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

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522 views

Distribution of results of a biased coin

the mass distribution of a coin is so that the chance of getting a head s only 40% .The coin is tossed a 100 times what is the chance -to get more than (inclusive ) 50 heads ? -to get more than ...
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1answer
97 views

Is the term Gaussian distribution the preferred term?

Is "Gaussian" the term preferred over "normal" when speaking of the distribution to which these names have been attached? Are they both referring to the same thing?
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1answer
226 views

Looking for Anthony?

You are trying to locate an old high school friend who lives in Chicago. Unfortunately, your friend's name is Anthony Smith and the Chicago phone book lists phone numbers for 24 different people named ...
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1answer
55 views

Independence of Random samples

I have a some questions that have been bothering for a while now. First, how does one obtain the joint probability distribution function of $X_{1},\cdots ,X_{n}$? Would it be $\prod\limits_{i=1}^n ...
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1answer
159 views

Independence of discrete random variables

Suppose $X,Y$ are uncorrelated random variables, $\mathbb{E}(XY)=\mathbb{E}(X)\mathbb{E}(Y)$, taking on two values $m,n\in\mathbb{R}$, that is, $P(X\in \{m,n\})=P(Y\in \{m,n\})=1$. How should I go ...
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2answers
88 views

If $X, Y \sim N(0,1)$, find the CDF of $\alpha X + \beta Y$ [duplicate]

Possible Duplicate: Proof that the sum of two Gaussian variables is another Gaussian Let $X,Y$ be independent normally distributed $N(0,1)$ random variable, and $\alpha,\beta\in ...
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2answers
754 views

Showing $\cos(t^2)$ is not a Characteristic Function

Usually when we try to show a function is not a characteristic function, we would prove it is not uniformly continuous. I am wondering if there is any other way to show $\cos(t^2)$ is not a ...
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1answer
192 views

Sequence of characteristic functions

Let $\eta_k(t)$ be the characteristic function of a random variable $X_k$, for $k=1,2,...$ Consider a sequence of positive real numbers $c_1,c_2,...$ Take a function $g(t)=\sum\limits_{k=1} ^\infty ...
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2answers
628 views

Poisson distribution question, reading from the table

The number of cars passing over a certain bridge between 11pm and midnight has a Poisson density with lambda = 4. In what proportion of nights would you expect more than one car to pass over the ...
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0answers
160 views

Convergence in Distribution

Consider $$\zeta_n=\max_{1\le k \le n} X_k ,$$ where $X_k\sim \exp(1)$. Let $\eta_n=\frac{\zeta_n}{\ln(n)}$. What is the limit of $\eta_n$ as $n\to \infty$? (Convergence in distribution).
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1answer
391 views

Double exponential distribution

Let $\zeta$ and $\eta$ be independent random variable with $\exp(\lambda)$ distribution. What is the distribution of $Z=|\zeta-\eta|$ . I am trying to calculate it by finding $\Pr(\zeta-\eta>x)$, ...
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1answer
161 views

Convergence of Random Variable

Let $X_1, X_2, ...$ be identically distributed nonnegative random variable with $EX_1<\infty$. Prove that $\frac{X_n}{n}\rightarrow 0$ almost surely. I am trying to use Borel Cantelli Lemma (part ...
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1answer
79 views

transforming a cdf with a function of $x$

I have a cdf $F(x)$ defined over 0,1. I have a function, $q(x)$, which returns a number between $x$ and $1$. I would like to define a new cdf, $G(x)$, such that $G(q(x)) = F(x)$. I would think ...
5
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1answer
249 views

expectation of function of exponential

I need to compute the following integral $$ \int_{0}^{\infty}x\,\left\{\vphantom{\LARGE A}% 1- \left[\vphantom{\Large A}1- \exp(-a\,x^{\alpha}) \right]^M \right\}\,{\rm d}x \qquad \mbox{with}\quad ...
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2answers
227 views

Assessing discreteness of the random variable by its characteristic function

It is easy to spot a discrete integer valued random variable by looking at its characteristic function, as that is periodic with period $2 \pi$, i.e. for binomial distribution it is $\phi(t) = (1-p+p ...
4
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2answers
263 views

Bound 1D gaussian domain in the interval $[-3\sigma, 3\sigma]$ so it still is a probability density function

I need to bound a 1D gaussian/normal (or similar) probability density function in the domain interval $[-3\sigma, 3\sigma]$ in a way that still integrates to 1. I would need something like this: $$ ...
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4answers
1k views

