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

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2
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1answer
62 views

How to invert finding marginals of a rotationally symmetric distribution?

Given $$f(x) = \int_{-\infty}^{+\infty} g(x^2+y^2) \,\mathrm{d}y,$$ is it possible to reconstruct $g(u)$ analytically? I tried differentiating $f(x)$ but only found $$f'(x) = \int_{x^2}^{+\infty} ...
0
votes
1answer
40 views

What's the name of this distribution?

I have two random variables $X$ and $Y$, both distributed uniformly over the interval $(0, 1)$. Is there a name for the distribution of $$Z = \frac{X}{Y}?$$ I think there is one, but I cannot ...
0
votes
0answers
323 views

The mean and variance of the random variable with Rician distribution

What is the mean and variance of $Z_0=\frac{\sum\nolimits_{i = 1}^{{n}}R_iZ_i}{\sum\nolimits_{i = 1}^{{n}}R_i}$, where $Z_i$ is a constant and $R_{i} =\sqrt{X_i^2+Y_i^2} $ ${(i=1,\ldots ,n)}$? $X_i$ ...
1
vote
0answers
68 views

how can I calculate the mean and variance of random variables?

There are two dependent (correlated) random variables with Rice (Rician) distributions $R_1\sim (u_1,s_1)$ and $R_2\sim(u_2,s_2)$. Then, how can I calculate the mean and variance of random variable ...
0
votes
1answer
67 views

What is the mean and variance of the averaged value over $n$ random variables with Rice distributions?

Assuming there are $n$ uncorrelated random variables (RVs) with Rice (Rician) distributions $R1~N(u_1,s_1) \ldots R_n~N(u_n,s_n)$, with non-zero mean and different variance, what is the mean and ...
2
votes
2answers
581 views

Sufficient Statistic for a Geometric R.V.

I have a problem that I know I am very close to the solution for, but I think I just need some more formatting to make it a really clean proof. The problem goes like this: Suppose X is a discrete ...
0
votes
1answer
302 views

Finding variance of joint probability function of discrete random variables

I've been working on a problem, and it's quite lengthy, so I'm going to restrict the problem while not losing too much information. I'm given a table of the discrete random variables $X$ and $Y$. I'm ...
1
vote
1answer
96 views

Proving that a function is monotone

Here is the setting: We have a middleman that buys a product from the producers, and sells the product to the customers. The middleman charges a price $R$ to the customers, and pays a price $p(R)$ to ...
1
vote
0answers
55 views

How to make this inference: Degree of a node in a graph is significantly diffenrent from poisson distribution

I am working on Gene-Gene interaction graphs. I build a graph by adding edges between genes (nodes) which show statistical interaction in predicting a quantitative parameter value (say, brain volume) ...
1
vote
1answer
89 views

Integrating out $\sigma^2$

My question is with regards to integrating on $\sigma^2$ when working with normal distributions. How does one handle the $^2$ aspect of $\sigma^2$? I am unsure whether I should treat $\sigma^2$ as an ...
1
vote
2answers
58 views

Is E[|X+Y|]\leq E[|X| ] + E[|Y|]

Let $X , Y$ be two independent Random variables then is $E[|aX+bY|] \leq |a|E[|X|]+|b|[E[|Y|]$ ? where $a ,b \in \mathbb{R}$. Thank you.
1
vote
1answer
25 views

$\mu_{2r+2}=\sigma^3\frac{d\mu_{2r}}{d\sigma}+\sigma^2\mu_{2r}$

How do I go about proving this? If $\mu_x$ be the $x'th$ order central moment of the distribution, then for a normal distribution with mean $0$ and standard deviation ...
0
votes
1answer
47 views

Generating log-distributed random variates

I'm looking for a numerically stable way to generate random variates that are distributed like $\log(U)$ with $U \in (0,1)$, to be stored in IEEE 754 floating-point variables. My idea is: Generate ...
1
vote
1answer
863 views

Is the Sigmoid Function a Probability Distribution?

This could be a stupid question but, since sigmoid function maps values between $-\infty$ and $\infty$ to values between 0 and 1, I thought it could be a probability distribution. However when I take ...
0
votes
0answers
37 views

first order auto regressive

Hello how to do the following: Suppose that $X =(X_1,X_2,...,X_n)$ follows the following: $X_t - \mu = \eta (X_{t-1} - \mu) + \epsilon_t,$ $t= 1,2,...$ where $\mu \in R$ and $\eta \in (-1,1)$ ...
0
votes
1answer
155 views

Find the expected value of the largest piece of a stick.

A stick of length 2 is broken into two pieces at a uniformly random chose point. What is the expected value of the largest piece? Here is what I have $U(X) = 2-x$ if $0\leq x \leq 1$ $U(X) = x$ ...
2
votes
1answer
64 views

Berry-Esseen theorem and test functions

We use the Berry-Esseen theorem to prove the closeness of the two distributions. In the proof of the theorem people have used the notion of test functions (functions which are smooth and fourth ...
1
vote
0answers
134 views

How does standard deviation depend on data?

