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
51 views

Does the area under the curve remain the same in this variable transformation?

$X$ is a continuous random variable with probability density function $f(X)$. Let $Y = f(X)$. Let $g(.)$ be the pdf of $Y$. Intuitively I think below relation holds true, how to prove it does or ...
1
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0answers
116 views

Bayesian updating of multivariate normal?

Let $\bf x$ be an unobserved realization of $\tilde{\bf x}\sim\mathcal{N}(\pmb\mu,\pmb\Sigma)$, where $\pmb\mu\equiv\begin{bmatrix}\mu_1\\\mu_2\end{bmatrix}$ and ...
2
votes
1answer
231 views

Can someone explain the intuition behind this moment generating function identity?

If $X_i \sim N(\mu, \sigma^2) $, we know that: $\bar{X} \sim N(\mu, \sigma^2 /n)$. But why does: $$\exp\left({\sigma^{2}\over 2}\sum_{i=1}^{n}(t_{i}-\bar{t})^{2}\right)= ...
1
vote
1answer
290 views

Random vector with uniform distribution.

Let $(X,Y)$ be a random vector with uniform distribution at $0 \leq x \leq 1$, $x \leq y \leq x+h$ with $0<h<1$. Find $E(X)$ and $E(XY)$. What i did: (1) Find densities: $f_X(x) = \left\{ ...
2
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1answer
52 views

Find the probability distribution.

A game is to choose a random real number $x$ between 0 and 10. The earnings are given by $|5-X|$ being X the number chosen. (a) - Find the earning distribution and (b) - If you play twice with $X_1$ ...
1
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1answer
117 views

How to vary lambda in exponentially distributed numbers

I am implementing an exponentially distributed random number generator (RNG) based on George Marsaglia's Ziggurat algorithm. I previously used the algorithm to create a normally distributed RNG. By ...
0
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2answers
101 views

Convergence of random variables defined by the normal distribution.

I'm trying to prove this: Given $\{\mu_n\}$ and $\{\sigma_n\}$ sequences of real numbers such that $\mu_n \rightarrow \mu$ and $\sigma_n \rightarrow \sigma$, if $X_n \sim N(\mu_n, \sigma_n^2)$ and $X ...
0
votes
1answer
69 views

Bernoulli process failures rate

I have seen an unproved claim, which states that given an infinite Bernoulli process with probability $p$ of success, for every $c<p$, the probability that at any given time the success rate is ...
0
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1answer
165 views

What is the maximum entropy distribution of points on a sphere that has a fixed non-zero average cosine of the polar angle?

Suppose we have a unit vector in 3D space whose orientation has some unknown distribution $p(\theta,\phi)$. All we know about this distribution is the average value of $cos(\theta)$: ...
2
votes
1answer
638 views

Going from the Poisson distribution to the Gaussian.

In this lecture, at about the $37$ minute mark, the professor explains how the binomial distribution, under certain circumstances, transforms into the Poisson distribution, then how as the mean value ...
1
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1answer
71 views

Joint Distribution applied to quadratic equations.

If i pick random number $b$ and $c$ from $[0,1]$ and then define $p(x) = x^2+bx+c$, what is the probability that p has two real roots?. I've been thinking that it would be enough to know $P\{4c < ...
0
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2answers
78 views

Probability distribution of the sum of $N$ values from a set of values numbered $1$ to $K$

I have been trying to figure out how to determine the probability distribution function for the sum of $N$ values taken from a set of $K$ consecutive values (valued $1$ to $K$). For example, if I ...
1
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0answers
80 views

Characteristic functions of group-invariant probability distributions

Suppose that we have a probability distribution $\rho(\mathbf x)$ on a manifold $\mathcal M$, which is invariant under the action of a Lie group $G$, $\rho(g\mathbf x)=\rho(\mathbf x)$ for all ...
1
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1answer
1k views

Probability distribution of defective parts

Suppose there are 1 million parts which have 1% defective parts i.e 1 million parts have 10000 defective parts. Now suppose we are taking different sample sizes from 1 million like 10%, 30%, 50%, 70%, ...
0
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1answer
49 views

Trouble simplifying the pdf of minimum exponentially distributed r.v.

Given the following: $ X_i \sim EXP(1, \eta) $ Asked: show that $Q=X_{1;n}-\eta$ is a pivotal quantity. My approach: $\ \ \ f(x)=e^{-(x-\eta)} \Rightarrow F(x)=\int_{-\infty}^x ...
1
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1answer
81 views

Gebelein's Inequality and convergence of distribution

We know that for a bivariate standard normal vector $Z=(Z_1,Z_2)$ it holds that \begin{align*} \operatorname{Cov}(1\{Z_1\leq u),1\{Z_2\leq u))\leq \operatorname{Cov}(Z_1,Z_2). \end{align*} This ...
2
votes
1answer
326 views

Affine transform of multivariate gaussian

If $X_1, \ldots, X_n$ are iid $N(0,1)$ or in other words $\mathbf{X}=(X_1, \ldots, X_n)$ is distributed $N(\mathbf{0}, \mathbf{I})$, then $A\mathbf{X}+\mu$ is distributed $N(\mu, AA^t)$. Showing that ...
3
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1answer
223 views

Geometric distribution with unequal probabilities for trials

I am researching an engineering problem in which I want to model the probability distribution of the number X of independent trials needed to get one success. If the probability of success at each ...
0
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2answers
40 views

Common distribution with sharper edges?

