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

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2answers
38 views

Uniqueness of moments for probability distributions with infinite moments.

I was taught the collection of a distribution's moments uniquely defined the distribution. Recently, I have been studying Pareto distributions, which have infinite means for shape parameters less than ...
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1answer
2k views

How do I find if the probability of the sample proportion is greater than something?

I have this problem and I have no clue how to solve it. In 2012, 31% of the adult population in the US had earned a bachelor’s degree or higher. One hundred people are randomly sampled from the ...
9
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1answer
128 views

Math Intuition and Natural Motivation Behind t-Student Distribution

I am trying to understand with basic mathematical background how the $t$-Student distribution is a "natural" $pdf$ to define. So I hope that this not too-general a question, but given that the ...
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1answer
37 views

Probability that 5 out of 7 bulbs will produce white flowers

If you have a bag with 25 tulip bulbs that will grow into white, yellow or red flowers. You want to plant 7 bulbs. Each bulb in the bag, independently of the others, grows into a white tulip ...
2
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1answer
30 views

stochastic dominance definition

I was wondering if, for positive random variables $X$ and $Y$, $\Pr(X\geq Y)\geq 1/2$ implies $\Pr(X\geq x)\geq \Pr(Y\geq x)$. Intuitively it "makes sense", since $X$ tends to be more often bigger ...
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1answer
44 views

$\chi^2$ distribution Stoch. increasing in non-centrality parameter

i.e for fixed $\nu>0$ if we have $\gamma_2 > \gamma_1>0$ then $\chi^{2}_{\nu}(\gamma_2)\succeq\chi^2_{\nu}(\gamma_1)$ where '$\succeq$' denotes stochastically larger. The convention that I ...
4
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1answer
78 views

How long before the prey can escape?

I've (sort of) come across the following problem in my research. The actual scenario is a little abstract to explain, so I'm rephrasing the problem in terms of a predator/prey scenario. I'm tagging ...
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1answer
33 views

$\frac{\chi^2_n}{n}$ Stochastically increasing in $n$?

I was wondering whether $\frac{\chi^2_n}{n}$ is stochastically increasing in $n$. My main problem: Suppose $\hspace{5pt}\frac{(n-p)\hat{\sigma}^2}{\sigma^2} \sim \chi^2_{n-p}$. Then the expected ...
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1answer
26 views

Given the moment generating function of a continuous-type r.v, how to find the p.d.f?

Say for $t<1$: $$M(t) = \frac{1}{(1-t)^2}$$ How to find the p.d.f of the random variable? $$M(t) = E(e^{tx})=\int_{-\infty}^{+\infty}e^{tx}f(x)dx$$ How do we find: $f(x) = xe^{-x}$ on ...
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1answer
32 views

Proving that the variance is non-negative

$$(E(X))^2 = \left( \int_{-\infty}^{+\infty}xf(x)dx \right)^2 \le \int_{-\infty}^{+\infty}x^2(f(x))^2dx \le \int_{-\infty}^{+\infty}x^2f(x)dx = E(X^2)$$ Because of cauchy-schwarz inequality and $f(x) ...
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3answers
43 views

Distribution of sum of random variables

Let $X_1, X_2, . . .$ be independent exponential random variables with mean $1/\mu$ and let $N$ be a discrete random variable with $P(N = k) = (1 − p)p^{k-1}$ for $k = 1, 2, . . . $ where $0 ≤ p < ...
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2answers
38 views

Neighbor Interaction in a Random List

Assuming a random arbitrarily long list where each element has a $50\%$ chance of being a $0$ or a $1$, such as: $0001101101$ What is the chance of having a neighbor that isn't the same? For ...
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1answer
26 views

Can the independence of random variables hold for their functions?

Suppose $X$ and $Y$ are two independent continuous random variables on $\mathbb{R}$. Define: $f:\mathbb{R}\mapsto\mathbb{R}$ as a $C^\infty$ map on $\mathbb{R}$. Then is it possible to find the ...
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1answer
516 views

Moment Generating Function of Gaussian Distribution

Derive from first principles, the moment generating function of a Gaussian Distribution with $$PDF= \dfrac{1}{\sqrt{2\pi \sigma^2}}e^{-(x- \mu)^2/2\sigma^2}$$ MY ATTEMPT MGF= E[$e^{tx}$]= ...
2
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1answer
67 views

How to find the density of $Y=g(X)$ in this case?

I have a vector $X=(1,X_2,X_3)$, where $(X_2,X_3)$ is a random vector in $\mathbb{R}^2$. Now consider $Y=g(X)=X/\|X\|$. What is a density function of $Y$ with respect to the uniform spherical ...
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0answers
12 views

Approximate CDF of arbitrarily aggregated random variable

I would like to know if my solution for the following is mathematically correct in general: I have a random variable $Z$ that is an arbitrary function of two other rvs $X$ and $Y$, so: $Z = f_{arb}(X, ...
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1answer
14 views

Variance of Transformed Random Vectors

Consider an $n$-dimensional normal random vector $\mathbf X:= (X_1, \dots, X_n)^T$ with mean $\mathbf 0$ and covariance matrix $\mathbf \Sigma$. Now define a new random vector $\mathbf Y:= (a_1X_1, ...
0
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1answer
34 views

Finding expected number of trials until we get head given density function?

