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

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0
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1answer
159 views

sum of marked Poisson variables

I would like to calculate the expected value of the sum of $N+1$ (assume $N$ is large) marked Poisson variables $X$ with intensity $\lambda$ over subset $I$: $$ S=\left\langle\sum_{i \in I} ...
0
votes
0answers
75 views

Is it possible to get a closed-form expression?

I have an intergral which is given as \begin{align} T= \int \limits_{0}^{\infty} \prod \limits_{n=1}^{N} \left[ 1- \frac{ \exp\left( - z B_n \right) }{1 +z A_n}\right]^K ...
0
votes
1answer
282 views

Finding Probability using Moment-Generating Function

I am given a moment-generation function $M_x(t)= e^{t+t^2}$ and asked to find the probability that the random variable is greater than $2.5$ Any help would be greatly appreciated!
1
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0answers
42 views

$\chi^2$ distribution as sum

Hi how to show the following: $\chi^2$ $\frac{2(\sum_{i=1}^{r}Y_i + (n-r)Y_r)}{\theta}$ is a chi-square dristribution with $2r$ degrees of freedom where: $Y_i$ stands for the order statistics of ...
1
vote
1answer
180 views

How to create a probability distribution function for a given set of data

I was wondering how to create a probability distribution function for the following data. ...
0
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1answer
73 views

Probability from chi square distribution

How do I find a probability for a chi square distribution? I have a continous random variable from which I've got the chi square with the formula: $$\sum \frac{(o-e)^2}{e}$$ where $o$ is the ...
1
vote
1answer
258 views

Asymptotics of maxima of i.i.d. chi-square random variables

How to find the following: Let $X_1$, $X_2$, $X_3$,..., $X_n$, be i.i.d with chi-square distribution with one-degree of freedom. Find $a_n$ and $b_n$ such that $ a_n(\max_i X_i - b_n)$ converges in ...
1
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1answer
36 views

Prove that if $a_k=x_{k1}a_{k-1}+x_{k2}a_{k-2}+\cdots+x_{kt}a_{k-t},$ ($x_{ki}$~U(0,b)) then $\dfrac{\log{a_k}}{k}\to^{p} c$

Let $a_1=a_2=\cdots=a_t= 1,a_k=x_{k1}a_{k-1}+x_{k2}a_{k-2}+\cdots+x_{kt}a_{k-t},$ where $x_{ki}$~U(0,b), and $x_{ki}(k>t,i=1,2,\cdots,t)$ are independent each other. Prove that $\exists c\in ...
2
votes
1answer
786 views

Expected Value of Normal CDF

I am trying to calculate the expected value of a Normal CDF, but I have gotten stuck. I want to find the expected value of $\Phi( \frac{a-bX}{c} )$ where $X$ is distributed as $\mathcal{N}(0,1)$ and ...
0
votes
1answer
60 views

Exact form of pdf of maximum of normal random variables

$$ z = max(x+b,y) $$ where x ~ N(m1,s1) and y~N(m2,s2), b is a contant What's the pdf of z? Or exact form of E(z)? (E is expectation operator) To the best of my guessing from the literature it is ...
1
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1answer
60 views

Poisson Process: Density of Quotient

The question is: Let $\{N(t): t \geq 0\}$ be a Poisson process of rate $1$ and let $T_1 < T_2 < \cdots$ denote the times of the points. Then what is the pdf of $Y = \frac{T_1}{T_3}?$ What ...
0
votes
1answer
147 views

When does the difference of two random variables follow a symmetric distribution?

Setup: Let $X_t$ and $Y_t$ denote two (possibly dependent) random variables with cumulative distribution functions (cdf) $F_X$ and $F_Y$. Assume the support of $F_X$ and $F_Y$ is $\mathbb{R}^+$. Let ...
1
vote
1answer
309 views

What is the distribution of $Y = e^X$ when $X$ is normal?

What is the distribution of $Y = e^X$ when $X$ is normally distributed? Am I supposed to use characteristics function of normal random variable ?
4
votes
2answers
395 views

Distribution of $Y = \sin X$ when $X$ is normal?

Assume $X$ is Normally distributed : $X\sim N(\mu,\sigma)$ What is the distribution of $Y = \sin X$ ? I think we should start with $F_Y(y)=P(\sin X < y)$. But how to continue?
1
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1answer
75 views

show convergence of sum towards integral — not Riemann partition but using another convergence

$X_1, X_2, ...$ real random variables. $P(X_n=\frac{k}{n}) = \frac{1}{n}$ for $k\leq n \in \mathbb{N}$. Let further $X \overset{d}{=}U_{[0,1]}$. I showed $X_n \overset{d} \to X$ when $n \to \infty$ ...
0
votes
1answer
138 views

Central Limit Theorem application where mean is not $0$

I am given a problem that says to use the central limit theorem, where the mean of the random variable is non-zero. However the version of then central limit theorem in my text has the assumption that ...
2
votes
1answer
69 views

A method to test for uniform distribution over a convex polytope

Assuming I have a convex polytope defined as the intersection of $Ax=b$ and $x>0$ and I have a way to sample points from this object, is there a way I can test for uniformity of these sampled ...
0
votes
1answer
192 views

Expectation of normal distribution as expectation of chi-squared distribution

This is Lemma 11.3 from the book the theory of linear models and multivariate analysis , which says Let $U\sim N_p(\theta,I)$ and K have a Poission distribution with mean $||\theta||^2/2$, then ...
1
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1answer
62 views

Is it possible to “customize” the multinomial distribution to your specifications?

