Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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0
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2answers
28 views

Checking whether the given statistic is sufficient

A random sample is drawn from a Bernoulli distribution with $X_i = 1 $ with unknown probability $p$ and zero otherwise. Examine whether the following statistics are sufficient for the parameter $p$ ? ...
2
votes
1answer
75 views

How to prove it has a $\chi^{2}$ distribution

I tried to make $T$ close to $$T_1=\left(\frac{(x_1 - \mu_1)^2}{\sigma_1^2}+\frac{(x_2 - \mu_2)^2}{\sigma_2^2}-2 \frac{\rho}{\sigma_1 \sigma_2} \frac{x_1 - \mu_1}{\sigma_1}\frac{x_2 - ...
0
votes
0answers
22 views

Does there exist an uncertainty (entropy) monotonic pmf combination rule?

Assume that I have two probability mass functions (pmf's): $p:=[p_1, p_2, p_3]$ and $q:=[q_1 ,q_2, q_3]$. Further, I assume that the uncertainty of these pmf's is quantified by the Renyi quadratic ...
0
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0answers
18 views

from probability density function (pdf) to the probability of one event

I use a kernel density estimator for multivariate data (5 to 8 dimensions) on data in a database (numerical computations of a theoretical model, all those points provide a good estimate of a weak ...
1
vote
0answers
59 views

minimum of dependent random variables

can someone give me a hint with following problem. Need to find PDF of $X-\min(Y,Z)$, where only available info is that $X,Y,Z$ are iid. Or, in other words PDF of minimum of two dependent variables: ...
1
vote
0answers
21 views

Maximum entropy distribution given second order marginals

Let $p(x,y,z)$ be a probability distribution over 3 variables (suppose them discrete, but it shouldn't matter). I know that the distribution with maximal entropy which preserves the first order ...
1
vote
2answers
51 views

issue in figuring out how to calculate probability [duplicate]

A fair, 6-sided die is rolled 6 times independently. Assume that the results of the different rolls are independent. Let $(a_1,\ldots,a_6)$ denote a typical outcome, where each $a_i$ belongs to ...
-1
votes
1answer
17 views

How do I find the mean of this problem?

$$ F(x) = \begin{cases} 0,\quad x < 1 \\ \frac{x^2-x}{2}\quad1 \leq x \leq 2 \\ 1, \quad x > 2 \end{cases} $$ How would I find the mean in this problem? Can somebody help me ...
0
votes
1answer
41 views

Probability distribution of number of ordered items in a permutation

I have a simple algorithm to check if a series of numbers is sorted: if the first two numbers are sorted, move to the next two. Else, stop and return false. I want to figure out the average case ...
0
votes
0answers
46 views

How can I find the rejection region of a test so it has significance $α$, if $T(X)$ is a sufficient statistic with a known distribution.

Suppose that $X_1, . . . , X_n$ form a random sample from a density function, $f (x|θ)$, for which $T$ is a sufficient statistic for $θ$. Define $H_0: θ = θ_0$ and $H_A: θ = θ_A$. If the distribution ...
0
votes
1answer
28 views

Probability that the detector lasts longer than 2 years

There is a detector in a satellite orbiting the earth. The lifetime (in years) of the detector is a random number $X$, that has exponential distribution with parameter 1/2. What is the ...
0
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0answers
32 views

Marginal density functions

Given a disc with center at origin and radius one, where $$f(x,y) = 1/\pi$$ $$\sqrt{x^2+y^2} =1$$ the marginal density function of $X$ is $$2/π * \sqrt{1-x^2}$$ Can the marginal density function ...
1
vote
1answer
62 views

Join Probability of three exponential random variables?

I'm trying to derive the probability $Pr[Y<a+bZ,Y<X,Z<X]$, where $X$, $Y$, and $Z$ are independent exponential distributed random variables with parameters $1/\lambda_x$, $1/\lambda_y$, and ...
0
votes
0answers
61 views

What is the distribution of sum of a Gaussian and and 2 r.v. Rayleigh distributed?

Let $Z=X+Y+W$; where $X∼N(0,σ_1^2)$ i.e. a Gaussian random variable and Y and W follow the Rayleigh distribution: $f_w(w)=\frac{w}{σ_2^2} . exp(−\frac{w^2}{2σ_2^2})$, $y\ge0$ What will be the ...
1
vote
1answer
177 views

Expectation and Variance of random walks

Consider random walks of fixed length (e.g. $5$) starting at node $u$ in an undirected and connected graph with $N$ vertices. If a node $k$ has $N_k$ edges, the probability of the walk reaching any of ...
0
votes
2answers
33 views

How to get started on this statistics problem?

