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

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0
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0answers
11 views

Using the symmetry assumption in this familiar probability problem

I'm revising some probability and have run into this old problem (context: Monte Carlo tests): Suppose there are random variables $t_0,t_1,\ldots,t_B$ that are independently and identically ...
0
votes
2answers
34 views

Double exponential function

Suppose $X$ has an exponential distribution with parameter $1$ and $Y=ln(X)$. The distribution of $X$ will be $f(x)=e^{-x}$. Now I want to find the distribution of $Y$. So say: ...
16
votes
3answers
7k views

Expectation of the min of two independent random variables?

How do you compute the minimum of two independent random variables in the general case ? In the particular case there would be two uniforms variables with difference support, how should one proceed ? ...
4
votes
1answer
281 views

Probability distribution of a coordinate of the random point on a hypersphere with given radius

If $(x_1,x_2,...,x_N)$ is a uniformly randomly chosen point on a hypersphere of a dimension $N$ with the radius $R$ (center in origin). What is the probability distribution of any coordinate? Done so ...
1
vote
1answer
23 views

Distribution function?

Let $F(x) = e^{-1/x}$ for $x>0$ and $F(x)=0$ for $x\leq0$. Now I am investigating if $F$ is a distribution function. Say: \begin{align} \int\limits_0^\infty e^{-1/x} \, dx = \left[ ...
3
votes
1answer
26 views

Distance between a Poisson and Normal distribution.

Let $X_a$ be a random variable Poisson distributed with intensity $a$. That is $$\mathbb{P}(X_a=k)= e^{-a} a^k / (k!)$$ for any $k\in \mathbb{N}$. Let $$Y_a=(X-a)/\sqrt{a}$$ the normalization of ...
0
votes
0answers
5 views

limit distribution min random vector

Let $A_n$ be a matrix whose dimension increases as $n$ increases. We also have a mean zero stochastic vector $X_n$ which consists of independent blocks of which the terms are dependent. So ...
0
votes
0answers
21 views

What is $dx/dF(x)$ where $F(.)$ is a continuous, increasing function.

I was wondering if it is possible to find $dx/dF(x)$, that is, the derivative of $x$ with respect to $F(x)$, which is an increasing, continuous function. Does it involve finding the derivative of the ...
1
vote
0answers
20 views

Compute (a) P(X = 2|Y = 3) and (b) P(Y = 3|X = 3) for the following joint distribution

I have some trouble understanding a question from my testbook. The question is as follows: Compute (a) P(X = 2|Y = 3) and (b) P(Y = 3|X = 3) for the following joint distribution: Y X=1 2 3 1 ...
0
votes
0answers
26 views

Deriving a difficult PDF

A traveller is lost in the Wild West and he is standing at a random point in a square of side length 1 mile. Suddenly a herd of buffaloes charging in a straight line at a constant speed of ...
0
votes
0answers
4 views

Which argument in KL Divergence minimization?

The KL divergence $D_{KL}(p||q) = p^T\ln(\frac{p}{q})$ is not a distance measure because first of all it is not symmetric. In applications, one usually has a prior distribution, say $y$, and wants ...
2
votes
0answers
17 views

Distribution of $\langle A,x\rangle\langle A,y\rangle + \langle B,x\rangle\langle B,y\rangle$ given $\langle x, y\rangle$

Let $A$ and $B$ be independent, normal distributed $N(0,1)$ normalized unit vectors, and let $x$ and $y$ be unit vectors with given inner product $\langle x, y\rangle=u$. Can we write the ...
0
votes
1answer
11 views

Continuum random variable distribution: integral and trapezium-rule methods give different result

Suppose p.d.f. is $\frac{1}{63}x^{2}$. Find the $P(4<x<5)$. I've tried with integration method and trapezium-rule, but they give me different result. With integration, ...
0
votes
0answers
11 views

Cost function for very sparse, real-valued data

Suppose the target output of my data prediction model is an MxN matrix where 95% of the values are 0.0 and the other values are anywhere between 0.0 and 1.0, what would be a good loss function to use ...
0
votes
1answer
18 views

Combinatorics-Summation doubt in the proof of the expectation of the Hypergeometric distribution.

The proof starts considering this equality: $(d/dx (1+x)^A)(1+x)^B = A(1+x)^{A+B-1}$ Then it keep on changing every $(1+x)^{A or B}$ for its binomial coefficient. That 's what I don't understand. If ...
0
votes
1answer
31 views

Probability and Transformation

X~unif(0,1) {Y | X = x} ~ unif (x,1) fx(x) = 1 ( 0 < x < 1) fy(y|x) = 1/(1-x) (0 < x < y < 1) f(x,y) = fx(x)fy(y|x) = 1 / (1-x) (0 < x < y < 1) fy(y) = - log (1-y) ( 0 ...
-2
votes
0answers
14 views

A Difficult Probability Question [on hold]

Jim has 30 blue gum balls, 40 red gum balls and 20 yellow gum balls. What is the probability of Jim picking a yellow gum ball if he picks a gum ball 15 times from a bag?
-2
votes
0answers
23 views

