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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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Distribution of column of random orthogonal matrix

Suppose $O \in \mathbb{R}^{n \times r}$ with $r < n$ is a random matrix whose distribution is uniform on the set of $r \times n$ matrices such that $O'O = I_r$. Is is true that the columns of $O$...
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If $X,Y$ are random variables dependent on $Z, W$, but $Z$ depends on $W$, what is the proper way to represent the entire joint distribution?

Suppose that we have the following random variables $Y,X$ which are dependent on $Z, W$. However, $Z$, which is defined on a finite set $Z \in \{z_1, \ldots, z_n\}$ is further dependent on $W$. I ...
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1answer
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$X$, $Y$ independent if and only if $X$, $Y$ uncorrelated.

Suppose that the joint probability density function of $(X, Y)$ is given by $f_{X,Y}(x, y) =[1 - \alpha(l-2x)(l-2y)]I_{(o,1)}(x)I_{(o,1)}(x)$ where -1 < $\alpha$ < 1. Prove or disprove: $X$ and ...
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How to show that $E(\hat{Y}| Y = y) = E(\theta |Y=y )$?

Show that $$E(\hat{Y}| Y = y) = E(\theta |Y=y ),$$ where $\hat{Y}$ is conditionally independent to Y given $\theta.$ Note that $E(\hat{Y}| Y = y) $ is equal to the mean of the predictive ...
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Joint Probability Density Function of two discrete random variables

The problem I am facing can be described like this: I have two matrices, as an example, let's assume two sample velocity data which were obtained from bottom of a lake at an instance t: Stream-wise ...
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2answers
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Computing CDF from an exponential r.v. X and Bernoulli r.v. Z

I need to compute the CDF of Y=ZX and i am struggling to compute this when Y=ZX Information: X is an exponential random variable with parameter 1 Z a Bernouilli random variable taking its values ...
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1answer
28 views

Asymptotic distribution of the MLE of $\theta$ such that $\log X_i$ is distributed as $N(\theta, \theta)$

Let $X_1, . . . , X_n$ be a random sample such that $\log X_i$ is distributed as $N(θ, θ),$ $θ > 0$ is unknown. I've calculated the MLE and I got $$\hat\theta = \frac{-1 + \sqrt{1 + 4n^{-1} \sum_{...
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1answer
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KL-divergence of two distributions using probability measure

I am reading an article stated that: Given $\mu$ is a probability measure, a measurable set $A$, and $\hat{\mu}(\cdot) = \mu(\cdot \bigcap A)$, then $D_{KL}(\hat{\mu}||\mu) = - \log \mu(A)$. How do ...
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Identifying, Describing a Binomial-type Distribution

Fix $p \in (0, 1), N, M \in \mathbf{N}_{\geqslant 1}$. It is well-known that if $X \sim \textbf{Bin} (N, p)$ and $Z \sim \textbf{Bin} (M, p)$, then $Y = X + Z$ is marginally distributed as $Y \sim \...
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Looking for a random variable with a specific probability distribution

I am trying to find a random variable $X$ with (non-degenerate) density function $f(x)$ such that for any nonzero constant $A$ with $P(X=-A)=0$, $$ \mathbb{E}[(A+X)^{-1}X]=0 \text{.} $$ Can anyone ...
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Need book recommendation to read more about this, $\frac{1}{2\sqrt{\pi t}}\int_{0}^{1}e^{{-(x-\omega)^2}\over 4t}d\omega$

I am solving a problem and I am stuck on this integral $$\frac{1}{2\sqrt{\pi t}}\int_{0}^{1}e^{{-(x-\omega)^2}\over 4t}d\omega$$ I have seen this integral many years ago in some probability or ...
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Proving t distribution in small sample size and population variance known case.

I am asking a follow up question to this question. Why prefer the t-score when the sample size is low? I have seen mathematical proofs that in variance unknown case, the t-statistic follows a t-...
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1answer
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CDF of derived distribution $|X-Y|$ when $X$ and $Y$ are exponential random variables

I recently had to solve this same problem, except $X$ and $Y$ were uniform on $[0,1]$. The joint probability distribution was uniform, so I just needed to find the proportion of the area inside the ...
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Where does poisson distribution formula come from?

I know that the the punctual probability function of a random variable X with a Poisson distribution is: $$P(X=k)= e^{-\lambda }\frac{\lambda ^{k}}{k!}$$ Also, I've learned that the formula can be ...
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A Binomial bound for the CDF of the Hypergeometric distribution?