I've got a problem: I take a random variable, assign values to it and than I have to invent two different probability distribution functions with similar mean values (+-5%), but in the second function ...
1
vote
1answer
68 views

Finding $EX$ of a density function (integrating $\ln u$ over infinity)

I've been given a density function as: $f(x) = 1/4e^{-|x|/2}$ where $-\infty < x < \infty$ and need to show that $EX = 0$ I understand that to find $EX$ I must calculate $\int xf(x)~dx$ ...
1
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0answers
57 views

Need advice: what should be my next step?($2$) (does Cauchy-Schwarz help here?)

This question is based on the question that I asked here Need advice: what should be my next step? I did a little more progress and wanted to share with you. As this is a new question, without any ...
1
vote
1answer
398 views

Rayleigh distribution

I have this question from my statistical theory course: A sniper shoots at a target. X and Y measure its deviation on the x and y axes. X and Y are independent and are distibuted normally with mean=0 ...
2
votes
1answer
118 views

Estimating the number of tickets bought in a lottery

A national lottery has the format where $7$ numbers are chosen from $45$ without replacement. The first $6$ numbers chosen constitute the "winning numbers", while the last number chosen is the ...
1
vote
0answers
85 views

What discrete distribution is completely determined by its mode and variance, is easy to sample, and has nice border properties?

I need to generate random ordered unranked trees that will be used to test some computer program. I'd like to incorporate some kind of control into the generation process, so that the generated trees ...
8
votes
1answer
279 views

Repeatedly Toss Balls into Bins

$n$ balls are randomly tossed into $m$ bins, each bin can hold $k$ balls. If a ball is tossed into a full bin (already has $k$ balls in it), it can be tossed repeatedly and randomly into the $m$ bins ...
0
votes
1answer
222 views

Empirical Bayes estimator for a Beta-Binomial parameters

Let $X_t$ be collected from a Binomial distribution with parameters $N_t$ and $P_t$, where $N_t$ is known for $t= 1, 2, \dots , T$. On the other hand, $P_t \sim \operatorname{Beta}(\alpha_t, ...
1
vote
1answer
155 views

What is the physical meaning of the output/ y -value of a normal distribution? (not the area under its curve)

Forgive me for my lack of knowledge regarding math terminology. I'm learning basic statistics right now, and I can see pretty intuitively that the area under a normal distribution on a certain ...
0
votes
2answers
98 views

Poisson distribution: trucks and cars

This is some probability problem that I conjured up. Can anyone check whether this problem makes sense and has a solution? Assume that the traffic on Spooner street follows a Poisson process with a ...
2
votes
1answer
256 views

CDF of $X+Y$,$X−Y$,$XY$ for $(X,Y)$ Chosen Uniformly Inside Triangle

Let $(X,Y)$ be chosen uniformly on the triangle $\{(x,y)\in\mathbb R^2:x+y\leq1,x\geq0,y\geq0\}$. What is the joint density function of $(X,Y)$? Find the CDFs of $X+Y$, $X-Y$, and $XY$. What I've ...
5
votes
0answers
148 views

Need advice: what should be my next step?

I am dealing with a quite algebraic question and I arrived at some good point. I had $2$ equations with $2$ unknowns and I was able to eleminate one of the variables. My final equation still seems ...
0
votes
2answers
44 views

Algorithm for integral of standard distribution

I need help in producing random data that follows standard distribution. Since it is to be used in a computer application, I would prefer an algorithm before a table. So, this is what I need. The ...
1
vote
1answer
202 views

How to show that $Z = |X|$ and $-|X|$ has a a standard normal distribution? ($X \sim$ Folded standard normal distributed)

Given the probability density function of $X$ (folded standard normal distributed) is: $$f(x) = \frac{2}{\sqrt{2 \pi}} \exp\left(-\frac{x^2}{2}\right),\quad x \geqslant 0 $$ How can one show that $Z ...
0
votes
1answer
22 views

Is there any bound for $\int_{\mathbb{R}}\sqrt{f_0[x]f_1[x]}\mbox{d}x$

I wonder if there is an upperbound for following the expression: $$\int_{\mathbb{R}}\sqrt{f_0[x]f_1[x]}\mbox{d}x$$ where $f_i$, $i=0,1$ are some density functions. Thanks in advance.
1
vote
2answers
151 views

Distribution of event times of binomial process $(Y_n)_{n \geqslant 0}$ conditioned on $Y_n = m$

Consider a sequence of independent $0$-or-$1$ outcomes $X_1, \ldots, X_n,\ldots$, such that $\mathbb{P}(X=1)=p$. The discrete time process $\left(Y_n\right)_{n \geqslant 0}$, such that $$Y_n = ...
1
vote
1answer
147 views

Weak Convergence to Exponential Random Variable

Assume that $X_1$, $X_2$,... are independent random variables uniformly distributed on $[0,1]$. Let $Y^{(n)}=n\inf\{X_i,1\leq i\leq n\}$. I am asked to prove that it converges weakly to an exponential ...
0
votes
1answer
34 views

What is this linear-fractional distribution formula missing?

