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

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32 views

If $X$~$U(0,1)$, and $Y=2x-4$. What is the density function of Y?

If $X$ is uniformly distributed $\mathcal{U}(0,1)$ , then what is the distribute density function of $Y$? I thought that if $$fx(x) = 1/(1-0), \; \mbox{for} \; 0<x<1$$ then ...
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0answers
14 views

Understanding compound distribution and plotting the mixed distribution graph

If $N$ takes the values $0, 1$ and $2$ with probabilities $½, ¼ $ and $¼ $ respectively, and the $X_i$ ’s have a $U(0,10)$ distribution, draw a sketch of the frequency distribution of $S$. $N$ is the ...
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0answers
41 views

Asymptotic conditional distribution of $\bar{Y}\mid\bar{X}=x$

I'm reviewing for my qualifying exam and I'm stuck on part of a problem. Setup Suppose that $(X,Y)$ are two random variables with joint distribution $ \begin{equation} ...
3
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2answers
78 views

Let $Y=1/X$. Find the pdf $f_Y(y)$ for $Y$.

The Statement of the Problem: Let $X$ have pdf $$f_X(x) = \begin{cases} \frac{1}{4} & 0<x<1 \\ \frac{3}{8} & 3<x<5 \\ 0 & \text{otherwise} \end{cases}$$ (a) ...
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1answer
50 views

Malalanobis distance between two multivariate Gaussian distributions

Let $\mathbf{x}\in\Bbb{R}^n$ be an $n$-dimensional real vector distributed normally with mean vector $\mu\in\Bbb{R}^n$ and covariance matrix $\Sigma$; i.e. $\mathbf{x}\sim\mathcal{N}(\mu,\Sigma)$. ...
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2answers
16 views

probability mass function of a random pick from either of the other two random entities.

This is not a hypothesis testing problem. Let $Z$ be discrete non-negative random variable, such that it picks either the value of random variable $X$ with probability $p$ or the random variable $Y$ ...
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0answers
14 views

What are some error measures used for fitting PMFs?

I have a given PMF, $f_X(x)$, and am trying to create a fitted PMF, $g_X(x)$, that comes "as close as possible" to it, but am not sure what to use as a measure of fit. Simply minimizing standard error ...
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1answer
36 views

Multinomial Distribution- expected number

Suppose you have a box with five balls of different colors. If you draw a ball 100 times and replace it, what is the expected number of different colors you would have after 100 trials?
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29 views

What is the proof of $\mathbb{E}\Phi (X) = \Phi\left(\mu\sqrt{\frac{1}{1+\sigma^2}}\right)$, where $X \sim \mathcal{N}(\mu,\sigma^2)$?

Let $X \sim \mathcal{N}(\mu,\sigma^2)$. I think it's true that $$\mathbb E \Phi(X) = \Phi \left(\mu\sqrt{\frac{1}{1+\sigma^2}}\right)$$ where $\Phi$ is the cdf of standard normal. This holds up ...
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1answer
73 views

Calculating Probability (P) given Z bounds.

I need to program a simple Probability calculation function for any given Z boundaries (Area P under the normal distribution curve): I know we can use the The Z table, but I want to actually ...
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1answer
22 views

How does the probability change when changing the decission?

Let's say you play roulette and you set 2 coins of 2 of the 3 columns or dozens. Then your chance is 2/3rd, so about 66% to win one coin. Let's assume that you loose because the other dozen or column ...
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1answer
33 views

What is the probability of occurence of natural numbers?

Suppose that humankind had a ∞-ary number system so that no psychologically distinguished numbers like 1000, 250, or 333 existed. What is the probability of a number n ∈ ℕ (including 0) to occur when ...
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1answer
16 views

Bayes' Rule for Parameter Estimation - Parameters are Random Variables?

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $\mathbf{X}: \Omega \to \mathbb{R}^n$, $\mathbf{Y}: \Omega \to \mathbb{R}^m$ be jointly continuous random vectors. That is, there exists ...
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0answers
20 views

finding two Zipf distributions out of one Zipf distribution

$\newcommand{\pop}{\operatorname{pop}}$Assuming $ G = \{g_1, g_2, g_3, \ldots , g_{N_G}\}$ is set of all the globally existing data objects. The total request rate for $G$ is shown by $\lambda_G$. The ...
2
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1answer
52 views

Formula to find possible number of combinations

A committee of 5 is to be selected from a group of 6 men and 9 women. If the selection is made randomly, what is the probability that the committee consists of 3 men and 2 women. We can solve ...
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1answer
25 views

Derive State Transition Matrix from an unscented transformation

I have an application where I am using an unscented Kalman filter to process data. While the unscented transformation eliminates the linearization assumption used with the typical state-transition ...
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1answer
53 views

How to perform mathematical operations using mean and standard deviation.