Normal probability distribution is common, but it has only mean and stdev parameters being well defined. There are no singular points which could serve as minimum and maximum (all suggestions like 3 ...
3
votes
1answer
336 views

Physical meaning of “probability density”

Is there some way of describing the co-domain of probability density functions? Does it relate in some way to something physically meaningful? I was given that question today - and I was at a loss. ...
3
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2answers
139 views

That Brownian Motion's increments are gaussian is “not surprising”?

In section 1 of chapter 1 of Continuous Martingales and Brownian Motion, the authors claim that the fact that the increments of of Brownian motion are gaussian random variables "is not ...
1
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0answers
50 views

How to conserve probability using a numerical integration scheme?

I have an iterative operator that conserves probability given by $P_{n+1}(z_j) = \int_a^b P_B(x+z_j)P_n(x) dx$, where $P_n$ is the PDF at time step $n$ and $P_B$ is a PDF that is fixed with compact ...
1
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0answers
115 views

Normal distribution inequality

Let $n(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) = \int_{-\infty}^x n(t)dt$. Prove the following inequality. $$(x^2+1)N + xn-(xN+n)^2>N^2$$ where the dependency of $n$ and $N$ on ...
3
votes
1answer
153 views

Can I identify the distribution of a random variable given a related distribution function?

Let $X_1$, $X_2$ be i.i.d random variables. Suppose we know the distribution function of $X:=|X_1-X_2| =\max\{X_1,X_2\} - \min\{X_1,X_2\}$. Can we find the distribution of $X_1$? I realize that ...
-1
votes
1answer
204 views

Jensen inequality

Does Jensen inequality, which is $\mathbb{E}(g(x)) \geq g(\mathbb{E}X)$ if $g$ is convex, assume that $\mathbb{E}X$ (expected value of random variable $X$) must belong to $R(X)$ (range of random ...
3
votes
3answers
170 views

Derivative of the maximum of two random variables

For any two real numbers $a$ and $b$ and any two random variables (with no mass points in their distributions) $x$ and $y$, why is it that the derivative of $E[\max\{a+x,b+y\}]$ with respect to $a$ is ...
0
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3answers
78 views

I do not understand this integral,please help…

$$\int_0^{\infty} P(y > z) \, dz = \int_0^{\infty} \int_z^{\infty} h(y) \, dy \, dz = \int_0^{\infty} \int_0^y \, dz \, h(y) \, dy$$ Why do we have the last equality? I used Fubini and derived the ...
1
vote
1answer
42 views

Kronecker delta for multivariate distributions?

I have found a formula (Theorem 2.1 here: http://arxiv.org/pdf/0905.4131v1.pdf) which shows the covariance matrix of a multivariate normal distribution $\Sigma_P$, but I'd like some help interpreting ...
2
votes
1answer
356 views

Convergence in distribution of Gaussian processes

Assume given a sequence $(W_n)$ of Gaussian processes indexed by, say, $\mathbb{R}^p$, with mean zero and covariance function $R_n$. This means that for each $n$, the finite-dimensional distributions ...
2
votes
2answers
136 views

Comparing the relative entropies of some stochastically ordered distributions

Motivation of this question: This question is related to the expected stopping time of a stochastic process under two hypotheses. Especially, it answers the question "how many more samples are ...
0
votes
1answer
92 views

What is the probability distribution for the number of trials between successes?

I am a bit of a novice at statistics, so please forgive me for sounding dumb. I was looking for a probability distribution describing the distribution of the number of trials between successes. I ...
2
votes
1answer
192 views

SOA Exam P Question: Exponential Distribution

Here is an Exam P problem as I have it. That is, it was passed down to me from someone else and I am unsure if the wording is exactly as it was originally posted. I've tried searching for this ...
0
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1answer
69 views

What is the meaning of this exercise?

The daily quantity demanded of unleaded gasoline in a regional market can be represented as Q=100-10p+E, where p belongs to [0,8], and E is a random variable having a probability density given by ...
4
votes
1answer
310 views

Is there a way to standardize the Poisson distribution?