Let $H \sim Hyp(N,K,n) $, where $Hyp$ denotes the hypergeometric distribution, $N$ the number of objects, $K$ the number of "good" objects, and $n$ the number of draws. I am interested in a particular ...
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1answer
43 views

Conditional expectation of the number of phone calls

There are twenty individuals numbered $1,2,...,20$.Each individual chooses 10 others from this group in a random fashion,independently of the choices of the others, and makes one phone call to each of ...
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1answer
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How to show that the Airy function is Lorentzian

The Airy function used to describe the reflected/transmitted intensity of a Fabry-Perot interferometer has the general form: $$\frac{F\sin^{2}\left(\theta\right)}{1+F\sin^{2}\left(\theta\right)},$$ ...
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PMF of the Binomial distribution multiplied by a constant [on hold]

I need to find theoretical PMF for a binomial distribution multiplied by a constant. How do I do that? (I am not a statistician) Thanks!
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Maintaining probability distribution under transform

The context of this question is correct dithering of color $c$ with gamma correction. We can only ever output integers $\lfloor c \rfloor - 1,\lfloor c \rfloor,\lfloor c \rfloor + 1, \dots$, but wish ...
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When can we Central Limit Theorem approximation with good approximation?

I think we an use it when n(no. of trials) is large. But my textbook used this approx. by stating that since the expectation is large, we use the approx. I'm unable to understand this, would ...
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1answer
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joint distributions of mins and maxs

Let $X$ and $Y$ be two independent geometric random variables, both with parameter $p$. Thus, $$ \mbox{for each}\,\,\, k = 1, 2, 3,\ldots;\quad \mathrm{P}\left(X = k\right) = \mathrm{P}\left(Y = k\...
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1answer
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Finding conditional density functions for the following scenarios

I'm trying to find the conditional probability density function for the following scenarios and wondering if they are correct A number x is chosen at random in the interval [0, 1], given that x > 1/...
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central region of Normal distribution and Exp distribution [on hold]

This is the question in my homework i do not know how to solve it enter image description here
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Sketching residual plots

A set of data has been analysed by fitting the model $Y_i = x_i^{'}\beta + \epsilon_i$, where $\epsilon_i$ follows a normal distribution with mean 0 and variance $\sigma^2$. Sketch a residual plot for ...
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1answer
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Entropy of $\max(0, \mathrm{uniform}(-1, 1))$

I'm trying to figure out how to deal with distributions that are mixtures of discrete and continuous. A simple example is max(0, uniform(-1, 1)) -- draw a (real) ...
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CDF for the generalized t distribution?

I found that there is a (symmetric) generalization of the Student-t distribution. This generalized t distribution has two parameters p and q, and allows to model many kind of symmetric distributions, ...
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False-positive in a mixture distribution

This is an example of a photoluminescence intensity signal. Suppose you have a mixture of two normal distributions for the signal data. One normal distribution is due to negative results while the ...
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1answer
22 views

How can I prove that a complex variable does not follow a normal distribution from $R$ and $\Phi$ distributions

I am trying to prove that a complex variable $Z = R.\exp(i.\Phi) = R.\cos(\Phi) + i.R.\sin(\Phi) = X + i.Y$ does not follow a normal distribution when $R\sim \mathcal{N}\left(\mu_R, \sigma_R^{2}\...
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Finding the conditional expectation of independent exponential random variables

Let $X$ and $Y$ be independent exponential random variables with respective rates $\lambda$ and $\mu$. Let $M = \text{min}(X,Y)$. Find (a) $E(MX|M=X)$ (b) $E(MX|M=Y)$ (c) Cov$(X,M)$ (a) I first ...
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Sufficient statistics and natural parameters of exponential family

I am studying some properties of exponential family distributions, i.e., distributions whose pdf/pmf can be written (in its "natural" form) as $$f_X(\mathbf{x}\mid\boldsymbol \theta) = h(\mathbf{x}) \...
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1answer
26 views

Distribution of arrival times of Poisson point processes

Let $(M_{t})_{t\geq 0}$ and $(N_t)_{t\geq0}$ be two independent Poisson point processes with rate $\lambda$ and $\mu$ respectively. Let $\tau$ be the first arrival time for the process $N_{t}$. Find: ...
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Sow that one probability distribution can't be preferred to others.