So according to the multinomial distribution, the probability function $\Pr(X_1 = x_1, X_2 = x_2, \dots, X_k = x_k)$ is equal to $\dfrac{n!}{x_1! x_2! \cdots x_k!} \cdot p_1^{x_1}\cdot p_2^{x_2} ...
0
votes
2answers
361 views

Determining Type of Probability Distribution

Suppose that pollen spores are randomly scattered in a home, at a density of $8$ sports per cubic cm. What is the probability of finding at least two spores in a space of $0.2$ cubic cm? Here is ...
4
votes
2answers
155 views

Proving that the expectation is always negative

I am interested in if the expected value $$E_{X_3}\left[\log\frac{f_1(y)}{f_2(y)}\right]<0$$ is always be negative or not. Here, I have $3$ random variables $X_1,X_2,X_3$, corresponding to the ...
1
vote
2answers
178 views

Number of draws required for ensuring 90% of different colors in the urn with large populations [duplicate]

My problem is: An urn contains $10000000$ ($10^7$) different colored balls, namely $K_1, K_2,\dots,K_n (n=10^7)$ with $K_1=1000, K_2=1000,\dots,K_n=1000$. My question is: How many balls do I need ...
0
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1answer
103 views

drawing at least 90% of colors from urn with large populations

My problem is: suppose I have an urn containing balls of $n = 10000000$ (i.e., $10^7$) different colors, with $1000$ balls of each color (so the total number of balls is $1000n = 10^{10}$). Suppose I ...
2
votes
1answer
180 views

drawing at least one colored ball of each from urn in a case of large populations

My problem is: If an urn contains balls of $10^7$ different colors, namely $K_1, K_2, \ldots K_{10^7}$, and there are 1000 balls of each color, so that the total number of balls in the urn is ...
2
votes
1answer
549 views

drawing different colored balls from one urn without replacement and at least

I have this problem (numbers in the example are much smaller than reality, so it would help to get a general equation): One urn contains $10$ red, $10$ yellow, $10$ black, $10$ green, and $10$ orange ...
1
vote
1answer
97 views

Poisson Process with a Random Variable

I really couldn't wrap my head around this basic concept so I'm looking help for some basic calculations to solidfy my understanding: Suppose we have $T\in (0,\infty)$. Say we have $E(T)=\mu$, ...
0
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0answers
49 views

Matching pdf with the Inverted Gamma Distribution

So the Inverted Gamma probability density function is: $\displaystyle{f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1}\exp\left(-\frac{\beta}{x}\right)}$ The equation I'm ...
1
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1answer
145 views

Conditions for positive dependence

Consider two random variables $X$ and $Y$ with joint distribution $F_{X,Y}$ and strictly positive density function $f_{X,Y}$. Additionally, let $x^*$ be the value of $x$ that solves: $$ \Pr[Y\leq ...
1
vote
1answer
2k views

Continuous uniform distribution over a circle with radius R

I started to do this problem with the standard integration techniques, but I cant help but think that there has got to be something I am not seeing. Since it is a uniform distribution, even though x ...
0
votes
1answer
252 views

Atomless probability measure

Assume that $m$ is an atomless probability measure on $\mathbb{R}^{d}$. Let $\left( X_{1},\ldots ,X_{d}\right) $ be a random vector with law $m$. Are the marginal cumulative distribution functions of ...
1
vote
1answer
50 views

Determine which mean is smaller over two non-normal distributions

Let's say I have a non-normal distribution A and another non-normal distribution B, the mean and std deviations of each distribution are different. I then randomly sample 100 values from A, SampleA, ...
1
vote
1answer
301 views

Weibull Scale Parameter Meaning and Estimation

Wikipedia: http://en.wikipedia.org/wiki/Weibull_distribution gives a nice description on what the shape parameter (they call it k) means in the Weibull distribution, but I can't find anywhere what the ...
1
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0answers
144 views

Reciprocal Shifted Log-Normal Distribution

Let $X$ be a log-normal distribution, let $k\geq0$ be a real value and let $Y=\frac{1}{X+k}$. What is the name of the $Y$ distribution other than 'reciprocal shifted log-normal'? What is the mean of ...
0
votes
1answer
919 views

