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

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71 views

Does this figure represent a cumulative distribution function?

Is this a c.d.f.? I have no problem for random variable $X$ at $-\infty<X<x_2$. But if p.d.f. were continuous in interval $x_2\leq X<\infty$ , then c.d.f. should have been continuous. If ...
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1answer
307 views

Why left continuity does not hold in general for cumulative distribution functions?

Definition: The c.d.f. $F$ of a random variable $X$ is a function defined for each real number $x$ as follows:$$F(x)=Pr(X\leq x) \text{ for } -\infty<x<\infty$$ Let ...
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1answer
219 views

Minimum and maximum of random variable

Let $X$ be random variable such that $\begin{align} F_X(x) = 1- e^{-x} \end{align}$ if $x \ge 0$ and $F_X(x)=0$ in other case. Find distribution function $Y= \min(1,X)$, $Z=\max(1,X)$. If I have to ...
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1answer
75 views

If X and θ are both random variables and θ is the parameter of the distribution of X, are X and θ independent?

The answer appears to be no because the distribution of X is defined conditionally by θ which is also assumed to have a distribution as opposed to be a constant. Essentially, the formulation of the ...
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1answer
293 views

How to describe a “sum percentile”

I have values $w_1 \ge w_2 \ge ..\ge w_n$. I want to know the the highest possible threshold $w^{th}$ so that $$ \sum_{i:w_i>w^{th}} w_i \ge \alpha \sum_{i=1}^n w_i $$ where $\alpha \in [0,1]$. ...
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1answer
41 views

distribution of the product of a poisson and a bolzmann

What is the distribution of the product of two variables for which each of them has its own distribution(specifically one poisson and one bolzmann)? I found on wikipedia that for the sum of the two ...
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1answer
27 views

Probability distribution of halving a segment

Suppose you cut a unit segment ("a stick") in half. I'd like to find out theoretically the distribution of the length of, say, the left piece based purely on some plausible assumptions, such as ...
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1answer
35 views

How to sample N points between 0 and R if they are exponentially distributed?

The density of my points x $\in$ [0,R] is exponential: $\rho(x) \approx e^x$ How can I sample N points from there? Thanks,
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3answers
265 views

Probability question: given $P(A|B)$ and $P(B)$ how do I find $P(A)$?

I have a probability distribution for some quantity $A$ given a fixed $B$, i.e. $P(A|B)$. I also have a prior distribution $P(B)$ for $B$. I'm trying to find the distribution $P(A)$. I had thought ...
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0answers
53 views

Problem showing a limit ( involving the limit that equals the exponential function )

I am to show that if $(p_n)_{n \geq 1}$ is a sequence in (0,1) and $X_n\sim Bin(n,p_n)$ and $n\cdot p_n \to \lambda$ for $n\to \infty$ with $\lambda\geq0 $. Then $$ X_n(P)=\mu_n ...
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1answer
68 views

Probability Density and Distribution of a Sphere

I am given that the density function for the radii of a sphere is constant over 0 < r<5 and zero elsewhere and am asked for calculate the density function f(r) and the cumulative distribution ...
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1answer
28 views

Probability of orderings

Let $t_1,\ldots,t_n$ be a set of $n$ intervals, in the form $[l_i,u_i]$. I define the precedence $t_i \succ t_j$ as the event in which a sample drawn from $t_i$ is greater than a sample drawn from ...
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1answer
183 views

confidence interval of binomial disribution using standard deviation

Just as the normal distribution has the 68–95–99.7 rule with 68% of the data within +- 1 standard deviation and so on, does the binomial distribution too has something like that. Or does its being a ...
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2answers
139 views

Finding the conditional probability density using a joint PDF

I have a joint PDF $f_{X,Y}(x,y) = cxe^{-x}$ for $x > 0, |y| < x$ First, to determine $c$ I solved the double integral $$\int_0^\infty \int_{-x}^x cxe^{-x}dydx = 1$$ which gave me $c = 1/4$ ...
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1answer
59 views

$E[X^4]$ for binomial random variable

For a binomial random variable $X$ with parameters $n,p$, the expectations $E[X]$ and $E[X^2]$ are given be $np$ and $n(n-1)p^2+np$, respectively. What about $E[X^4]$? Is there a table where I can ...
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2answers
37 views

Finding the mass generating function of a continuous random variable given a pdf

The pdf is given, $$f_X(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{\frac{-(x-\mu)^2}{2\sigma^2}}$$ Where $x\in(-\infty,\infty)$, $\sigma>0$, $\mu\in(-\infty,\infty)$. ...
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1answer
345 views

Probability of a “run” of coin tosses.

Given probability of tails is p, so heads is 1-p. Define X as a random variable for the length of a run (X=5 is either TTTTTH or HHHHHT). Find pmf. So I think a run of just heads or just tails is ...
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1answer
319 views

Determine if a set of points on a sphere come from a uniform distribution?