Suppose we flip a coin with a random probability of Heads $P$ that has density $f(p) = 6p(1−p),\; p \in [0, 1]$. If we keep on flipping this coin until we get a single Heads, what is the expected ...
0
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2answers
37 views

Get the distribution of $X|Y=y$ given this joint probability density function

Given the joint probability density function $f(x,y) = \lambda^2 \exp(-\lambda y)$ with $0 < x < y.$ How do I get the distribution of $X|Y=y$ ? Thanks in advance!
3
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1answer
150 views

Find a probability density

I am going through a paper trying to understand all the single steps, but I got stuck. I need to calculate $$p(x+\delta t) \mid x(t), t)= \int p(x(t+\delta t) \mid \mu , x(t), t)p(\mu\mid x(t), t) ...
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1answer
60 views

Mixture of Discrete Binomial Distributions

Let $B\left(p,N\right)$ be a Binomial distribution with parameters $p$ and $N$. We define a Mixture of Discrete Binomial Distributions by $\left\{ \left(B\left(p_{i},N\right),\alpha_{i}\right)\right\} ...
3
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1answer
30 views

Cancellation law of equal in distribution

I came across this gem while discussing with my friends, If $X$ and $Y$ are two real valued random variables (not necessarily independent) that satisfy $$X =^d X+Y$$ (where $=^d$ means equal in ...
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0answers
84 views

Of strings and substrings: A problem of probability

Problem Let $\Sigma=\{a, b\}$. Let $\Sigma^*$ denote the Kleene star of $\Sigma$: \begin{equation*} \Sigma^* = \{\varepsilon, a, b, aa, ab, ba, bb, aaa, aab, \ldots\} \end{equation*} where ...
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1answer
315 views

Joint probability of sum of two random variables and one of its terms

Let $X$ and $Y$ be two independent random variables (weibull distributed) and $Z=X+Y$. I am trying to find $\mathbb P\big(Z\geq z ~\cap~ X\leq x\big)$. I know that $$ \mathbb P(Z\geq z \cap X\leq x) = ...
2
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1answer
29 views

Degree of Polynomial in Centered Moments of Gamma$(n,1)$

I'm interested in the degree of the polynomial in $n$ of the expression for the $k$-th central moment $$ E((X_n - n)^k) $$ where $X_n$ is a Gamma$(n,1)$ random variable, that is, the sum of $n$ ...
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1answer
33 views

Interchange of limit and integral for a positive random variable with finite moments.

Let $f$ be the pdf of a non-negative random variable $X$ with finite moments of all orders, i.e. $E[X^n]<+\infty$ for all $n\in \mathbb N$. May I interchange the limit with the integral and infer ...
0
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1answer
56 views

Find the CDF of a function of two random variables

The joint probability density function of two continuous random variables $X$ and $Y$ is: $$f(x,y) = \begin{cases} 6x,& 0\leqslant x\leqslant y,\ 0\leqslant y\leqslant 1\\ 0,& \text{ ...
3
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1answer
100 views

Warren's proof for Benford's Law

Warren has a little proof of Benford's law in Hacker's Delight. To quote: Let $f(x)$ for $1 \leq x < 10$ be the probability density function for the leading digits of the set of numbers with ...
2
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1answer
52 views

$p$-stable Random Variables for $p>2$?

I will preface this by saying I am certainly no expert in Probability theory. My actual problem is an interpolation one, in which I am considering interpolation of bandlimited functions with shifts ...
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2answers
28 views

Weighted sum of identical distributed random variables

Suppose $X_1$, $X_2$, $\ldots$ ,$X_N$ are identically distributed (not necessarily independent). Then, given $a_1+a_2+\ldots+a_N=1$, and let $S=a_1 X_1 + a_2 X_2 + \ldots + a_N X_N$. Does $S$ follow ...
2
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2answers
93 views

Convergence in distribution for $\frac{Y}{\sqrt{\lambda}}$

Given a sequence of independent r.v's $\{X_n\}_{n\geq 1}$ such that $P(X_n=x)=\frac{1}{2}$ if $x=-1$ and/or $x=1$ Let $N\in Po(\lambda)$ be independent of $\{X_n\}_{n\geq 1}$ and we set that ...
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1answer
54 views

Cards probability problem

Two players; the dealer and a player. The player is given three cards face down. The dealer turns over a 2 (let's say of hearts). Before the player turns any cards over, what is the probability that ...
6
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1answer
82 views

Multivariate normal density function of function of random variable

Let $X_1,\dots,X_n$ be i.i.d random variables and $g$ be a symmetric function such that $$g(X_i,X_j)\sim N(\mu,\sigma^2)$$ for all $1\le i<j\le n$. I wish to know the density function of the joint ...
4
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1answer
139 views

Central Limit Theorem is incorrect - where is my mistake?