Exponential Random Variables

QUESTION: Let $X$ and $Y$ be exponentially distributed random variables with parameters $a$ and $b$ respectively. Calculate the following probabilities: (a) $P(X>a)$ (b) $P(X>Y)$ (c) ...
4
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0answers
883 views

Uniform distribution on the surface of unit sphere

It is known that given $X=(X_1, X_2, \ldots, X_n)$ iid $\sim N(0,1)$, then $X/\sqrt{X_1^2+\cdots+X_n^2}$ is uniformly distributed on the surface of unit sphere. Intuitively, I know that that's ...
0
votes
3answers
22 views

Distribution of an additional random variable

I need a little help for the following exercise: Let $X_1, X_2,\ldots$ be sequence of iid rv with values in $\{1,2,3 \}$ and $p(i):=P(X=i) \gt 0$ for $i \in \{1,2,3\}$. Define an another rv ...
7
votes
0answers
129 views

How well can the maximum of a Gaussian process be approximated by a finite-dimensional Gaussian variable?

Consider a compact set $K$ in $\mathbb{R}^p$, and let $W$ be a mean zero continuous Gaussian process on $K$, meaning that $W$ takes its values in the space of continuous functions from $K$ to ...
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0answers
33 views

Convolution of independent distributions

Let F be a distribution on $R$ and X be a randomvariable with distribution F. If $x\geq0,$ $\overline G(x)=P(X>x|X\geq0)$ (i.e. conditional distribution), then ...
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0answers
54 views

Best Distribution to Approximate this Histogram

Given the histogram below, what common distribution would be well fit to this data i.e. Beta, skewed normal. I want to use the data as a prior in Bayesian analysis so want to approximate it by a ...
1
vote
1answer
1k views

If $X$ and $Y$ are independent then $f(X)$ and $g(Y)$ are also independent.

Knowing that if you have two independent $X$ and $Y$, and $ f $ and $ g $ measurable functions, how to show that then $ U = f (X) $ and $ V = g (Y) $ are still independent.
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3answers
66 views

Basic statistics - Calculate distribution of winning

I have a 100 sided fair dice with each side labelled 1 thru 100. I win if the number rolled is 49 or higher (1% advantage). 1. What is the probability of me winning exactly 500 rolls if the dice is ...
1
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0answers
32 views

Holder norm of Wiener process

Show that if $W(t),\ 0 \le t \le 1$\ is a standard Wiener process, then its Holder norm $\sup\limits_{0 \le s,\ t \le 1}\frac{|W(t) - W(s)|}{|t - s|^{\alpha}},\ 0 < \alpha < \frac{1}{2}$\ has ...
2
votes
0answers
244 views

Variance of minimum of N random variables

Let $X_1,X_2,\dots,X_N$ be i.i.d. random variables with support $[0,M]$, and with density and distribution functions $f_X(x)$ and $F_X(x)$ respectively. Given $Y= \min_{i\in \{1,\dots,N\}} X_i$ , ...
1
vote
1answer
58 views

Need an easy CDF for Inverse transform sampling

I want to use inverse transform sampling to generate some random numbers, which all fall into a given interval $(0,x_{max})$. The numbers are not necessarily distributed evenly but can be "skewed". I ...
0
votes
1answer
2k views

Expected Values and CDF

I have a piecewise function and I have to find the expected value of $x$ and the cdf. If I have three different pieces for the function, how do I find the expected value? Do I integrate each piece ...
3
votes
1answer
84 views

Is it possible to sample the Dirac delta function?

The Dirac delta function can be a probability measure with the unit/Heaviside step function as its cumulative distribution function. Is it possible to sample such a distribution? If a random variable ...
1
vote
1answer
153 views

How to calculate probability using multinomial distribution?

So according to the multinomial distribution, the probability function $\Pr(X_1 = x_1, X_2 = x_2, \dots, X_k = x_k)$ is equal to $\dfrac{n!}{x_1! x_2! \cdots x_k!} \cdot p_1^{x_1}\cdot p_2^{x_2} ...
0
votes
2answers
64 views

Noise pdf Gaussian

Why the probability distribution function of the noise in a channel is Gaussian (normal distribution)? Intuitive discussion is appreciated.
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2answers
285 views

How to prove uniform distribution of $m\oplus k$ if $k$ is uniformly distributed?

All values $m, k, c$ are $n$-bit strings. $\oplus$ stands for the bitwise modulo-2 addition. How to prove uniform distribution of $c=m\oplus k$ if $k$ is uniformly distributed? $m$ may be of any ...
3
votes
1answer
64 views

Convergence in distribution and normality of the limit

Let $Z=(Z_1,Z_2)$ be a bivariate standard normal vector and $Y_{1,n},Y_{2,n}$ two sequences of real valued random variables with finite variance such that $Y_{1,n}\xrightarrow{d}Z_1$ and ...