Calculate expectation from empirical cdf

I have a empirical cumulative probability distribution function for a random variable. The random variable is "time to failure" and I have the full curve i.e till the probability reaches 1. I want to ...
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2answers
308 views

Definite integral of cdf of the form $\Phi(\alpha+\sqrt{d^2-\frac{x^2}{2\sigma^2}})$

Any solution for the following definite integaral? Here $\Phi(x)$ represents the cumulative distributive function of standard normal distribution ...
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0answers
40 views

Determining the significance of a set ordering

First some preamble: Let A be a set of unordered elements, and let A' and A'' be subsets of A, where |A'| = |A''| = k. It’s given that as k approaches |A|, for every x in A' there is an increased ...
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1answer
1k views

Multivariate Normal Difference Distribution

Since the distribution of a difference of two normally distributed variates X and Y with means and variances $(\mu_x,\sigma_x^2)$ and $(\mu_y,\sigma_y^2)$ respectively is given by another normal ...
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1answer
191 views

Does “2 bumps” in a histogram suggest 2 underlying populations?

I've plotted a histogram of data collected in real life, not generated data. It looks like a negative binomial binomial distribution with a 2nd bump, lower from the peak bump of the curve. Here is a ...
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3answers
908 views

How to obtain the Standard Deviation of a ratio of independent binomial random variables?

X and Y are 2 independent binomial random variables with parameters (n,p) and (m,q) respectively. (trials, probability parameter)
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2answers
403 views

QQ plot explanation

The figure shows the Q-Q plot of a theoretical and empirical standardized Normal distribution generated through the $qqnorm()$ function of R statistical tool. How can I describe the right tail (top ...
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0answers
413 views

Verifying Normalization

I have encountered an exercise question asking the reader to verify that a wavefunction is normalized. So I calculated the probability density -- $|{\psi}|^2$, then verified that the integral does ...
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1answer
189 views

Signal extraction from multivariate normal

Define: $y= \theta + \varepsilon + a,$ where $a$ is a choice variable in a behavioral economic model, with equilibrium solution $a^e$, and $\theta$ and $\varepsilon$ are independently distributed ...
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1answer
93 views

Reasoning error in “maximin” type problem

I have a problem that I have approached two different ways. The approaches give me two different answers. I have matched the answers against a simulation, so I think I know which one is right. But, ...
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2answers
131 views

a question about Erlang/chi square distribution

Suppose you have $$ Y = X^2+Y^2 $$ where $X$ and $Y$ are both Gaussian with zero mean and variance $\sigma^2/2$ (you can think of $y$ as the square norm of $Z = X + jY$). The pdf should be ...
3
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1answer
97 views

Rate of change of $\mathbb{E}X_{\frac{2^n}{2},2^n}$ as $n$ increases?

I am trying to get an equation that will show the rate of change of the expected value of $\frac{2^n}{2}$th lowest of $2^n$ draws from $X$ as $n$ increases (where $n >1$). Let's call the order ...
4
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0answers
209 views

Equivalence of two sequences

I'm having some trouble showing that two things I really want to be the same are in fact the same. I want to show that these two sequences are, in fact, the same thing: $$a_0=1,a_1=-1, ...
3
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1answer
202 views

Computing quadratic variation for stable Levy flights with $0<\alpha<2$?

The wiki page on semi-martingales states that Every Lévy process is a semimartingale. and that The quadratic variation exists for every semimartingale. Let $X_t$ be a stable Levy process ...
3
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1answer
127 views

The limit of the expectation of the top half+1 order stats of $n$ draws of $X$ as $n\to\infty$?

Can anyone help me compute the limit of the average of the top half +1 of order marginal order distribution of $n$ draws from $X$, as $i\to\infty$? Specifically, the limit as $i\to\infty$ of ...
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2answers
236 views

Transformations that leave a binomial distribution invariant

The binomial distribution is written as $$p(r|n,\theta )=\binom{n}{r}\theta ^r(1-\theta )^{n-r}$$ where $n$ is a positive integer, $0\leq\theta\leq1$, and $r$ is an integer taking values from $0$ to ...
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4answers
286 views

Can a vertical line be a valid PDF?

Consider a random variable X that always takes a single value, c. I would think that a valid PDF for X would be $$f(x) = \begin{cases} 1/c & x = c \\ 0 & \text{Otherwise}\end{cases}$$ ...
3
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2answers
143 views

Limit of a nested probability function?

