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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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Probability distribution of a moving particle

I am having a issue with the wording of this question. Find the probability of the following. The velocity $v$ of a randomly selected particle, whose distribution obeys the probability density ...
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11 views

Distribution of life time of a serial circuit with bulbs

Assume that we have a serial circuit with three bulbs. Each bulb's life time is exponentially distributed: $$f_{bulb}(t) =\left\{ \begin{aligned} &\lambda e^{-\lambda t} & t \ge 0\\ &0 &...
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1answer
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Linear transform of bivariate normal distribution

Suppose that $Y_1$ and $Y_2$ follow a bivariate normal distribution with parameters $\mu(Y_1)= \mu(Y_2)= 0, {\sigma^2}(Y_1)= 1, {\sigma^2}(Y_2)= 2$, and $\rho = 1/\sqrt 2$. Find a linear ...
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1answer
15 views

Asymptotic distribution of median estimator when density doesn't exist

We know that when density (say $f$) exists at the median(say $\theta$) then the median estimator(say $\hat{\theta_n}$) has the following property: $$ \sqrt n(\hat{\theta_n}-\theta) \to^d N(0,1/\{4f(\...
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1answer
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Is P(A≤−B)= 1− P(A≤B) an correct equation?

Is P(A≤−B)= 1− P(A≤B) an correct equation? If yes, kindly provide the derivation of the same. As I get it, P(A≤−B)= 1− P(A>−B) i.e. 1−P(−A
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Find the P.M.F. of $X_{1}+X_{2}$ given $(X_1,X_2,X_3,X_4) \sim \text{Mult}(n,4,p_1,p_2,p_3,p_4)$

I can find the marginal P.M.F.s of $X_1$ and $X_2$ but then I am lost on how to convolute the two PMF's into one PMF. I should be using convolution formulas right? because that is the only way I can ...
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2answers
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Shortcut to finding the distribution of a specific random variable

Question: A dice is rolled 3 times. Let X denote the maximum of the three values rolled. What is the distribution of X (that is, P[X = x] for x = 1,2,3,4,6)? You can leave your final answer in terms ...
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1answer
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Transforming sum of exponential variables to chi-squared distribution

Assume $X_i$ are generated with the following distribution: $$ f(x; \theta, c) = \theta^{-c}cx^{c-1}e^{-(x/\theta)^c}$$ $\theta>0$ and $c>0$ is known. Further, assume $T(X)=\sum^{n}_{i=1} X_i^...
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Why does $ \mathbb{P}\left(X < -z\right) = \alpha \Rightarrow -z = \chi^2_{1 - \alpha}(2n) $ hold?

Assume $X_i$ are generated by $\Gamma(\theta_0,n)$ distribution, and $S_n = \sum X_i$. Further, it is known that $2 \theta_0 S_n$ follows a $\chi^2(2n)$ distribution, $\theta_0$ is known, $\theta_1 &...
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Reverse engineering distributions

Suppose I am given a measurable function $f:\mathbb{R^n}\rightarrow \mathbb{R}^n$ and a probability distribution $\mathbb{P}$ on the Borel or Lebesgue sigma algebra of $\mathbb{R}^n$. Assume that the ...
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1answer
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Can we write $P_{(2\lt X \le 4)}$ as $P_{(2 \lt X)} \cap P_{(X \lt 4)}$? [on hold]

Asking this becuase, I came across the explanation like $$P_{(2\lt X \le 4)}=P_{(2 \lt X)} \cap P_{(X \lt 4)}$$ in a lecture by Marc Tagoba in Statlect. If this is valid enough, where I am going ...
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how to measure the peak of a distribution?

As mentioned and explained in detail in this Math Exchange here, particularly by Peter Westfall, Kurtosis only measures "extremity of the tails", not the "peak" of the distribution which can even ...
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24 views

Calculating $P(2\leq X\leq 4)$ for an exponentially random variable

While calculating P(2≤X≤4), for an exponential random distribution, the solution says, $P(2\leq X\leq 4) = F(4)-F(2)$, where F denotes the CDF. My version is, P(2≤X≤4) = P(22) and P(X≤4), i.e. 1-P(...
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0answers
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Sufficiency in the exponential distribution

I am trying to show that given a random sample $\{X_i\}_{i=1}^n$ where $X_i\sim exp(\lambda^{-1})$, the statistic $T(\mathbf{X})=\sum_{i=1}^n X_i$ is sufficient by using only the definition. I have ...
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What is the expected value: [on hold]

How would I calculate the expected value of: There is a game involving opening doors. There are 10 doors and 3 contain normal balls while one contains a gold ball. One gold ball is worth 3 points ...
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19 views