Suppose $X$ and $Y$ are random variables that are independent also $X$ and $Y$ are uniformly distributed on the interval $(0, 1)$. If $Z=\max \{ X, Y \}$. Then find the probability that $Z \leq z$ and ...
0
votes
1answer
29 views

Distribution of student $t$ ratio under the wrong mean

Suppose that we have an i.i.d. sample of size $n$: $X_1,\ldots,X_n\sim N(\mu_0,\sigma_0^2)$. Define: $$ ...
1
vote
1answer
90 views

Probability that your return is positive for the week, given its distribution per year

You make an investment. Assume that returns are normally distributed with a mean return of .20 per year and a standard deviation of .10. Suppose you check on your returns once a week. What is the ...
0
votes
1answer
23 views

The probability of winning the lottery is $0.0186$. What is the $P(X<10)$?

For $P(X<10)$, I summed up $(1-p)^{n-1} p$ from $1-9$. $(0.0186)$ $(1-0.0186)^1 . (0.0186)$ $(0.9814)^2 . (0.0186)$ $ \vdots $ $(0.9814)^8 . (0.0186)$ and then added it up. Would that be ...
2
votes
0answers
23 views

Convergence of time-slice measures to convergence in law

Question: Are there some conditions that allow one to go from finite dimensional distribution convergence to convergence in path space? Motivation: Consider a sequence of measures for e.g. solutions ...
0
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0answers
14 views

Question on the distribution of miles run monthly…

I am working on a project related to the habits of people who are jogging/running on a regular basis. I wonder if there are known models based on real life statistics (or else) regarding the ...
0
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0answers
49 views

Regarding distribution of product of matrix and its transpose

Let $\mathbf{H}$ be a matrix of size $m\times n(n>m-1)$ where each column of the matrix indicates an $m-$ dimensional measurement. and there are $n$ measurements in all. Let us assume that ...
0
votes
1answer
18 views

Statistical distribution where location of peak changes depending on variables

I need a distribution that satisfies the following requirements: Area under the graph from -3 to 3 is close to 100% (like the normal distribution) Area under the graph can be made to focused on to ...
0
votes
1answer
54 views

How does one in general decide to use a binomial distribution vs a Poisson distribution?

I saw the following problem and thought Poisson would be the correct distribution, but the solution manual says otherwise. My textbook says an event is rare if the number of trials is at least 50, ...
1
vote
2answers
1k views

How is logistic loss and cross-entropy related?

I found that Kullback-Leibler loss, log-loss or cross-entropy is the same loss function. Is the logistic-loss function used in logistic regression equivalent to the cross-entropy function? If yes, can ...
1
vote
0answers
18 views

product of independent random variables

Let $X \sim U(-10,10)$ and $Y$ have a pdf $f_Y(y) = \dfrac{1}{t} y ^ {(1/t) -1} $, $0\leq y\leq 1$. If $X$ and $Y$ are both independent, what is the pdf of $XY$? Here is my attempt: Let $W=XY$, ...
0
votes
0answers
51 views

How to minimize the Kolmogorov-Smirnov statistic with Power-law distribution

I'm fitting data into a power-law distribution. In order to do this I need to solve two parameters: the scaling parameter $\alpha$ and the lower bound $x_{\min}$ In order for me to solve $\alpha $ I ...
0
votes
0answers
53 views

How to deduce two function of many independent random variables is independent?

Let: $Z_1,\dots , Z_n$ be a random sample from a population with a standard normal distribution. $\bar{Z}_n=\frac{1}{n}\sum\limits_{i=1}^nZ_i$ $S^2_n = ...
1
vote
0answers
32 views

Proving a process is a P Brownian Motion

Let $X_t = tW_{\frac{1}{t}} \forall t>0$ and $X_0 = 0$. I am trying to show that this process is a brownian motion under some measure P. I have shown that it is continuous and that it is ...
2
votes
1answer
54 views

Difference in pdf formula between Dirichlet and Multinomial distributions

The pdf for Dirichlet distribution seems to be $$ Dir(\alpha_1,\alpha_2,\ldots\alpha_k) \text{ is defined as} $$ $$ pdf(θ_1,θ_2,\ldots,θ_k )= \frac{Γ(\alpha_0)}{Γ(∝_1 )Γ(∝_2 )\cdots Γ(∝_k )} ...
2
votes
0answers
63 views

A tough double integral

I am looking at an ugly ratio of two random variables and am interested in the density of the the ratio. So what I did was to write down the joint characteristic function of the numerator and ...
0
votes
1answer
357 views

How to calculate this joint PMF?

Consider three random variables X, Y, and Z, associated with the same experiment. The random variable X is geometric with parameter p∈(0,1). If X is even, then Y and Z are equal to zero. If X is odd, ...
1
vote
1answer
124 views

What's the probability distribution of the difference between two consecutive sorted random numbers generated by a known distribution?

