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. Use this tag along with the tags (probability), (probability-theory) or (statistics).
27,699
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19
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Does the invariant distribution depend on the initial distribution of a Markov Chain?
I have just learnt about the invariant distribution, and was wondering if such depends on the initial distribution of a Markov chain in anyway, and if so, whether the following is the correct ...
- 2,145
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14
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Can you measure differences in self-similarity?
For background, I have a rudimentary understanding of self-similarity as it applies to observing network traffic as a stochastic process. I also have a robust mathematical background but not in ...
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-3
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1
answer
40
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Let $2k$ be an even integer number and $r+s=2k$. What is the probability that none of $r$ and $s$ is not prime? [closed]
Let that $k$ be an integer number and $S$ defined as:
$$
S=\lbrace (r_i,s_i)| r_i+s_i=2k, 1<r_i\leq k \leq s_i<2k-1 \rbrace
$$
No, Let that choose an element of $(r_j,s_j)\in S$ randomly by ...
- 759
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40
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Given $n$ uniform random variables, probability no two differ by less than $x$
Given $n$ random variables uniformly distributed from $0$ to $1$, what is the probability that no two differ by less than some $x$?
My initial thought was that for $n$ random variables there are $n \...
1
vote
0
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30
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Statistics to detect employee/customer collusion
I'm an analyst at an agency that operates landfills. There was a recent discovery that certain trash haulers colluded with 2 or 3 weigh-masters to undercharge in return for kickbacks. I'm tasked ...
- 11
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0
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16
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P(X=Y) for jointly cumulative probability distribution function
Let (X,Y) be a random discrete vector with probability function given as:
$\mathbb{P}(X=x, Y=y) = \frac{\lambda^ye^{-2\lambda}}{x!(y-x)!}\mathbb{I}_{\{0,...,y\}}(x)\mathbb{I}_{\mathbb{N}}(y), \lambda &...
0
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1
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24
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Given bivariate bernoulli with an integral as a parameter prove that are marginally identically distributed and correlation is positive
So this is a question from a past exam. The joint density function is
\begin{equation}
\begin{aligned}
P\left(X_1=x_1, X_2=x_2\right) & =I_{\{0,1\}}\left(x_1\right) I_{\{0,1\}}\left(x_2\right) \...
- 125
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23
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Convolution of two PDF
I am probably overlooking like, all the important details, but when trying to work out how to take the convolution of two pdfs I am going as follows:
according to https://en.wikipedia.org/wiki/...
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-1
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37
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Question About Derivation of Variance of Kaplan-Meier Estimate of the Survival Function: Why is Degree of Freedom Not Removed?
The variance of a sample proportion $\hat{p}$ is given by
$$ \text{Var}[\hat{p}] = \frac{p(1-p)}{n} $$
where $n$ is the sample size and $p$ is the true probability of success. Now to the best of my ...
2
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1
answer
65
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Fisher Information Matrix for Weibull Distribution...
I wish to find the Fisher Information Matrix for the Weibull Distribution... I face two difficulties,
I can't find any sufficient guide in internet to lead me to derive the Fisher Information Matrix.....
- 159
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0
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17
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Conditional distribution of multivariate normal random vector
To calculate the conditional distribution of multivariate normal random vector $(X,Y)$, since that for all $a \in \mathbb{R}$ the random vector $(X-aY,Y)$ is normally distributed, we can determine $a$ ...
- 13
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0
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30
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Need help calculating the density of $(a,b)X$ where $X$ has density $f$
I’m currently working on a problem where I need to calculate the density of $(a,b)X=(aX,bX)$ where $a,b>0$ and $X$ has density $f$ .
However, I’m facing some difficulties as the Jacobian method, ...
- 85
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2
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55
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What is the value of favorable outcomes in probability distribution
the given question is this:
In the ii) part what should be the probability that the age is a prime number given that the age is greater than 15 years.
I have two contradicting methods of solving this ...
- 123
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0
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32
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Correct definition of random variable and (discrete) probability distribution?
I defined it as follows:
Let us consider a random variable $X$ defined on a countable sample space $\Omega$ over $K$ possible outcomes, and $P(X)$ its discrete PD defined as the set of probabilities ...
- 139
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1
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53
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Find a function such that f(x)/P(X > x) is a constant.
What do you think of the following question?
"Let X be a continuous random variable such that $P(X\le 0) = 0$. Let $f(·)$
be its probability density function. Suppose that $f(x)/P(X > x)$ is ...
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-1
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0
answers
22
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Derivation of gamma function to obtain the PDF from the CDF
if $Z$ is a continuous random variable taking values from zero to infinity having a gamma distribution with shape parameter $A$ and scale parameter $B$ with the following CDF
\begin{equation}
F_{Z}(z) ...
3
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6
answers
63
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$P(X\geq{5})$ in a continuous uniform distribution
I had an exam in which I think I selected the right answer but in the Quiz is wrong.