How to code Kolmogorov test in R [on hold]

I have a little problem writing in R language.If there is anyone to know how to code the next problem in R i'll be pleasant Let $H_0:F=N(0,1)$ Get $10000 $ samples of size $n = 30$ from the ...
2
votes
2answers
49 views

Finding variance

Given a sapmle $(X_1, X_2 , \ldots , X_n)$ from normal distribution with parameters $(a , \sigma ^ 2)$, find $$ \operatorname{Var}\left( \frac{1}{n} \sum\limits_{n=1}^n(X_i - \overline X)^2\right)$$ ...
0
votes
0answers
17 views

Density of probability measure including transition probabilities

If $P_1$ is probability measure on $(\mathbb R, \mathcal B^1)$ with pdf $f_1$, $P_{1, \ldots, k-1}^k$ transition probability from $\mathbb R^{k-1}$ to $\mathbb R$ with pdf $f_{1, \ldots, k-1}^k ...
-1
votes
1answer
28 views

Probability Distribution Function and Expectation in Practice [on hold]

There is 1m bar. It is divided randomly, long part is X, short part is 1-X. long part X is divided randomly again, long part is V, short part is Y. (so, V+Y=X) let W is the shortest bar among 1-X, ...
1
vote
1answer
15 views

Find distribution mean from the mean and sd of the log

I have a distribution with a long tail and use a model to predict the mean and standard deviation of its log. Given the mean and standard deviation of the log, how do I find the mean of the actual ...
0
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0answers
9 views

How to compute the normalizing constant $B(\alpha)$ of the Dirichlet distribution for high value of $\alpha$?

The normalization constant of the Dirichlet distribution has the following form: $B(\alpha) = \frac{\prod_{i=1}^{K}\Gamma \left ( \alpha_i \right )}{\Gamma \left ( \sum_{i=1}^{K} \alpha_i \right )}$ ...
-2
votes
0answers
28 views

Probability Distribution Function and Expectation [on hold]

Question is (X,Y)~f(x,y)=2x (0<=x<=1, |y| < x^2) W=X+Y 1) find probability distribution function, pdf fw(w). 2) E(W) using fw(w). How do I solve the problem ??
0
votes
2answers
19 views

Logarithm of Gaussian function is whether convex or nonconvex?

I have a gaussian distribution such as $$P(x)=\frac {1}{\sqrt {2\pi}\sigma}e^{-\frac {(x-\mu)^2}{2\sigma^2}} $$ As my knowledge, $P(x)$ is non convex function interm of $x$. However, if I map it to ...
2
votes
1answer
46 views

Probability calculation involving order statistics

There are $n+1$ independent and identically distributed random variables with the same distribution as $D \sim \text{Exp}(\mu)$, denoted by $D, D_1, D_2, \ldots, D_n$. Define event $E_1$ as "$D$ is ...
1
vote
2answers
28 views

Link between exponential distribution and poisson probability mass function

Customers arrive randomly and independently at a service window, and the time between arrivals has an exponential distribution with a mean of 12 minutes. Let X equal the number of arrivals per hour. ...
3
votes
1answer
410 views

The PMF of the larger of two numbers selected at random from $1,\dots,12$

Two balls are chosen at random from a box containing 12 balls, numbered 1;2; : : : ;12. Let X be the larger of the two numbers obtained. Compute the PMF of X, if the sampling is done (a) ...
-2
votes
1answer
40 views

find distribution function and pdf [on hold]

when X~uniform (0,1) Definition of Y is as following Y=e^X (0<=X<0.5) =logX (0.5<=X<=1) 1) find Definition function of Y, F(y) and pdf(probability distribution function) f(y) ...
0
votes
1answer
16 views

distribution of a linear combination of $n$ independent exponential random variables [on hold]

I want to know what's the distribution of a linear combination of $n$ independent exponential RV, when the coefficients are distinct and positive. for example, I want to get the distribution of U ...
0
votes
0answers
28 views
+50

Bayesian statistics and Basis for continous functions

I was thinking about Bayesian statistics, and one thought bothered me: In Bayesian statistics, we assume that the pdf $p(x)$ can be described as: $p(x)=\int f(x|\theta)g(\theta)d\theta$ usually ...
1
vote
1answer
27 views

Existence proof for two random variables

I have two probability measures $\nu_1,\nu_2$ on a measurable set $(E,\Sigma)$ and a probability measure $\mu$ on $(E \times E, \sigma(\Sigma \times \Sigma))$ with $$ \nu_1(A) = \mu(A \times E) ...
0
votes
1answer
42 views

Why do characteristic functions use $e^{ix}$ and not $e^{-ix}$? Does it matter?

I've heard the characteristic function be described as the Fourier-Stieltjes Transform of the distribution measure of a r.v., but I was curious as to why it's written as $E[e^{ix}]$ and not the ...
1
vote
1answer
23 views

Normally distributed variable with normally distributed mean.

