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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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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
15 views

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
11 views

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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0answers
7 views

central region of Normal distribution and Exp distribution

This is the question in my homework i do not know how to solve it enter image description here
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12 views

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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0answers
10 views

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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8 views

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
19 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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1answer
28 views

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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0answers
17 views

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
25 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
30 views

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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0answers
19 views

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
23 views

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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2answers
45 views

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
45 views

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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0answers
31 views

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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0answers
44 views

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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14 views

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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1answer
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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
45 views

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
30 views

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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0answers
32 views

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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2answers
32 views

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
40 views

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
46 views

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
12 views

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 ...
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1answer
18 views

given exponential Cumulative distribution function, finding another Cumulative distribution function with functionl connection

There is given $X$ a random variable with exponential cumulative distribution such that $X~Exp(1)$ so the exponential Cumulative distribution function is: $P(x\le t)= F_x(t)=(1-e^{-t} , 0\le t) \...
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1answer
66 views

Cutting a rope randomly and taking the longer piece, cutting the longer piece and taking the shorter piece.

Cut a rope with unit length into two pieces randomly. Cut the longer piece of the first cut into two randomly again. Take the shorter piece from that second cut. What would be the PDF and the expected ...
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0answers
23 views

A random variable formed by two Normal random variables, under a conditioning process. [on hold]

Imagine two independent random variables, $X$ $\sim$ $N$ $(\mu_1$,$\sigma_1^2)$ and $Y$ $\sim$ $N$ $(\mu_2$,$\sigma_2^2)$. Now imagine a process whereby one observation of $X$ and one observation of $...
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27 views

Change of variable to calculate expected value

$X$ and $W$ are independent random variables. $$ Z=X+W $$ $$ W \sim \mathcal{N}(0,\sigma) $$ $$ E[X]=\bar{x} $$ I want to calculate $E[Z]$ with respect to the joint pdf $p(z,x)$ $$ E[Z]=\int\int (x+w)...
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1answer
7 views

Existence of a sequence of independent $E$-valued random variables with distribution $\mu$ given $\mu$ and $E$ Polish

I know that the following question is true for $E=\mathbb{R}$. I would like to know if it can be extended to Polish spaces. Suppose that $(E,d)$ is a Polish space. Write $\mathcal{B}(E)$ for the ...
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1answer
19 views

Expectation of sum is less than the second moment

Given $E[f^2(X)] < \infty$ and $X_i \sim_{iid} X$, need to show $$ E\left[ \frac{1}{n} \left( \sum_{i=1}^{n} (f(X_i) - E[f(X)] \right)^2 \right] \leq E[f^2(X)]. $$ My try: $$E\left[ \frac{1}{n} \...
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0answers
46 views

Does MLE really care about PDF?

I am wondering, whether MLE really cares whether it operates on proper distributions. Lets take a look at the following situation: likelihood: $$L(\theta \mid x) = \prod_{n}^{N}{f(x_n \mid \theta)}$$ ...
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0answers
19 views

Independence of sample mean and sample variance

It is well known that under normality assumption, the sample mean and sample variance are independent, by Basu's Theorem. My question is that, is the normal distribution the only distribution whose ...
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1answer
52 views

Joint distribution of absolute difference and sum of two independent exponential distributions

If $X\sim \rm{Exp}(1)$ and $Y\sim \rm{Exp}(1)$ are two independent random variables. What is the joint distribution of $U = |X - Y|$ and $V = X + Y$? I used the Jacobian transformation to obtain ...
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1answer
53 views

What is the probability to be after $n$ random jumps of unit length in space within a distance of radius $r$ from the start?

Assume a particle, at instant 0 at the origin of three dimensional euclidean space jumps at each tick of the clock exactly one unit from its current position into a random direction. By this we ...
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2answers
22 views

Expectation of sum of geometric random variables vs. expectation of Pascal r.v.

Let $\{X_i\}$ be a Bernoulli process, i.e. $X_1, X_2, X_3, \dots$ are i.i.d. Bernoulli variables with parameter $p$. Let $T_k$ be the time at which the $k$th success occurs. I can reason about the ...
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Convergence of Random Variable (convergence in probability)

Let $(x_{n} )$be a sequence of real random variable defined on probability space ,converge in Probability to $x$ . Let $y$ be a random real variable on probability space. For $\varepsilon>0$ ...
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Convergence of random variable ! (Probability) [on hold]

Let $(x_{n} )$be a sequence of real random variable defined on probability space ,converge in Probability to $x$ . Let $y$ be a random real variable on probability space. For $\varepsilon>0$ ...
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1answer
79 views

MLE of simultaneous exponential distributions

Given the $X_i\sim \text{exp}({\theta})$ and $Y_i\sim \text{exp}(\frac{1}{\theta})$, where $X_i$ and $Y_i$ are indpendent, with the same $\theta>0$. I have to find the MLE and its distribution. I ...
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1answer
62 views
+50

Inference regarding the mean lifetime of a bulb using a new technique

The lifetime in hours of each bulb manufactured by a particular company follows an independent exponential distribution with mean $\lambda$. We need to test the null hypothesis $H_0: \lambda=1000$ ...
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3answers
26 views

Probability generating function of exponential distribution

The exponential distribution is given by: $$PDF: \lambda e^{\lambda x}$$ And the formula for probability generating function is given by: $$G(z) = \sum_{x=0}^\infty p(x)z^x$$ where $p(x)$ is a ...
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0answers
9 views

Wasserstein distance between centered Gaussian mixtures

We use $\mathcal{W}_2(\cdot, \cdot)$ to denote the quadratic Wasserstien distance as defined here. Now, let $X,Y = \mathcal{N}(0,1)$ be two standard normal random variables and for $ a \in[0,1]$ let $...
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2answers
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How to determine Y(n)

A random variable $x$ from the set $\{1, 2, ... ,n\}. $ Let $x$ has distribution function $f(k) = Y(n) · g^k$ where $g$ is a fixed number within $0$ and $1$. Find $Y(n)$ which is a constant term in ...
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0answers
17 views

Lexicographical order in context of identifiability of mixture of two Normal distributions

I want to understand a method used in a paper on identifiability of mixture of two Normal distributions. This is Teicher 1963 "Identifiability of finite mixtures", fragment of the proof The author ...
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0answers
14 views

Correct formula to calculate Chi square statistic [closed]

I came across two formulas for calculating Chi square statistic. Method-1: Chi square statistic, X²= [(n -1)*s² ]/σ², where n is the sample size, s denotes standard deviation of the sample and σ is ...