It's given that a particular parameter, say base-time $T_b$ follows a lognormal distribution with the mean of $10$ years and the standard deviation of $5$ years. Now, how do I estimate the value for ...
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1answer
15 views

Obtaining a distribution of values from an equation

Given any equation and range, for example, $y = x^2 + x$ where $x$ is a value from $0$ to $1$ (inclusive) Is it possible to determine the distribution of values outputted by this function between a ...
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2answers
40 views

Show $Cov(X+Y, Z+W)=Cov(X,Z)+Cov(X,W)+Cov(Y,Z)+Cov(Y,W)$

Show $cov(X+Y,Z+W)=cov(X,Z)+cov(X,W)+cov(Y,Z)+cov(Y,W)$ I want to use this theorem I just proved to show this: $cov(X\pm Y, Z)= cov(X,Z) \pm cov(Y,Z)$ For the previous proof I re-wrote both sides in ...
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1answer
36 views

Difference of Ordered Uniform Random Variables

Let $X_1, X_2,..., X_n$ be $n$ random variables distributed uniform(0,1) and $X_{(1)},X_{(2)},..., X_{(n)}$ be the ordered statistics of $X_1,...,X_n$ such that: $X_{(1)} < X_{(2)} < ... < ...
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1answer
30 views

Compound of two exponential distributions

What is the distribution of a exponential distribution, whose expected values is drawn from the expontial distribution $$X\sim\mathrm{Exp}(\text{mean}=\alpha) f(x\mid α) = (1/α) \exp(-x/α) $$ ...
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1answer
22 views

A doubt regarding KL-divergence for estimating a distribution

Suppose given a probability distribution $q$. I'm trying to estimate it by $p$ such that the KL-divergence between $p$ and $q$ is minimized. Now which one of the two: $KL(p||q)$, $KL(q||p)$ should be ...
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1answer
39 views

Distribution of the maximum of absolute value of multivariate Gaussian

I am currently working on some simulations. However, I encounter a statistical problem as following. Suppose $ 0 < t_1 < t_2 < \dots < t_m < 1 $ and $ B(t) $ denotes Brownian bridge. ...
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1answer
29 views

Understanding degrees of freedom for the chi-square component of the t-distribution

In looking at the t-distribution of $\sqrt{n}(\bar{x}-\mu)/S$, I can see that this equates to the following: $$\frac{\sqrt{n}(\bar{x}-\mu)}{S} = \frac{(\bar{x}-\mu)}{\frac{S}{\sqrt{n}}} = ...
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1answer
44 views

Hypergeometric distribution exercise!

A store has $20$ guitars in stock but 3 are defective. Claire buys $5$ guitars from this lot. (a) Find the probability that Claire bought $2$ defective guitars. I use $N=20,n=5, k = 3,x=2$ where $N$ ...
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1answer
28 views

Bernoulli trials with at least 1 success and 1 failure

Independent Bernoulli trials are performed, with probability $1/2$ of success, until there has been at least one success. Find the PMF of the number of trials performed. How is this different from ...
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1answer
29 views

Let X and Y be continuous random variables with joint PDF of the form $f(x,y) = c(x+y)$. Find the joint CDF

Let $X$ and $Y$ be continuous random variables with joint pdf of the form $f(x, y) = c(x+y)$ $0 < x < y < 2$ and zero otherwise. a. Find c so that f(x, y) is a joint pdf. I answered this ...
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2answers
32 views

Are there two different notions of “conditional probability”?

This question comes from reading the discussion here. (1) If one is given a "probability measure" $P : F \rightarrow [0,1]$ mapping a Borel $\sigma$-algebra $F$ to $[0,1]$ then for two ``random ...
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2answers
50 views

Probability that one chi-squared random variable is less than other chi-squared random variable

I have two random variable $X=\mathcal N(\mu,\sigma^2)$ and $Y=\mathcal N(0,\sigma^2) $ independent to each other. Now, $Z=X^2$ and $W=Y^2$, are chi-square random variable with first degree of ...
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0answers
21 views

The uniqueness of solution of an equation that involves CDFs

I have two monotone CDFs $F(x)$ and $G(x)$. The functions are symmetric in a sense that $F(x)=1-G(1-x)$, $f(x)=g(1-x)$. I am trying to show that equation $xF(2x)+(1-x)G(2x)=1/n$, $n\geq2$ has a unique ...
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1answer
13 views

Cumulant-Legendre

I have a short question: So suppose $b=\text{ess sup} X<\infty$, where $X$ is a random variable on $\mathbb{R}$. Now take $\Lambda (u)=\ln \mathbb{E}[e^{uX}]$, the cumulant, and ...
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1answer
19 views

Let X and Z form a random sample from a poisson dist.If Y=min( X,Z), what is P(Y=1)??

