0\leq BC(p,q)\leq 1$) and the ...
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Find a bound $R(n,k,\tau)$ such that $P\left\{ \|\ A\epsilon \|^2 \leq R(n,k,\tau)\right\} \geq 1-n^{-\tau}.$

Let $\epsilon\in\mathbb{R}^n$ be a Gaussian $N(0,I_n)$ vector and let $A$ be a projection matrix of rank $k$. Find a bound $R(n,k,\tau)$ such that $$P\left\{ \|\ A\epsilon \|^2 \leq R(n,k,\tau)\...
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Convolution formula, trouble with limits

Good day. I have been working on this for some time and seem to keep stumbling. I think I have a grip on this concept but my trouble seems to lie in the limits of integration. So here goes: Given X ...
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Functions of exchangeable random vectors

Consider the random vector $\begin{pmatrix} X_0\\ X_1\\ X_2 \end{pmatrix}$ with joint cdf $F$. Consider the random vector $ \begin{pmatrix} Y_3\\ Y_4\\ Y_5 \end{pmatrix}\equiv \begin{pmatrix} X_1-...
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Constraining parameters of a linear dynamical system with a Bayesian prior

I have a stochastic linear dynamical system: $x_{t} = A x_{t-1} + w_t,$ where $x_{t}$ is a latent vector, $A$ is a linear state transition matrix and $w_t$ is a process noise vector drawn from $\...
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Distribution consistent with an empty graph

In Daphne Koller's book "Probabilistic graphical models" there is a sentence: "the family of distributions consistent with G, the empty graph." I have two questions: 1.What does "Distribution ...
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How to show $U$ and $V$ are not independent random variable?

$U$ stands for the number of trials to get the first head, $V$ stands for the number of trials to get two heads. I used hand-waving proof, saying that you could not have the two heads trials without ...
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How to distribute N approximately equispaced points with a given probability density?

Let $x_i$ be points in $R^D$ space, $i = 0\ ..\ N-1$, where $N$ is fixed. The problem is to distribute the $N$ points in the space so that their density is equal to given probability density $p(x)$, ...
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Given $X ~$ exponential(1) r.v. Define $Y\sim$ as the integer part of $X+1$, solve for the distribution of $Y$ and derive the conditional distribution

$$Y=[X+1]=i+1$$ iff $$i\le X \lt i+1,i=0,1,....$$ My approach to derive the distribution of $Y$ is to look at $P(Y=y)=P(Y\le y)-P(Y\le y-1)$ Usually, I am ok to find $g^{-1}$ given $Y=g(X)$ ...
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How do you solve for lambda in an exponential distribution?

I am working on a problem and am unsure how to solve it. The problem: Find an exponential distribution such that P(Z $\ge$ 3) = .04 What I have done so far: P(Z$\ge$3) = 1 - P(Z$\lt$ 3) We are ...
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1answer
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How do you solve for the mean in a Normal Distribution?

I am working on a problem and am a little bit stuck on how to solve it. The problem: Find a Normal Distribution with SD 2.5 and 5% Quantile at -15.2. What I have done so far: $$X=\mu+2.5Z$$ $$.05=P(\...
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Probability of Choosing a person that likes cake.

You have $20$ people in a population. $5$ people prefer cupcakes over cake and $15$ prefer cake over cupcakes. you will choose $5$ people at random. What is the probability that you will choose $2$ ...
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Type of Distribution

There are 3 yellow balls in barrel and 4 blue balls. What distribution can be used to calculate the probability that out of a sample of 4 balls chosen, what is the probability that 2 balls are yellow ...
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probability density function usage in a problem

The maximum electric power required in a city with a random variable X is modeled has a probability density function Below: $$f(x) = c^{2}xe^{-cx}U(x)$$ $$c= 5\times10^{-6} per kilowatt$$ First if ...
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Binomial probability question : bits transmission error probability calculation [Check my answer please]

Q. A communications channel transmits the digits 0 and 1. However, due to static, the digit transmitted is incorrectly received with probability 0.2. Suppose that we want to transmit an important ...
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1answer
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Why can we substitute any arbitrary variable in for this expression of the CDF?

I'm trying to follow this example of a variable transformation, which defines the PDF and CDF as follows: PROBLEM: The above definition is used in solving this problem: Question: Now, where I get ...
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Relationship between Rademacher distribution and Normal distribution

How to find the relationship between Rademacher distribution and Normal distribution, where Rademacher distribution is given as The probability mass function of this distribution (https://en....
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1answer
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How do you solve a normal distribution for an expected value?