See Definition 1 in this paper: http://arxiv.org/pdf/1111.4689v3.pdf The left-hand-side of the second formula appears to suggest some kind of recursion, but the right hand side is not a recursive ...
2
votes
1answer
77 views

Inequality For Subgaussian Distributions

For my research I am trying to bound some exponential moments of subgaussian r.v.'s. And I am stuck with proving one of such inequalities. More specifically: Let $a$ be unit vector in ...
3
votes
1answer
125 views

Expectation of a compound random variable

Let $X_{1},X_{2},...,X_{n},...$ be independent random variables following the uniform distribution on $[0,1]$. Let $N$ follow the negative binomial distribution with the probability function ...
0
votes
1answer
94 views

Cumulative Distribution Fit

I am looking for a monotonically decreasing function to fit a cumulative distribution. The distribution is the number of values of a random variable X, that are greater than Y as a function of Y. In ...
2
votes
0answers
229 views

Central limit theorem - speed of convergence in center vs tails

I've been told that one of the implications of the central limit theorem is that as we increase the sampling of random variables, we converge faster to a normal distribution in the center and slower ...
2
votes
1answer
53 views

Inferring possible future distribution given observed events.

I have a process which continues for a possibly infinite amount of time, where some event can either happen or not happen, with equal probability (call it $p$) for each time step. I've observed $n$ ...
0
votes
1answer
250 views

If $x = y$ what is $p(x\mid y)$?

I don't speak maths too well (engineer) so simple language preferred or could you describe it as a graph please? This has probably been asked but I have no idea what to search... Follow up question: ...
2
votes
1answer
513 views

Show estimators of P(X=0) for X~POISSON are biased/unbiased

Let $X \sim \operatorname{Poi}(\mu)$ and $\theta = \Pr[X=0] = e^{-\mu}$. Show that $\tilde{\theta} = u(X)$ is an unbiased estimator of $\theta$ where $u(0) = 1, u(x) = 0$, for $x=1,2,3,\ldots$ Is ...
2
votes
1answer
210 views

Product of three Poisson distributions

Product of two Poisson distributions is a Bessel function: $$ \sum_{r=0}^\infty \frac{e^{-f} f^r }{\Gamma(r+1)} \frac{e^{-g} g^r }{\Gamma(r+1)} = e^{-f-g} I_0\left(2 \sqrt{f g} \right) $$ What I ...
3
votes
1answer
341 views

Uniform measure on the rationals between 0 and 1

I am trying to think of a probability measure on the set of rationals between 0 and 1 ($X:=\mathbb{Q}\cap[0,1]$). I want to achieve something like a uniform measure, i.e. every number should have the ...
2
votes
1answer
176 views

Problem with coupling (basic probability)

If I have two probability spaces : $\\\Omega_1=\{w^1_1,w^1_2,w^1_3\}$ with $P_1$ defined to be $P_1(w^1_1)=P_1(w^1_2)=P_1(w^1_3) = 1/3$ and $\Omega_2=\{w^2_1,w^2_2,w^2_3\}$ with $P_2$ defined to be ...
1
vote
1answer
163 views

Lower bounds of laplace transform of characteristic functions

I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function of some undetermined probability ...
0
votes
1answer
2k views

Likelihood function of a gamma distributed sample

I missed the day of class where we went over likelihood functions, and I had a quick question. If $X_1,...X_n$ are i.i.d. $ {\Gamma}(\alpha,\beta)$ r.v.s, I'm trying to find the likelihood function ...
2
votes
1answer
582 views

Negative Binomial Question Without Exact Values

The question I am working on is: Three brothers and their wives decide to have children until each family has two female children. What is the pmf $X=$ the total number of of male children born to ...
1
vote
0answers
2k views

Finding the mean and variance of the sum of three exponentially distributed random variables

Given: $A = X + Y + Z$, the parameter of $Y$ and $Z$ is $\mu$, the parameter of $X$ is $\lambda$. The coefficient of correlation of Y and Z is $\beta$, $X$ and $Y$ and $X$ and $Z$ are pairwise ...
0
votes
1answer
179 views

Probability question involving two independent poisson random variables?

A restaurant has two waiters. suppose that the number of customers serviced daily by each waiter can be viewed as independent Poisson random variables with parameters $\lambda$ and $\mu$. whats the ...