Let X and Z form a random sample of poisson distribution and define Y=min( X and Z) What is P(Y=1)?? I think Y is minimum of two. If X=1, then Z can be any number except 0 If Z=1, then X can be ...
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2answers
34 views

The size of a fish in a lake follows a normal distribution

I have a homework question that I wasn't positive about. This is the first probability course I have taken and the class is only taught using excel so I apologize for the lack of formulas in my ...
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1answer
22 views

Lifetime of light bulbs is modeled as a Poisson Process - using excel

I have a homework question that I can't seem to figure out. Any help is appreciated! The lifetime of light bulbs (in days) is modeled as a Poisson Process with expected lifetime of beta = 200 days. A ...
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1answer
17 views

Using PMF and CDF to calculate probability

Given the following CDF what is $$P(T > 3)$$ and according to my answer key it's 1-1/2 = 1/2. Can someone explain to me why it is 1-F(3), and would subtracting F(3) be subtracting 4 as well? ...
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1answer
26 views

Comparing Percentiles of 2 Samples Drawn from the Same Distribution

Suppose I have two sets of numbers: $A=\{a_1,a_2,...a_{N_1}\}$ and $B=\{b_1,b_2,...b_{N_2}\}$ with $N_1<N_2$. WLOG assume that $a_i<a_j$ for all $i<j$ and similarly for $b_i$ and $b_j$. ...
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2answers
85 views

Distribution of minimum absolute value

Consider $K$ independent Laplace variables $X_k, k=1,\ldots,K$, with mean 0 and scale $\lambda$ (so that their PDF is $f(x)=\frac{1}{2\lambda}e^{-\frac{|x|}{\lambda}}$. Let $Y$ be the variable taking ...
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1answer
22 views

Given Nd6, what is the probability that the two highest are minimum 4?

So, my statistics knowledge is rather poor, so I would welcome a formula explanation to the question: given Nd6 (6-sided dice) what is the probability that the two highest numbers are at least a 4? ...
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2answers
44 views

Does the distribution of a process on $\mathbb{R}^{[0,\infty)}$ uniquely define it?

Question: Can I have two different stochastic processes $(A_t)_{t \in [0, \infty)}$, $(B_t)_{t \in [0, \infty)}$ having the same distribution on $\mathbb{R}^{[0, \infty)}$ differ in some ways? ...
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2answers
18 views

Probability of X given the sum

I am given that $X \sim P(\lambda)$, $Y \sim P(\gamma)$, and told to calculate the distribution of $P(X | X+Y = n)$ I proceed as follows $$ \begin{equation} \begin{split} P(X=i|X+Y=n) &= ...
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1answer
32 views

Exponential distribution question!

Suppose that the time between calls from your best friend has an exponential distribution with a mean time of $3$ days. (a) If you just received a call from her, what is the probability that you will ...
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1answer
25 views

How to show that $E(X^k)=npE((Y + 1)^{k-1})$ where $X\sim\mathrm{Bin}(n,p)$ and $Y \sim \mathrm{Bin}(n-1,p)$.

Show that $$E(X^k)=npE((Y + 1)^{k-1})$$ where $X\sim\mathrm{Bin}(n,p)$ and $Y \sim\mathrm{Bin}(n-1,p)$. I am looking for suggestions on where to start? Or any resources someone may have. I am not ...
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0answers
25 views

convolution of two probability density functions

Please no one call me dumb - I am not a mathematician and haven't done proper math for the last ten years. But I have a problem at work where I need to perform a convolution of two probability density ...
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0answers
26 views

distance distribution in Poisson point process

Consider a homogeneous Poisson point process in 2D space with density $\lambda$ per unit area. Let $\mathcal{B}(o,R)$ denote a disk centered at origin with radius $R$. Let $n$ be the number of points ...
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1answer
21 views

Finding this Probability Density Function

I would much appreciate if you help me out with this problem Let $X \sim Unif(0,1)$ Find the density of $Y = -\lambda^{-1} \log(1-X)$ with $\lambda > 0$ Then calculate $P(Y>t+s|Y>t)$ for ...
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0answers
36 views

Why does symmetry happen in reset-based random walks?

I am studying the basic concepts about random walks / brownian motion, and based on the idea of a Möbius-based walk in Wolfram's website, I wanted to try my own version of it in Python to compare it ...
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0answers
24 views

Number of same degree vertex pairs between two random graphs

I am considering the random graphs generated by the Erdős-Rényi model for this question. Random Graphs as Models of Networks by Newman is a reference on this topic. A random graph $\Gamma_{n,p}$ has ...
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2answers
24 views

Essential supremum via cumulant

Let $p(t)=\log \mathbb{E}[\exp (tX)]$ for $X$ real valued random variable. Now it holds (assuming that $p$ is smooth and finite on $\mathbb{R}$) that $p'(\infty)=\text{ess}\sup X$. How can I prove ...
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1answer
21 views

If $E(|X|)<\infty$, how do we show that it can be expressed as below

$F(x)$ is the distribution function of $\mathbb X$, and $f(x)$ is the derivation of $F(x)$, Prove that $\int_{0}^{\infty}(1-F(x))dx-\int_{-\infty}^{0}F(x)=E(X)$. Note that ...
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2answers
59 views

Why birthday distribution is not uniform. [closed]

I was reading about birthday problem and I found a statement that real-life birthday distributions are not uniform since not all dates are equally likely (last line ...