Say I flip a coin 80 times and I ask for the probability to get over 48 heads. I then flip a coin 800 times and ask for the probability to get over 480 heads. Translating this into Central Limit ...
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1answer
47 views

Probability and Name of Distribution

Suppose $X$ has PDF $f_X$ given by \begin{align*} f_X (x) = \begin{cases} \frac{\alpha x_0^\alpha} {x^{\alpha+1}} &\text{if $x ≥ x_0$,}\\ 0 &\text{if $x < x_0$,}\end{cases} \end{align*} ...
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0answers
34 views

Transition density of an AR(1) process?

If we have an AR(1) process, i.e: $X_{t+1} = \alpha X_t + e_{t+1}$ with $X_0=0$ then what is its Markov Chain transition density? We know that for a Markov chain, the following holds: $P(X_{t+1}\leq ...
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1answer
39 views

Moment generating functions…which distributions to use?

Q: You hired a terrible programmer and the moment generating function for the distribution of software bugs is M(t) = (1 - $\theta$t)$^{-\alpha}$. Groups of bugs can be detected within $\mu$ = 47 ...
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0answers
28 views

Weighting the data by the history

I have a input stream 3D data that comes every time frame. Each point is defined by 3D vector of x,y,z. There is a evaluation function [say f(x)] that computes if the point at time t is valid or ...
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1answer
33 views

Normal distribution calculations

We have a gaussian distribution $$ X \sim N(\mu,\sigma^2)$$ where $\mu = 4$ and $\sigma^2 =1.5$ . Probability is given by : $P(x<c)=0.35$ $c$ needs to be calculated. And we got ...
6
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1answer
64 views

Convergence of $n^{-\gamma}T$ where $T$ a hitting time for uniform rvs, can I use CLT?

Let $X_1,X_2,\dots$ be iid uniform on $\{1,\dots,n\}$ and define $T=\inf\{k:X_k=X_r \text{ for some }r<k\}$. The objective is to figure out when $n^{-\gamma} T$ converges weakly to some ...
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1answer
20 views

Linear regression relationships

Velocity $= X$, distance to stop $= Y$ $\beta_0= -17.5791$, $\hat{\operatorname{se}}(\beta_0)=6.7584$ $\beta_1 = 3.9324$, $\hat{\operatorname{se}}\beta_1 = 0.41.55$ degrees of freedom $=48$ (a) is ...
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1answer
17 views

What am I plugging in wrong to my normal distribution calculator?

I am trying to find the probability of the following question: Cans of regular Coke are labeled as containing 12 oz. Statistics students weighed the contents of 7 randomly chosen cans, and found the ...
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1answer
27 views

Distribution Technique Question of two independent Exponential Distributions

If $X_1$ and $X_2$ are two independent random variables having exponential densities then $f(x_1,x_2)$ is defined as $$f(x_1,x_2)=\exp(-(x_1+x_2))\,{\bf 1}_{(0,\infty)}(x_1){\bf ...
1
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1answer
55 views

Runs of white balls in sampling without replacement

There are $m$ white balls and $n$ black balls in a box. Balls are randomly drawn from the box with no return. Denote $X_1$ : number of white balls that been drawn before the first black. For $2 \leq i ...
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1answer
33 views

Seeking an example for Bayes estimator of two unknown parameters

I searched the web, taking advantage of several search approaches; however, due to redundancy of the existing information about Bayes estimator of one unknown parameter of random variables (either in ...
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1answer
535 views

Problem on continuous probability distribution

Problem:We pick two random numbers, x and y, between 0 and 2. What is the probability that x*y<1 AND y/x<1. I am familiar with continuous probability distributions for one variable, but it ...
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2answers
37 views

Find the required Chi-square score for an arbitrarily low p-value (2 degrees of freedom)

I'm trying to use the Chi-Square test to find the significance of data that suffers from the multiple testing problem. Because I have this multiple testing problem, the required p-value to view a test ...
2
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1answer
57 views

Intuition for probability density function as a Radon-Nikodym derivative

If someone asked me what it meant for $X$ to be standard normally distributed, I would tell them it means $X$ has probability density function $f(x) = \frac{1}{\sqrt{2\pi}}\mathrm e^{-x^2/2}$ for all ...
2
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2answers
70 views

Probability distribution of number of waiting customers in front of a counter [closed]

The number of customers arriving at a bank counter is in accordance with a Poisson distribution with mean rate of 5 customers in 3 minutes. Service time at the counter follows exponential distribution ...
1
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2answers
39 views

Derivation of mean and variance of Hypergeometric Distribution

I need clarified and detailed derivation of mean and variance of a hyper-geometric distribution. If a box contains $N$ balls, $a$ of them are black and $N-a$ are white, and $n$ number of balls are ...