Simplifying a marginal likelihood function

I have the following likelihood function $$ p( z \big| x, \lambda, \sigma) = \frac{1}{\sigma^{2}} \cdot \exp \bigg( -\frac{\big( z^{2} + \lambda^{2} \cdot x^{2} \big)}{ \sigma^{2} } \bigg) \cdot I_{0}...
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2answers
31 views

proof that $Y=μ+σX$ if X∼N(0,1),

I want to proof that If$$X∼N(0,1)$$, then$$Y=μ+σX$$has the normal distribution with mean $μ$ and variance $σ^2$. I searched it before, but I don't understand why I have to calculate the probability ...
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2answers
26 views

Comparing two normal distributions

Given a normal distribution $X$~$N(60,9^2)$ with a random variable $A$ and a normal distribution $Y$~$N(50,7^2)$ with a random variable $B$, how do I go about finding the probability $P(B>A)$? (...
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1answer
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Urn problems: find mean and variance - stuck

I am stuck in a problem, and I can't think of a next step to find the solution. The question is the following: Suppose an urn has $k$ balls, numbered from $1$ to $k$, $k \in \mathbb{N}$. A sample ...
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0answers
14 views

Minimum of an unknown distribution

I wanted to find $min_x P_x$ but I don't know the distribution $P_x$. I know I can find it empirically. Is there any other way to find this value? I only know that $P_x$ is discrete. Will the same ...
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2answers
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Find the pdf , distribution function of $X$ and $E[(X-2)^2]$

I 'll be very grateful if you can help me , here is the question : When a person sends an email, the probability that there is an attachment is 0.5. If there is an attachment then the size of the ...
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9 views

Bitcoin price Distribution: GBM or Not?

Is the Geometric Brownian Motion (GBM) a suitable model to describe the Bitcoin price over time? In my opinion it is NOT and a distribution which changes over time is more appropriate model (Btc is ...
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1answer
47 views

A group of $200$ persons consisting of $100$ men and $100$ women is randomly divided into $100$ pairs of $2$ each

A group of $200$ persons consisting of $100$ men and $100$ women is randomly divided into $100$ pairs of $2$ each.Find the maximum chance that at most $30$ of these pairs will consist of a man and a ...
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0answers
20 views

Escape time probability distribution

I have a system where a random walker is moving on $\mathbb{Z}$. However, at each point in $\mathbb{Z}$, there is a probability $q$ that an escape route exists along which the walker can escape. I ...
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19 views

Why does the unordered arrivals in Poisson process iid uniform when conditioned by $N(t)=k$?

Let $N(t)$ be the number of arrivals in the Poisson process of rate $\lambda$. I already know that the 'ordered' arrival times are uniformly distributed on the region $0<t_1<t_2<\cdots<...
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1answer
18 views

Product of distribution of independent random variables

Suppose that $X_1,\ldots,X_n$ are independent random variables defined in some probability space $(\Omega,\mathcal F,P)$. Suppose that $f:\mathbb{R}^n\to\mathbb R$ is Borel measurable. I think that ...
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1answer
31 views

Exponential distribution MLE with lifetime and frequency table

\begin{array}{c|c} \hline \text{Lifetime (months)} & \text{Observed frequency} \\ \hline 0-2 & 50 \\ \hline 2-4 & 35 \\ \hline 4-6 & 25 \\ \hline 6-8 & 15 \\ \hline 8-10 & 5 ...
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Distribution for a sequence of dependant variables

Let’s say I have a sequence of numbers: (1, 1.5, 2, 2.5, 4, 6) Which is sampled from some distribution. Let’s say that the numbers are not independent; I know a priori that the interval between the ...
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10 views

Tail difference of quantiles of (symmetric) distribution functions

Assume, for example, $z_\alpha$ are $\Phi^{-1}(\alpha)$ quantiles from standard normal distribution, $\alpha > 0$. If we are interested in the sum$$z_\alpha + z_{1 - \alpha}$$ for standard normal ...
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7 views

Distribution of Singular Values of Subunitary Matrix

Let $U$ be a random $n \times n$ unitary matrix (w.r.t. the Haar measure) and let $M$ be a $k \times l$ submatrix. What is the distribution of the singular values of $M$?
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Map a random variable to a Gaussian

If I have a random variable $X \in \mathbb{R}^n$, under which conditions is there a $C^1$ function $\varphi: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $\varphi(X) \sim \mathcal{N}(0, I_n)$ (...
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Introduction of shape parameters in the formulation of probability distribution

I'm familiar with the definition of location, scale, and shape parameters, and the type of distributions they parametrized. I'm interested in understanding how shape parameters became part of the ...
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1answer
15 views

How to create a covariance between two distributions?