It is a multiple choice problem that goes as follow:
"The waiting time until a train passes is a uniform ...
- 93
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1
answer
23
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Find the distribution of $X|Y=y$ where X and Y look like a a bivariate poisson.
It isn't a bivariate Poisson precisely, but has a pmf of:
$$
\mathbb{P}(X=x, Y=y)=\frac{\lambda^y e^{-2 \lambda}}{x !(y-x) !} I_{\{0, \ldots, y\}}(x) I_{\mathbb{N}}(y), \lambda>0$$
When I tried to ...
- 125
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55
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Describe a probability distribution for $X$, such that $E[\sin(X)] = E[\cos(X)] = 2/\pi$, with certain probability bounds.
I was given an interesting problem by my mentor at university. I am currently investigating the Markov Inequality, and was recently challenged to find a random variable $X$, taking values in $[0,\pi/2]...
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7/7-t is an example of what MGF is it uniform or geometric? I'm confused. [closed]
𝑀𝑊(𝑡) = 7/7-t ∙ ( 3/4 + 1/4 𝑒^𝑡) 5 I know that ( 3/4 + 1/4 𝑒^𝑡) 5 is a binomial but my problem is what is 7/7-t.
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14
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Under what conditions does an exchangeable extension to a conditional distribution exist?
Informally:
Under what conditions can a conditional distribution (conditional density function/conditional mass function) be extended to a full joint distribution that's exchangeable? Intuitively, let'...
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2
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54
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Number of heads $\geq 15 \space + $ number of tails or number of tails $\geq 25 \space + $ number of heads
Suppose you have a fair coin. What is the expected number of flips that you need until you have at least $15$ more heads than tails OR you have at least $25$ more tails than heads?
Given that the coin ...
- 177
-1
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0
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59
views
Does this probability distribution have a name? (generalization of beta)
I am interested in the following generalization of the beta distribution:
$$f(x; \alpha, \beta) = constant * \prod_{i=1}^k x^{\alpha_i-1}_i (1-x_i)^{\beta_i-1}$$
where the support is the set of ...
-1
votes
0
answers
38
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Let X be a continuous random variable and u a measurable function. Show that u(X) is not necessarily a continuous random variable:
I would like to know if my proof is valid, because I am new to probability theory and not sure if my reasoning is valid for this proof:
Thrm: Let $X:Ω → \mathbb{R}$ be a continuous random variable and ...
- 1
1
vote
0
answers
14
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Is there any way to parametrize log-normal distribution with ranking and $\mu$, $\sigma$?
In a paper, I see a way of Pareto distribution's parametrization:
"
Specifically, I assume that varieties can be indexed from highest to lowest benefit on a continuum from $[0, n]$, with
$$
z_i ...
- 31
1
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1
answer
48
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To what family of densities does $e^{-u}(u^k-k!) \log u$ belong?
Apparently $\int_0^{\infty} e^{-u} (u-1) \log u du = 1$, $\int_0^{\infty} e^{-u} \frac{1}{3}(u^2-2) \log u du = 1$ etc.
Does these densities belong to a known family of densities? The closest I found ...
- 49
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0
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27
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Average probability of sampling all clusters
A system consists of $N$ nodes. The nodes are distributed into $M$ clusters such that each node belongs to a unique cluster. Each cluster $i$ has one unit of weight, $w_i = 1$. Thus the initial weight ...
- 113
2
votes
1
answer
29
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Lebesgue integral to get cumulative distribution function of random variable
In the following, I tried to get cdf of random variable $X$, which has an $\operatorname{Exp}(1)$ distribution, by lebesgue integral.
Probability space $(\Omega, \mathcal{F}, \mathbb{P})$: $\mathbb{P}...
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1
answer
61
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Another formula for Expectation
I came across the following question:
If $ X $ is a continuous random variable $ (-\infty<X<\infty) $ having distribution function $ F(x) $, show that $$ E(X)=\int_{0}^{\infty}[1-F(x)-F(-x)]\,\...
- 187
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0
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78
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Can I think of probabilities as proportions instead?
I am new to probability theory, so bare with me if I do not nail all of the terminology (I will still try my best)! Also, I gave a short "What is my question" sentence, but I invite you to ...
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Find standard normal random variable [closed]
For the standard normal random variable z find z for The area between 0 and z is 0.4750 find me the value of Z of this example.
2
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1
answer
47
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Trouble understanding order statistics
Order statistics were introduced in my text as follows:
I am trying to understand what this means.
$X_1 , \dots , X_n$ is a random sample, i.e. an independent and identically distributed sequence of ...
- 271
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votes
1
answer
43
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Find variance and expectation with given probabilities of variable values
I've got random variable $x$ and probabilities: $P(x=0)=1/4, P(x\in[a,b])=\cfrac{b-a}{2}, P(x=3)=1/4$ where $1\leq a < b \leq 2$.
What I did:
First I tried to calculate the mathematical expectation ...