Let $F_0(x) =x$. Then, let $F_i(x) = (1-(1-F_{i-1})^2)^2$ where $i>0$. Is there a way I can calculate $F_{\infty}(x)$? (Terminology-wise, would I say that is a limit of $F_i(x)$ as $i \to \infty$ ...
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1answer
97 views

Order statistic distances as a function of moments?

Let $X$ be a random variable defined over $0-1$ and let $X_1 . . . X_4$ be the order statistics of $4$ draws from $X$ (where $X_1$ is the minimum). I am looking to see if I can identify any general ...
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0answers
86 views

Asymptotic behaviour of the solution to a certain PDE

$\delta M=\beta(x-y)M_y+\mu x(y-1)M_x+\delta y$, where $M(x,y)=\sum_{n=0}^{\infty}{\sum_{k=0}^{\infty}{s_{n,k}x^ny^k}}$ is the generating function for a certain probability distribution $\{s_{n,k}\}$ ...
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1answer
165 views

How to prove an inequality involving averages of different order statistics?

Let $X$ be a continuous random variable with values ranging from 0 to 1. Let $X_{kn}$ be the random variable representing the $k$th smallest order statistic of $n$ draws from $X$. Note that ...
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1answer
1k views

what is the cumulative distribution function of a logistic function?

I've found on wikipedia for Logistic function (http://en.wikipedia.org/wiki/Logistic_function) they have the formula for a Logistic curve: $P(t) = 1 / (1 + e^{-t})$ and they have a diagram of the ...
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3answers
289 views

Concept of Probability distribution

Sorry for a silly question, but it seems like only you can answer this question. What's is the concept of Probability distribution, what's the meanining behind this term. Why we need probability ...
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2answers
136 views

Drawing random values from a distribution

If I have a set of $n$ elements, and I want to assign to each-one some value $\phi$, drawn at random from a distribution $f(\phi)$ such that $\int_0^1f(\phi)\;d\phi\:=\:1$ Does this mean that the ...
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2answers
2k views

PDF of product of variables?

could anyone please indicate a general strategy (if there is any) to get the PDF (or CDF) of the product of two random variables, each having known distributions and limits? After having scanned ...
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1answer
498 views

PDF of $f(x)=1/\sin(x)$?

What is the probability density function (PDF) of $f(x)=1/\sin(x)$ when $x$ is uniformly distributed in $(0,90)$? $f(x)=\sin(x)$ has a known PDF, which has the form $2(\pi\sqrt{1-\sin(x)^2})^{-1}$, ...
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1answer
209 views

Probability in a Bell Curve

Lets assume that a phone number lies in one of three pages with equal probability (1/3). The probability of selecting the pages follows a Bell Curve with probabilities as : .2 .6 .2 Now ...
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1answer
194 views

distribution of sum of independent gaussians

what is the probability density function of $|y|^2$, where $$ y = \sum_{i=1}^n a_i x_i $$ where $x_i$ are complex gaussian random variables with zero mean and unitary variance? With only two ...
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3answers
838 views

Is this a Delta Function? (and Delta as limit of Gaussian?)

I have a set of users that generate calls. If I assign the same probability to each user, they have identical call generation probability which can be defined as $\delta$. These callers are chosen ...
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1answer
408 views

which is the best distance function?

We all know there are distance functions, like Kullback Leibler distance, Bhattacharyya measure, Euclidean distance, Wasserstein distance, and so on. Take a sample distance: ...
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1answer
245 views

How can I prove this expression not to be a characteristic function

Let $\phi$ be a function of two real arguments defined as follows: $$ \phi\left(t_1, t_2\right) = \exp \left(-t_1^2-t_2^2 +i \frac{t_1}{3}\frac{ t_1^2-3 t_2^2 }{t_1^2+t_2^2} \right)$$ and whenever ...
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3answers
155 views

How to quantify uniform distribution?

I have a real-world situation with a machine that lays a layer of wires on a high-pressure hose. The machine has S "slots" (approx 200), and each slot could have one wire or be empty. Typically there ...
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1answer
130 views

mixture of complex gaussians

I would like to compute the following probability $$ P\left( ax \geq \sum_{i=1}^n b_i y_i \right) $$ where $a, b_i$ are constant coefficients (in my case, they are positive too) and $x, y_i$ are ...
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1answer
307 views

Joint and Marginal distributions of a random sample

Let $X_{1},X_{2},\ldots ,X_{n}$ be a random sample of size $n$ from a population distribution $F$. I want to find the following: 1. the joint P.d.f of $X_{1},X_{2},\ldots ,X_{n}$. 2. the marginal ...