I am working on a problem and am a bit stuck. It is: For X ~ $\mathcal N $(-2,2) find E(X$^3$) What I know so far is that: For X ~ N(0,1) EX = 0 and VarX = 1 I am confused though how we find ...
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Finding a joint probability

Not sure if this is the place to ask this question. In an acoustical noise cancellation system, the interference picked up in a master channel signal an at time n has to be cleaned up or removed by ...
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Condition on a probability distribution so that translating it leads to a multiplicatively different distribution, with increasing factor

Consider a real-valued random variable $X$. I want the following function: $$ t\longrightarrow\max_S\left(\max\left(\frac{P[X\in S]}{P[X+t\in S ]},\frac{P[X+t\in S]}{P[X\in S]}\right)\right) $$ to ...
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3answers
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How $P[X=x]= \frac2{3^x}$ can give an even value for $x =1,2,3,\dots$

I have a question that says “Let $X$ be a discrete random variable with probability function $P[X=x] = \frac2{3^x}$ for $x = 1,2,3,\dots$ What is the probability that $X$ is even? The thing is I don’...
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Independent random variables $(X_i)$ have the same law, then $(X_i,\sum X_i ) $ have the same law for any $i$

Let $(X_i)$ be sequence of real independent r.v.'s and having the same law. If we let $X=\sum X_i$, how can one show that $(X_i, X)$ have the same law for any $i$. In this case $(X_i, X)$ is a random ...
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27 views

Skewness of a squared random variable

Say we have some probability density function $f(x)$ which is unimodal (with the modus at 0) and also symmetric in the y-axis. It seems to me that 'in most cases', the square of a r.v $x \sim f(x)$ ...
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1answer
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Let $X_1\sim Laplace(0,\sqrt{1/2})$ and $X_2 \sim Laplace(1/2,\sqrt{1/2})$. Are $X_1$ and $X_2$ independent?

Let $X_1\sim Laplace(0,\sqrt{1/2})$ and $X_2 \sim Laplace(1/2,\sqrt{1/2})$. Are $X_1$ and $X_2$ independent? I understand that in case of independence, the joint pdf is the product of the marginal ...
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1answer
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Given a random variable $X_1$, does there always exsist an i.i.d. sequence $\{X_n\}$?

Below is from Tao's lecture note and he says there exists an i.i.d. sequence of random variables $\{X_n\}_n$ such that each $X_i$ is uniformly distributed. For any given random variable $X_1$, does ...
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Question about random sample of $X^2$

Let $X$ be a Random Variable. Suppose I want to approximate $E[X^2]$ by taking M values. Can I draw a random sample of $X$ of size M, square every value from this sample, and then take the mean? Or ...
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Unbiased estimator of a square root of chi-squared distribution [duplicate]

Let $Y_1,Y_2,...,Y_n$ be a random sample from $N(\mu,\sigma^2)$. I need to show that $S$ is a biased estimator of $\sigma$. As from the definition, I see that $\frac{(n-1)S^2}{\sigma^2}\sim\chi^{2}_{(...
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Finding $P(XY \le a)$ when $x<y$ and the distribution of both is uniform

Let $X$ and $Y$ be random variables with joint pdf $$f_{XY}(x,y) = \begin{cases} 1, &0 < x <1,\ \ x < y < x+1 \\ 0, & \text{otherwise} \end{cases}$$ Find $P(XY \le a)$ ...
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1answer
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Beta distribution CDF to Binomial Survival Function

There is a claim in my book that there is a connection to the Beta CDF and a Binomial Summation without explaining further. "Integration by Parts can be used to show that for $0<y<1$, and $\...
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1answer
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Find marginal probability density function without the joint density function or the other marginal pdf

This is a question from exam review sheet. Please give me some guidance here. I do not know how can I find fY(y) without having information on f(x,y) or at least fX(x)? Consider a random variable Y ...
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2answers
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Joint Density Formulas for XY and X/Y

I used convolution to find a formula for $X+Y$, but am unsure on how to figure out formulas for $XY$ and $X/Y$. Any help would be appreciated.
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Finding P.M.F of maximum ordered statistic of discrete uniform distribution.