Let $R= ${ $r_1, r_2, r_3, ...$} be a countable set of rewards, and let $U$ be a utility function on R. Let $P_1, P_2, P_3, ...$ be a sequence of probability distributions on R. For each distribution ...
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1answer
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Probability of at least $n/2$ coin flips

We define p-coin as having a $p$ probability to land on Tails and $1-p$ to land on Heads. $X$ is a random variables that given $n$-flips results, gives the number of tails ($T$) that we got: $$X(...
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Connecting two interpretations of the negative binomial distribution

In my probability course, my professor derived the negative binomial distribution by reasoning about the probability that the time of the $k$-th success, $T_k$, takes some value $n$. If $p$ is ...
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1answer
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Is normal distribution the only class of distributions closed under addition?

When I add two random variables that have a normal distribution, I get again a normal distribution. Is a normal distribution the only one with this property?
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How to sample two values from a random variable X with the lesser to be a random variable Y?

The variable X has pdf $$f(x) = \frac18(6 - x)$$ for $$2 ≤ x ≤ 6$$ A sample of two values of X is taken. Denoting the lesser of the two values by Y, use the cdf of X to write down the cdf of Y. Hence ...
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1answer
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Moment generating function of two Poisson distributions

The time between accidents on the Riverfront Bridge follows a Poisson process with a mean time of 40 days between accidents. The time between accidents on the Overview Bridge follows a Poisson ...
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modeling of random variable

I have 2 random variables $U$, $V$ - uniformly distributed on $[0,1]$. How to model (simulate) random variable with distribution function $f(x) = \frac{1}{2}e^\frac{-x}{2} \mathbf{1}_{[0,\infty)}(x)$
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Random variable defined on the Lebesgue probability space

There is a random variable X defined on the Lebesgue probability space whose cumulative distribution function is F. We can find X(w) knowing that: $X(ω)=\inf\{x∈R:F(x)>ω\}$. 1) how do we prove ...
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Asymptotic distribution of MLE of joint exponential distributions

Given $X_1,...,X_n \sim \text{Exp}(\theta)$ and $Y_1,...,Y_n \sim \text{Exp}(\frac{1}{\theta})$, where $\theta>0$ and have the same $\theta$ in both distributions. $X$ and $Y$ are independent. I ...
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The relation between copula density and survival copula

I would appreciate if someone help me to solve the following problem. If I have copula $C(u_1,\ldots,u_n)$ it is obvious how to obtain copula density. It is $$c(u_1,\ldots,u_n)=\frac{\partial^n C(u_1,\...
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is it reasonable to consider a binomial random variable as a distribution conditional on a Bernoulli random variable?

Let $X$ denote a Bernoulli random variable represent the result of tossing one fair coin, one coin, one time. Let $Y$ denote a binomial random variable represent the result of tossing one fair coin $...
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1answer
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How to show the maximum likelihood of $\theta$?

Let $x$ have a uniform density $f_x(x\mid\theta) \sim U(0,\theta)=\left\{ \begin{array}{ll} \frac{1}{\theta} & 0 \leq x \leq \theta \\ 0 & \text{otherwise} \end{array} ...
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Probability of picking a Green ball? [on hold]

With an infinite number of bags S1, S2, S3, etc. S1 contains 3 yellow balls and 2 green ones. Each of the following bags contains 2 green and 2 yellow balls. We draw balls from S1 and put it in S2, ...
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2answers
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Having a problem on a paradoxical answer

Suppose that we have PN objects (disks for an example) and we have P slots (or boxes),how many ways can we distribute those PN objects on those P slots so that each slot has exactly N object? i saw a ...
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How to graph mathematically a normal distribution? [on hold]

Hello and sorry for the question, it is good that in most of the books I had seen graphics but I do not know what values can be given. The fact is that if the graph on the Cartesian plane wanted to ...
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Poisson Random Variable Question

A radioactive source emits certain particles with a Poisson distribution. The probability of no particle emissions during an hour of observation is $0.4$. What is the probability that the first ...
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1answer
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Find minimal sufficient statistic for truncated exponential distribution

Let $X_1, ..., X_n$ be iid $f(x; \theta, \lambda) = \dfrac{\lambda e^{-\lambda x}}{1-e^{-\lambda \theta}}$ for x $\in [0, \theta]$. I want to find A minimally sufficient statistic for $\theta$ ...
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2answers
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Finding $c$ when $X$ has pdf $f(x) = cx(1 - x)$ for $0<x<1$

Let $X$ be a random variable having density function as $f(x) = cx(1 - x)$, for $0 < x < 1$. Find the value of $c$. After solving this question I got the answer as $6$, whereas the answer ...
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1answer
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For game where first player to N points wins, find the distribution of win probability and total number of points between players

Two players, A and B, play a series of points in a game with player A winning each point with probability p and player B winning each point with probability q = 1 - p. The first player to win N ...