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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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A random variable formed by two Normal random variables, under a conditioning process.

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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1answer
12 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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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
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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
29 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: $$L(\theta|x) = \prod_{n}^{N}{f(x_n|\theta)}$$ or log-...
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0answers
11 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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Joint distribution of absolute difference and sum of two independent exponential distributions [on hold]

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 $V = X + Y$?
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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)

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 $\varepsilon>0$ ...
2
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1answer
35 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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0answers
16 views

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
22 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
15 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 [on hold]

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 ...
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0answers
27 views

Probabilities of choosing a red ball:Case 1) 10 distinct balls,8 red and 2 black Case 2) 10balls ,8 identical red and 2 identical black

I know the probability in both cases evaluate out to be 8/10. I want to know the intuition behind solving the probability of choosing a red ball when there are 8 identical red balls and 2 identical ...
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1answer
32 views

Problem to calculate a marginal function in probability

I have a problem in probability. I have $f(x, y) = \frac 14 \cos(y) $, if $x$ is between $0$ and $\pi$, and if$ y$ is between $-\frac x2$ and $\frac x2$. I have to calculate $f(y)$. I calculated ...
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0answers
12 views

Probability of having picked every item from the set at least once after n turns, while picking 3 per turn

Let's say I have a set of 100 items. Each turn, I pick three items at random, note which ones I've picked, and put them back. What is the probability I've picked every item at least once after $n$ ...
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0answers
24 views

How to calculate Poisson probability with float $k$?

Ok, so I see the equation is $\frac{λ^k * e^{-λ}}{k!}$ but what if my $k$ is not an integer but a float possibility? for example, $k = 2.5$ and $λ = 5$. I have tried to multiply $k$ and $l$ with the ...
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0answers
10 views

Convergence of KL-divergence along a convergent sequence of measures

My question is about Lemma 12 and 13 (page 6) of of this paper https://arxiv.org/abs/1802.09583. The Lemma 13 in particular proves, ``Let $\log(g)$ be bounded. If $P_n \rightarrow P$, then $KL((P_n)_g ...
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0answers
22 views

When does $E[f(X_i)]=E[f(X_j)], i\neq j$?

Suppose we have random variables $X_1, \dots, X_N$, with joint probability distribution $F_{X_1,\dots,X_N}$. Under what conditions does the following equality holds? $$E[f(X_i)]=E[f(X_j)],\ \ i\neq ...
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1answer
23 views

Does this discrete probability distribution have a name?

I was wondering whether the probability distribution $$P(X = k) = \frac{\lambda}{(1+\lambda)^{k+1}}, \quad k= 0, 1, 2, \dotsc,$$ where $\lambda$ is a fixed positive number, has a name.
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1answer
24 views

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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0answers
17 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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2answers
40 views

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 ...
1
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1answer
18 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
23 views

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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0answers
19 views

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
21 views

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
19 views

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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1answer
24 views

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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0answers
28 views

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
27 views

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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0answers
12 views

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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0answers
33 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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0answers
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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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0answers
21 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
38 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
28 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
22 views

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
15 views

Minimum of an unknown distribution [on hold]

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
54 views

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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0answers
10 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 ...
1
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0answers
22 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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0answers
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<...