I have a large distribution of points on the unit sphere $S^2$ and I want to determine if those points came from a uniform distribution on the surface. Essentially, I'm looking for a two dimensional ...
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1answer
296 views

Standard normal density function

Please help me solve this problem: Let $X$ and $Y$ be independent standard normal random variables, let $Z$ have an arbitrary density function, and form $Q = \frac{X + YZ}{\sqrt{1+ Z^2}}$. Prove that ...
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1answer
42 views

Convergence to a delta distribution

Is it Okay to say that when I have a probability density $P(x;\mu,\sigma)$, with $\mu$ the first moment of the probability density and $\sigma$ the square root of the second central moment of the ...
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1answer
345 views

Expected value and variance of a trigonometric random variable

I have $$Y=A\cos(\omega t) + c $$ where $A$ and $Y$ are random variables. I know $E[Y]$ would look something like $$E[Y]=E[A]cos(\omega t) +c,$$ but how do I represent $V[Y]$ ? I'm having trouble ...
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1answer
33 views

How to show that $\mu$ is normally distributed

Suppose we have: $$p(\mu \mid \sigma, \boldsymbol{w}, \boldsymbol{y}) \propto \exp\left[-\frac{1}{2\sigma^2}\sum_{t=1}^T\left(\frac{(y_t-\mu)^2}{w_t^2}\right) \right]$$ where $\boldsymbol{w} = (w_1, ...
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1answer
859 views

Find the posterior distribution of θ

I have this problem Given the prior distribution is \begin{align}Pr(\theta=i)=\pi_i=\begin{cases} 0.5, & \text{for i=4}.\\ 0.3, & \text{for i=5}.\\ 0.2, & \text{for i=6}.\\ ...
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2answers
206 views

Conditional Expectation given joint distribution

Given 2 random variables $X,Y$, is it possible to write conditional expectation $\mathbb{E}[X|Y]$ in terms of their joint distributional function $F_{X,Y}(x,y)$?
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0answers
1k views

Probability Integral Transformation

I just attended an introductory course on Statistics and we came across the following: I know what random variables, the uniform distribution, etc. are but the notation from the proposition ...
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0answers
25 views

distribution of spaced normal variables

If we have samples $x_1$, $x_2$, ...,$x_M$ in $R^N$ obtained as follows: 1) $x_1$ is drawn from a normal distribution $N(0,\Sigma)$ 2) $x_k$ ($k>1$) are also drawn from $N(0,\Sigma)$, but only in ...
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1answer
72 views

Tail probability of the $\chi^2$ distribution

Ho to prove that $$ \int_{2s\epsilon^{-2}}^{\infty}\frac{1}{\Gamma(d/2)2^{d/2}}x^{d/2-1}e^{-x/2}dx \leq const.\epsilon^{-d}\exp(-\epsilon^{-2}s) $$ holds for $\epsilon >0$ sufficiently small? Here ...
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1answer
264 views

How to determine significance of a binomial test on a sample

I'm looking to determine the significance of the result of a binomial test of a sample of a population. Example: Given a group of 10,000 people, I ask 1,000 if they prefer iOS or android. 550 respond ...
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1answer
650 views

Joint Distribution of two correlated ito integral

I have a question regarding finding the joint distribution of two process$$dX_{t}=a_{t}dB_{t}$$$$dY_{t}=b_{t}dW_{t}$$where $B_{t}$ and $W_{t}$ are two Brownian motions with correlated increments, in ...
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2answers
389 views

Finding expected perimeter and area of rectangle using random distribution

Hi i have a question: A random rectangle is formed in the following way- Base X is generated from Gamma(1, λ) distribution and after generating the base, the height Y is chosen uniformly on the ...
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2answers
68 views

If $x_1,\cdots, x_n ~ N(0,\sigma^2)$ and iid and $T=\sum_{i=1}^{n}x_{i}^{2}$ What is the distribution of T

I was thinking using the mgf $M_{T}(s)=E(e^{Ts})=E(e^{s\sum_{i=1}^{n}x_{i}^2})=E(\prod_{i=1}^{n}e^{sx_{i}^2})$ What would be next?
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1answer
345 views

expectation of Gamma distribution help

If x∼Gamma(1,λ) how would i find the expected value E(e^bx) where b=aλ I'm kinda stuck as to how to approach the question. Some help will be greatly appreciated Thank you in advance
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1answer
966 views

Expectation of $\frac{1}{x+1}$ in poisson distribution(help needed)

as the title states, i'm trying to find the Expecteed value of $\frac{1}{x+1}$ $X~Poisson(\lambda)$ my attempt: $\sum \frac{1}{x+1} \cdot \frac{e^{-λ}\cdot λ^x}{x!}$ $\implies \sum ...
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1answer
308 views

expectation calculation in probability and statistics

2 four-sided dice are rolled. X = number of odd dice Y = number of even dice Z = number of dice showing 1 or 2 So each of X, Y, Z only takes on the values 0, 1, 2. (a) joint p.m.f. of (X,Y)? joint ...
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1answer
230 views

Help in Finding the Joint Pdf using Jacobian or C/C function approach

I am trying to find the joint pdf of the random variable z which is the summation of $n$ i.i.d random variables i.e.: $$Z= X_1 + X_2+ ....+ X_n$$ Where all the $X_i$'s are iid random variables with ...
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1answer
40 views

Why is it valid to use the PDF for a naive bayes classifier?