For example, first we generate $n$ random numbers based on an exponential distribution ($f(x)=\lambda e^{-\lambda x}$), then we sort these random numbers in ascending order which are represented by ...
2
votes
2answers
140 views

Bayesian Inference/maximum Likelihood

I am rather struggling with the gist part d) of this question. Why would I wish to compare the MLE with the posterior mean?
0
votes
0answers
44 views

Cauchy distribution

For a given value of the location parameter say it is 0. Why does the median equal the location parameter. I'm slightly confused by what the location parameter actually represents? Any help would be ...
0
votes
1answer
584 views

Probability of a binary communication system

A binary communication system is used to send one of two messages: message A is a sequence of 0s with probability 2/3, message B is a sequence of 1s with probability 2/3, The ith received bit is ...
3
votes
1answer
126 views

Expectation of maximum of two independent random variable with known distribution [closed]

Assume $X$ and $Y$ are two random variables such that $X\sim \textrm{Unif}(0,1)$ and $Y=e^{-t}\times a $ where $t\sim \mathrm{Exp}(\lambda)$ and $a\sim \textrm{Unif}(0,1)$. What is ...
0
votes
1answer
22 views

How to Validate this Statistical Model?

I have a statistical model and I want to test out how well it works. I have N data points, and I want to see if they come from a specific distribution that my model predicts. Each distribution is ...
3
votes
3answers
1k views

Integrating the tail exponent of a Pareto Distribution (difficult integrals).

Looking for two difficult integrals, whichever is solved first: With $\alpha>0,L>0, \mu>0,\sigma>0, a>0, b>0, x>L$ , $$\phi_l (x;\mu,\sigma)=\frac{1}{\sqrt{2 \pi } \sigma ...
0
votes
1answer
57 views

Understanding derivatives in simple terms

Im am trying to understand the idea of derivatives and how they relate to the real world. I understand if i have function, in pkysics first derivative is the velocity, and the second derivative is ...
0
votes
0answers
61 views

Solving infinitesimal operator in stochastic process

I am trying to understand a notion in a paper (p. 4) about identities in stochastic processes. The author uses the following infinitesimal generator of a diffusion $Y_{t}$, $t \geq 0$: $$ ...
0
votes
1answer
60 views

Show that the following function is not a cumulative distribution function

Suppose $F(x,y)=0$ if $x+y<2016$ and $F(x,y)=1$ if $x+y>2016$. Prove that $F$ is not a bivariate cumulative distribution function. My attempt: ...
1
vote
1answer
467 views

How to derive the Dirichlet-multinomial?

Common knowledge The multinomial: $$p(z|\theta) = \frac{n!}{\prod_i z_i!} \prod_i \theta_i^{z_i}$$ And the Dirichlet: $$p(\theta|\alpha) = \frac{1}{B(\alpha)}\prod_i \theta_i^{\alpha_i-1}$$ with ...
2
votes
1answer
91 views

Uniformly distributed points on spherical surface

Let $x=(x_1,\ldots,x_n)$ be uniformly distributed on the $(n-1)$-dimensional spherical surface $S^{n-1}(n^\frac{1}{2})$ of radius $n^\frac{1}{2}$. I'm trying to show that as $n\to\infty$, $x_1$ ...
0
votes
0answers
37 views

Maximum of a function relatred to relative entropy of Gaussian Mixture Distribution

If we have the following functions \begin{equation} g(x)=\sum_{1}^{T}\frac{\epsilon_{i}}{2\pi\sigma_{i}^2}e^{-\frac{|\mathbf{x}|^2}{2\sigma_{i}^2}} \end{equation} \begin{equation} ...
0
votes
0answers
46 views

Chernoff Bounds upper and lower tails

Most of the forms of the chernoff bounds i have seen bound from above either $$P(X>(1+\delta)\mathbb{E}(X))$$ or $$P(X<(1-\delta)\mathbb{E}(X))$$ are there any upper bounds on say ...
2
votes
1answer
72 views

Joint distributions

Let $X, Y$ be continuos random variables of densities $f_X, f_Y$. Let $Z = \begin{pmatrix} X \\ Y \end{pmatrix}$. When is $Z$ continuos? And in this case, how to express its density with respect to ...
0
votes
2answers
44 views

Probability exponential distributuion problem

Electronic store sells online two items, $A_1$, and $A_2$. Times between orders are independent variables with exponential distribution. In one hour there are $15$ orders of item $A_1$ on average, and ...
1
vote
1answer
83 views

the marginal pdf of a transformed variable from a joint distrubution

The questions tells us to let X and Y be random variables for which the joint p.d.f. is as follows: $$f(x,y)= \begin{cases} 2(x+y), & \text{for $0 \le\ y \le\ x \le\ 1$} \\ 0, & ...
0
votes
1answer
27 views

Confidence interval for these 2 questions.

I'm so lost here. I've studied the 2 examples in our notes but they don't seem to apply to these questions. Google is proving useless for me too. How can I find the confidence interval for question 3? ...