For example, a variable of Normal distribution, $T$, with mean $\mu$ and variance $\sigma^2$ can be standardized into $S$ like this: $$ ...
2
votes
1answer
94 views

Series of continuous random variables is continuous

We work on the usual $(\Omega,\mathscr{F},P)$. Suppose $X_i$ are independent random variables. Say the distribution of $X_i$ is $F_i$. Under what circumstances can I guarantee that $\sum_{i=1}^\infty ...
1
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2answers
64 views

Is the quotients of a group of triangular distributed numbers still following a triangular distribution?

I have a group of numbers (about 10000 numbers) between 0.8 and 1.0 which follows simple triangular distribution (for example, lower limit: 0.8, upper limit: 1.0, mode: 0.9). If I divide 2 by each ...
1
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1answer
66 views

How $\log p$ distributed?

From literature we know: If a number $n \le x$ is chosen at random, and choose $\lambda \ge 0$ and $j$ not too large (say $\lambda ,j \le 20$) then the number of primes in $[ n , n + \log(n) ]$ is ...
0
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1answer
101 views

Chernoff type Sum of independent random variables having exponential tails

Say I have n independent variables $\{X_1,X_2 \dots X_n\}$ with Expectation 0 such that $Pr(|X_n| > \alpha) < e^{-\lambda \alpha}$. Can we produce chernoff type inequalities for the sum of these ...
3
votes
1answer
199 views

Maximum Entropy Distribution When Mean and Variance are Not Fixed with Positive Support

I know when the mean and variance of $\ln x$ are both fixed, then the maximum entropy probability distribution is lognormal. When the mean of a random variable is fixed the MEPD is the exponential ...
0
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1answer
76 views

Does this discrete distribution have a name?

I encountered a discrete probability distribution $P_\lambda$ on $\mathbb N$ with $\lambda \in (0,1]$. It is of the form $$P_\lambda(n) = c \cdot \prod_{i=0}^{n} \frac{\lambda^{i}}{1-\lambda^{i+1}}$$ ...
0
votes
1answer
52 views

Range changes for functions of stochastic variables.

I have the stochastic variables $X=U~(-1,1)$, and $Y=2X^2+1$. I need to find the cdf of $Y$ ($F_Y(y)$). I have reasoned like this: $$ F_y(y) = P(Y<y) = P(2X^2+1<y) = P(-\sqrt{\frac{y-1}2} < X ...
1
vote
1answer
144 views

Convergence in distribution of the log-Gamma distribution

Suppose $X$ has density $f(x)=\exp(kx-e^x)/\Gamma(k)$, $x>0$, for some parameter $k>0$. Then the moment-generating function of $X$ has the form $$ M_X(\theta)=\frac{\Gamma(\theta+k)}{\Gamma(k)}. ...
2
votes
3answers
111 views

CLT for random variables with varying distributions

The following is not a homework problem. Here it is: Suppose $\{X_k\}_{k=1}^{\infty}$ is a sequence of independent random variables such that $$ P(X_k=1)=P(X_k=-1)=\frac{1}{2}-\frac{1}{2k^2} $$ $$ ...
1
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2answers
65 views

Can a probability density function be used directly as probability function?

This might be something basic but it confuses me greatly. I am reading a literature, where they use the probability density function of a Gaussian distribution, that is ...
2
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1answer
94 views

Variant of the hypergeometric distribution: throwing balls back into the urn

Following the notation from WP:hypergeometric distribution. We have an urn with $K$ red and $N-K$ white balls. When we find a red ball, we keep it, removing it from the population. But when we find a ...
0
votes
1answer
41 views

Exponential distribution for the lifetime of an LCD screen: probability that it functions for $50{,}000$ hours

If the probability that an LCD screen functions for $x$ hours is defined by the density function: $$f(x)=0.01*\exp(-x/100)I_{[0,\infty)}(x)$$ where $I$ is an indicator function and $x$ is measured ...
-1
votes
3answers
958 views

Find the probability using the poisson distribution?

Grandma bakes chocolate chip cookies in batches of 100. She puts 300 chips into the dough. When the cookies are done, she gives you one. What is the probability that your cookie contains at least 2 ...
5
votes
1answer
140 views

Survivor function of a variable that has discrete and continuous components

I'm currently reading The Statistical Analysis of Failure Time Data by Kalbfleisch and Prentice and had trouble at arriving at the expression for the survivor function of a random variable $T$ having ...
3
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2answers
830 views

Binomial theorem in probability

We know according to binomial probability theorem , $$P= \binom{n}{r} p^r (1-p)^{n-r} \tag{1}$$ Now If I flip a coin 10 times and want to get the probability for 4 heads then we get according to the ...
1
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0answers
90 views

What is the intuition behind $P(X={\lambda})\ {\equiv}\ P(X={\lambda}-1)$ for a Poisson distribution

Where $X_\lambda$ is a Poisson random variable with mean $\lambda$,$$P(X_\lambda=k\in\mathbb{N}) = \frac{\lambda^{k}e^{-\lambda}}{k!} $$ When it happens that $(\lambda-1)\in\mathbb{N}$, ...