In my understanding of a Naive Bayes Classifier, one takes the argmax of the probabilities that example $x$ belong to class $c_i$, that is $$\text{argmax}_{c_i\in C}P(C=c_i|X=x)$$ I understand that ...
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0answers
108 views

How to calculate (discrete) Pareto distribution. e.g., Probability that someone is in (an arbitrary jellybean giving) secondary sub set.

Assuming two equal sized populations (A & B), 20% of group A has been given one or more jelly beans (arbitrary distinction) by one or more people group B. The subset in group B (B1) who gives ...
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2answers
85 views

Dice sum probability

Simulate two separate dice (use random numbers with the appropriate range) being rolled 10 times. What are the percentage of rolls that resulted in a sum of 7, a sum of 2 and a sum of 11. I came ...
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1answer
75 views

what is the meaning to evaluate the variance of probability destribution for insurance in general?

what is the meaning to evaluate the variance of probability destribution for insurance in general? What does it do(setting the price, estimate cash reserve or else), also does evaluate the variance ...
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2answers
695 views

Joint PMF of two independent random variable

Let $X$ and $Y$ be independent random variables. Each of them has a geometric distribution with $E[X] = 2$ and $E[Y] = 3$. (a) Find the joint p.m.f. of $X$ and $Y$. (b) Compute the ...
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3answers
131 views

Homework on Binomial Distribution

I have another homework, and I don't get this particular sub-question. I need to understand it. I don't know what I'm supposed to do. For the sake of brevity, I won't post the entire problem. ...
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1answer
67 views

Probability and Card drawing

I got a deck with 10 red cards and 30 black cards. Now, I draw a couple of cards 20 times. I draw the couple without replacement, but the 20 drawings of couples are with replacement (I restore the ...
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2answers
333 views

Mean and variance calculation

An unfair coin has probability $p$ of heads. I flip it until I get heads, then I flip it some more until I get tails. Let $X$ be the total number of flips. So here are some possible outcomes: HT : $X = ...
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1answer
55 views

Approximation of a standard normal random variable in probability

Let $Z$ be standard normal random variable and $(a_n)_{n\in\mathbb{N}}$ a real sequence with $a_n\xrightarrow{n\rightarrow\infty}\infty$. Can we obtain something like $$P(Z>a_n)\leq \exp(- c_n)$$ ...
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1answer
159 views

Connection between Normal,Gamma and beta distribution

if $X,Y,Z\sim {N}(0,1)$ then $(X^2+Y^2+Z^2)\sim \text{Gamma} (0.5\cdot 3 , 0.5)=\text{Gamma} (1.5,0.5)$ Also, we define $T$, such that $T= \frac{X^2 }{ X^2+Y^2+Z^2} $ I don't understand why then ...
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2answers
618 views

Finding the distribution of a poisson distribution with random variable lambda

So suppose $X$ is a rv with a Poisson distribution with $\lambda$ being a random variable as well. $\lambda$ has an exponential distribution with mean $1/c$ and $f_\lambda(t) = c\times\exp(-ct)1_{[0, ...
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2answers
178 views

what is the probability of choosing $m$ numbers from the interval $[0,1]$

As far as I know there are infinitely many real numbers between $[0,1]\subseteq R$, what is the probability of choosing a given set of numbers $\{x_1,...,x_m\}$ where $x_i\in[0,1]$ from $[0,1]$? where ...
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2answers
891 views

Probability of Independent Events Happening at least twice

If the probability of hitting a target is $1/5$ and ten shots are fired independently, what is the probability of the target being hit at least twice? Would it be $$1 - \left[\left(\frac 1 ...
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1answer
29 views

Discrete Random Variable's PMF

I have a discrete random variable, X, that has a probability mean function defined as f(x) = c(2/3)^x for all x = 0, 1, 2... I'm trying to find the value for c to make this a pmf. There is apparently ...
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1answer
37 views

Regarding the Weibull Random Variable

So I'm plotting a Weibull RV . The PDF is $$f(x)=\begin{cases} B \, x^{B-1} e^{-x\,B} , & \text{if } x \ge 0\\ 0, & \text{elsewhere} \\ \end{cases} $$ Now when $B=0.5$ and $x=0$, what will ...