- 3
-1
votes
1
answer
99
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Laplace distribution CLT [closed]
I'm really stuck with this problem that my statistics professor gave me:
n = 81 measurements X = (X1,…,X81) were made according to a Laplace(θ),
$f_{\theta}(x) = \frac{1}{2} \theta e^{-\theta|x|} $.
...
0
votes
1
answer
32
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Finding a probability given a mgf
A viral meme starts in one account, and is re-shared by M other
accounts, where M is a non-negative discrete random variable. Assume that the future
behaviour following from each of the initial re-...
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2
votes
1
answer
62
views
Proof of weak convergence when domain of rv is N
I need to show:
If $X_i, i\geq1, X$ are random variables with domain $\mathbb{N}$, then $ X_n \rightarrow X$ weakly iff $\forall i \in \mathbb{N}: P(X_n = i) \rightarrow P(X=i)$.
The direction $\...
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48
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Prove that X is not always random variable in case that X^2 is random variable [closed]
Could someone give me an idea how to prove it?
- 636
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40
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Given the L2 norm of a Gaussian matrix, what distribution does the Gaussian matrix follow?
Given a random gaussian matrix X with zero mean matrix and covariance matrix Σ, and two deterministic matrices A and B. If I know the value of $||{\bf{AX}}||_F^2$, how could I get the pdf of $||{\bf{...
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1
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53
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Find distribution of $\sum_{i=1}^{n}\frac{X_{i}^{2}}{\sigma^{2}}$
Let $X_{1},...,X_{n}\sim f(x)=Kx^{2}e^{\frac{−x^{2}}{2σ^{2}}}$ i.i.d. I need find the distribution of $\sum_{i=1}^{n}\frac{X_{i}^{2}}{\sigma^{2}}$. To do this, calculate the normalizing constant, K, ...
1
vote
1
answer
29
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Conceptual doubt about the use of normal distributions
Let's say we have a set of scores from an exam that can range from 0-10. Given N=30 scores we can compute the mean ($\bar{x}=4.8$) and the standard deviation ($\sigma_{x}=2.0$). If we use the normal ...
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0
votes
1
answer
45
views
Expected revenue for overbooked hotel.
I am trying to solve the following problem:
A hotel has 120 rooms that can be booked for 60€ a night. Typically, 3% of the guests do not show
up to their booking. In this case, the hotel still gets ...
- 55
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0
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16
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Skew-Normal Distribution when one variate is not standard normal
I am studying about the skew-normal distribution and stumbled this work, where a skew-normal RV has the form [Eq. (3)]:
$X = \delta |Z_1| + \sqrt{1-\delta^2} Z_2$,
where $Z_1$ and $Z_2$ are ...
- 113
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0
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67
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Is there an analytical solution here?
Can we solve analytically for $b$ in the equation below?
$$A = \int_0^b (b - x) f_{X}(x) \mathop{d x},$$
where $A \geq 0$, $f_{X}$ is the probability density function of $X$ and
$$X \sim \operatorname{...
- 147
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votes
0
answers
63
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pdf transformation of periodic scalar random variables related by a derivative
Two random scalar variables, x and y, are related by the following expression: $y(t) = f[x(t)] = a\frac{dx(t)}{dt}$, where $x(t)$ is periodic and $a$ is a constant.
How can I calculate the pdf $g_{Y}[...
- 11
0
votes
0
answers
13
views
No joint distribution for set of random variables
What is a system described by a set of random variables for which there are distributions over subsets of these variables which are not marginal of a distribution over all random variables at once.
...
0
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1
answer
20
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Single-crossing property for stochastic dominance
A useful criterion for first order stochastic dominance of two random variables $X$ and $Y$, denoted $X\le Y$, is to check whether the densities (or pmf if discrete) cross at most at one point. ...
- 878
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0
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47
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Variation of St. Petersburg Paradox
I was discussing the the St. Petersburg paradox and the following question came up:
Suppose the game doesn't end within nine rounds, then the player directly receives $2^{10}$ dollars , while ...
- 335
2
votes
1
answer
31
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Distribution of an estimator
Let $X_1,\ldots,X_n$ be indenpendent identically distributed random variables with density $$f_X(x)=\theta(1+x)^{-1-\theta}$$ for some $\theta>0$ and $x>0$. I would like to know how to compute ...
- 551
1
vote
0
answers
50
views
Finding marginals of the uniform distribution on a triangle
The statement is:
Random vector $(X, Y)$ is uniformly distributed on a triangle $A = (0, 0)$, $B = (2, 0)$, $C = (1, 1)$. Find distribution and density functions of $X, Y$. Check if $X, Y$ are ...
- 11
0
votes
1
answer
43
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Negative binomial distribution with non integer parameter
I am solving an actuarial math problem.
Suppose that the number of claims, $N$, in a year follows a Poisson distribution $\mathrm{Po}(\Theta)$, where $\Theta$ is a gamma distribution $\Gamma(5, 1/2)$, ...
- 3