Let $X_1,X_2,X_3....X_n$ be a sample from PMF $P(X=x)=P_X(x)=\dfrac{1}{N} \ \ \ \ ;x=1,2,...N$ $X_n=$max($X_1,X_2,X_3....X_n$) I calculated P.M.F of $X_n$ from this formula $n(F_X(x))^{n-1}f_X(x)$...
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1answer
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Conditional distribution w.r.t a sum of exponential r.v's

Let $X$, and $Y$ be two i.i.d exponential r.v's with rate $1$. I'm supposed to find the distribution of $Y|X+Y = c$, where $c>0$ is some constant. I can prove this by using properties of arrival ...
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What does it actually mean by the term “Probability Distribution”? [on hold]

According to Google: To me, this is actually an incorrect definition. Why? Coz: They are surely talking about a PDF (Probability Mass Function). Before the definition of a function, we need to ...
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Why CDF is not coming to 1, when i put the maximum range?

Suppose, $F_X(x)=-\frac{x}{a^2}+\frac{2\sqrt{x}}{a}$ And, $f_X(x)=\frac{d}{dx}F_X(x)=\frac{1}{a\sqrt{x}}-\frac{1}{a^2}$, Here, $0\leq x \leq a^2$ Similar, $f_Y(y)=\frac{1}{a\sqrt{y}}-\frac{1}{a^2}$, ...
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1answer
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Function to calculate t-stat of similarity in survey answers

I have a survey of a large number of questions. Each question is multiple choice, and has three possible answers. Users get served random questions to answer. So they do not all answer the same ...
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What are some statistical distributions with the irrational numbers e and pi in their functions? (apart from the most common - Normal, Poisson)

I've been researching on the application and origin of irrational numbers in probability theory and statistical distributions, so far having derived a unique proof of Stirling's approximation, and ...
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What is the connectivity probability of a road segment?

Given a lane with length $l$ in a road segment $r_{ij}$. The vehicles arrive to the entry of the lane following exponential distribution with parameter $α_{ij}$. The number of vehicles in the lane ...
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Estimation in presence of signal dependent noise

Given a model as below: $$y_1 = x + \eta_1$$ $$y_2 = x + \eta_2$$ where $n_1 \sim N(0,\sigma_1^2)$ and $n_1 \sim N(0,\sigma_2^2)$, $N$ denotes a Gaussian distribution and $\sigma_1^2$ and $\sigma_2^2$...
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1answer
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Which gamma do I need for my Cauchy distribution?

I need a probability distribution which is the ratio of two normal distributions or $P=(N_1/N_2)$. The mean of both normals can be assumed to be zero, and the variance is known. Apparently the ...
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Failure Rate calculation based on manufacturing defects (electronics-capacitor)

Electronics failure rates typically follow the exponential distribution and models like Prism, Mil-Hdbk-217 or 217Plus (newer) are used to predict failure rates. I'm using 217Plus model for ...
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1answer
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Finding $a$ and $b$ such that $\hat u$ is an unbiased estimator

A chemist wants to decide the amount of a certain substance $\mu$ in a specific type of food. In the lab, the chemist has two measuring intruments $A$ and $B$. The results from the instruments can be ...
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1answer
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From the marginal to the joint cdf

I have some doubts on the relation between the joint cumulative distribution function and its marginals. Consider a random vector $X$ of dimension $L\times 1$ with cumulative distribution function $...
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Probability of average times called being inside an interval

Assume that when a man calls a restaurant to order some food, his call has a probability $0.2$ of reaching through. Every attempt is independent and we can assume that the restaurant is open every day....
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Sufficient conditions on the marginals for exchangeability of the joint distribution?

Consider a random vector $X$ of dimension $L\times 1$ with cumulative distribution function $F$ absolutely continuous. Let $F_1,..., F_L$ denote the marginal cdf's. Assume that the probability ...
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1answer
16 views

Intuition/proof that the input into chi-square goodness-of-fit test is standard normal.

Looking for an easy proof/intuition for the fact* that (assuming $E_i$): $\sqrt{\frac{({O_i-E_i})^2}{E_i}} = \frac{{O_i-E_i}}{\sqrt{E_i}} \sim N(0,1)$ A similar question exits but is not as ...
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1answer
40 views

Convolution without known pdfs in terms of z

I have looked at many sites with almost enough information for this to make sense... just not quite there yet. So I have this convolution formula (w instead of z and $dx$ instead of $dy$): $$f(w) = \...
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2answers
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Can identically distributed random variables $X$ and $Y$ have $P(X < Y) \geq p$?

Question comes from Joe Blitzstein's "Introduction to Probability". Let $X$ denote days of the week, encoded as $1, 2, ..., 7$ with equal probabilities. Set $Y = (X + 1)$ mod $7.$ It is easy to see ...
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Find the distribution of the error

Suppose we want to find a circle to fit the data points. The error of the data point can be calculated as $e_k=||\vec{x}_k-\vec{x}_c||^2-r^2$ ------------------(1) where the $\vec{x}_k$ is the ...