I have two distributions $A$ and $B$ that are i.i.d. I want to create two distributions $A'$ and $B'$, that have the "same distribution" as $A$ and $B$ (meaning the same probability distribution ...
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1answer
21 views

Approximate Probability With Beta Distribution

We are given a sample of 7 proportions for the percentage of cloud cover recorded at a set time every day for a week, where the sample mean is $\bar{x}=0.51$ and the sample variance is $s=0.3277$. We ...
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1answer
34 views

How do I prove that the SBAF activation function is not a probability density function?

The SBAF activation function is as follows - Note : 0<=x<=1 $$ f(x) = \frac{1}{1+ kx^a(1-x)^{1-a}} $$ Where k and a are constants. I know we have to show that integral $\int_{-\infty}^{\infty} f(...
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1answer
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Estimator of peakedness for a discrete random variable [on hold]

If anyone has come across any statistical measure that could measure how peaked a discrete random variable is peaked as compared to say a normal random variable or atleast a fair estimator that would ...
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2answers
21 views

Double gaussian integral with variable limits of integration.

$$\frac{1}{2\pi}\int\limits_{0}^{\infty}\int\limits_{-x}^{\infty}e^{-\frac{(x^2+y^2)}{2}}dydx$$ Is there a particularly nice way of working this to an exact value? The -x on the limits of ...
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1answer
22 views

Operational Meaning of Relative Entropy

Is there an operational meaning to understand the non-negativity of relative entropy between two probability distributions? I understand the mathematical argument/proof. But I want to know if there is ...
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0answers
35 views

Using Excel to Solve a Statistics and Probability Delivery Driver Problem [on hold]

Frankie has heard his delivery drivers complain that they don’t have enough time to complete their scheduled deliveries, and he observed that it has been a while since he hired a new driver. The ...
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22 views

Sum of two dependent random variables with copula

I'm trying to calculate the sum of two random variables by using Copula Theory in R or Matlab. However, I have very limited knowledge about probability. Actually I read a lot of theoretical ...
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How to prove that two random variables are independent

If $X$ and $Y$ are independent exponential random variables, find the joint density of the polar coordinates $R$ and $\Theta$ of the point $(X, Y )$ on 2-dim plane. Are $R$ and $\Theta$ independent? ...
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Maximize expected utility of reward [on hold]

This task is given from DeGroot Optimal Statistical decisions book.
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8 views

Entropy of a race and information of a message [duplicate]

Having 17 people running a race, the 1st runner have 3/4 probability of winning and all the others have (each) 1/64. The entropy of the marathon is 1.811bits.Knowing that the 1st runner didn't won ...
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1answer
24 views

What is the distribution of a nested Laplace?

We have three random variables $X_1, X_2, X_3$ having the following conditional distributions: \begin{align} p(X_2 \mid X_1=x_1) &\sim \mathrm{Laplace}(X_2;x_1, b) = \frac{1}{2b} \exp(-\frac{|X_2 ...
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0answers
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I assumed X follows bin(20,1/20) here and proceesed as required.

There are twenty individuals numbered $1, 2, \dots , 20$. Each individual chooses $10$ others from this group in a random fashion, independently of the choices of the others, and makes one phone call ...
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16 views

Conditional distribution two variables

I have to find conditional distribution $X_2\mid X_1+X_2$. I found marginal distribution for X1: if X1=0: 0.4, if X1=2: 0,6 and for X2: if X2=-1: 0.4, if X2=0: 0.4 and if X2=1: 0.2
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Determine the limit distribution of $S_N = X_1 + X_2 + · · · + X_N$

Let $X_1, X_2, . . .$ be independent, $L(a)$-distributed random variables, and let $N \in Po(m)$ be independent of $X_1, X_2, . . . . $ Determine the limit distribution of $S_N = X_1 + X_2 + · ...
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0answers
28 views

Expected max number of tries needed to get heads in n samples [duplicate]

Let's say I'm tossing a coin until I get the first heads. I repeat that experiment n times. Then I'll take the maximum value of tries that were needed to get heads in that experiment (out of the n ...
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1answer
37 views

Finding limiting distribution.

Consider the transition matrix $ P = \begin{bmatrix} 1-p&p\\ q&1-q \end{bmatrix} $ for general $2$-state Markov Chain $(0 \le p, q\le 1)$. (a) Find the limiting ...
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1answer
20 views

Geometric distribution of k elements [closed]

Count a distribution for the sum $k$ independent elements of geometric distribution.