What the idea behind the prove of following statement? I am pretty sure the statement it is correct. If $X \sim N(\text{mean}_x, \text{var}_x)$ and $Y \sim N(\text{mean}_y+X, \text{var}_y)$, then $Y ...
-1
votes
1answer
25 views

Trinomial Distribution - Cumulative Probability of (X-Y) [on hold]

In an election there are n voters. They can each vote for Candidate A (with probability p); Candidate B (with probability q) or neither (with probability (1-p-q) ). What is the Probability that ...
0
votes
1answer
20 views

Question on independent events in probability

The "on" temperature of thermostatically controlled switch for an air conditioning system is set at $60$ degrees, but the actual temperature X at which the switch turns on is a random variable having ...
1
vote
1answer
52 views

Strange sum of random variables

So guys, I'm having this hard proof to solve in probability. I don't really know how to tackle it! Hope that someone can help. Let $\{Z_i\}_{i\in\mathbb{Z}}$ be i.i.d. random variables with zero mean ...
0
votes
0answers
7 views

Dividing weight (increasing domain) while preserving log-concavity

I have a sequence of $n$ values $p_1,\dots,p_n \in [0,1]$ satisfying (a) $\sum_{j=1}^n p_j = 1$ and (b) $p_j^2 \geq p_{j-1}p_{j+1}$ for $1 < j < n$. I have no control on the values of the ...
0
votes
1answer
394 views

Probability generating function for logarithmic series distribution, support $k\geq1$

I'm trying to derive the probability generating function (pgf) for the logarithmic series distribution, and not getting the expected form $\frac{\log{(1-qs)}}{\log{(1-q)}}$. It seems that pgfs are ...
1
vote
0answers
34 views

PDF of product of two dependent r.v [on hold]

I have a random variable $q$ and another as the $\min(X,q)$. $q$ and $X$ are continuously distributed r.v. I have the pdfs for both, but now I need to find the co-variance of $q$ with $\min(X,q)$. ...
0
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0answers
30 views

Deal with circular standard deviation

I have a sensor that gives me a direction of the target in degrees with a certain standard deviation and I use a circular probability density function (PDF) to represent probable locations for the ...
2
votes
0answers
39 views

Convergence of Sum of Random Variables “Independent in Limit”

Consider a sequence of random variables $X_n\sim U[-n,n]$ and a random variable $Y\sim N(0,1)$, all independently distributed. In addition, consider a bounded, measurable function $f:\mathbb{R}\to ...
2
votes
1answer
27 views

Find the probability generating function $G(s)$ of this branching process.

Suppose that $X_n$ is size of the $n$th generation of a branching process started from a single individual, where each individual has a random number of children with probability mass function: ...
0
votes
0answers
198 views

CDF of sum of $3$ independent discrete uniform random variables on $\{1,2,\dots,n\}$

What is an approximate closed formula for this probability, with a derivation: $p(k,n)$ is the probability, that among $n$ PC disks and $k$ errors in sum on them, there will be at least $1$ disk ...
1
vote
1answer
23 views

Suppose that $N$ is an iid geometric RV and $X_i$ is an iid Bernoulli RV. Find the p.g.f. of $R=X_1+ \dots + X_n$.

Each year a tree of a particular type flowers once and produces a random number $N$ of flowers, where $\mathbb{P}(N=n)=(1-p)p^n$, $n=0,1,2,\dots $ and $0<p<1$. Each flower has probability $1/2$ ...
1
vote
1answer
51 views

Probability Generating Function of a Negative Multinomial Distribution

Derive the probability generating function (pfg) of a negative multinomial distribution with parameters $(k; p_{0}, p_{1}, ..., p_{r})$ where the k-th occurrence of the event with the probability ...
0
votes
1answer
30 views

Probability of long identical substring

You have a string of $20,000$ consecutive bits. Each bit is either a $1$ or a $0$ and has a $0.5$ chance of being either. Calculate the probability that there is at least one substring of at least ...
2
votes
0answers
23 views

Is Gaussian $(X_1, X_2)$ optimal for $h(a_1X_1+ a_2X_2+Z_1) - \mu \, h( b_1X_1+b_2X_2+ Z_2)$?

Let \begin{align} W &= h(X_1+Z_1) - \mu \, h( X_2+ Z_2) \quad (1) \end{align} where $h(\cdot)$ is the differential entropy function, $\mu\ge 1 $ is a scalar, and $Z_1$ and $Z_2$ are ...
2
votes
1answer
60 views

Distribution of the product of a Normal and an Exponential random variable

What is the probability distribution of $M$, given $M=V*X/k$, where $X$ is Normal, $V$ is Exponential, $k$ constant? Or, in the real world, the probability distribution of (Cost/k) where ...
1
vote
1answer
19 views

Joint probability distribution (over unit circle)

A couple of two continuous random variables $(X,Y)$ is distributed uniformly over the closed unity circle (so $-1\leq x \leq 1$ , $y$ analog). $U$ is defined as the distance from $O$ to the point ...