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

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0answers
7 views

Probability Scheme of Variable X [on hold]

from digits 1,2,.....9 we are making 3-digit number. Variable X is measuring number of 1-digits in this number. Write probability scheme for varibable X?
1
vote
1answer
274 views

joint probability distribution of one discrete, one continuous random variable

This is a problem on the joint distribution of a discrete and a continuous random variable. Kitty Oil Co. has decided to drill for oil in 10 different locations; the cost of drilling at each ...
1
vote
1answer
767 views

Understanding the difference between normal distribution and lognormal distribution

I'm having trouble understanding the difference between a normal distribution and lognormal distribution. Here's what I've done so far. Definitions of lognormal curves: "A continuous distribution in ...
0
votes
0answers
28 views

Make the sum of random variables converge, while the sum of the variances diverges

Suppose $X_n$, $n=1,2,3,...$, are independent and $Var(X_n)$ is uniformly bounded by finite constant $C>0$. Construct $X_n$ such that $\sum_nX_n$ converges a.s., but $\sum_nVar(X_n)=\infty$.
0
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1answer
28 views

Pdf of the product of an exponential r.v. and a beta r.v.

Let $X$ and $Y$ are 2 independent random variables, where $X$ has an exponential distribution with parameter $1$ and $Y$ is $\beta(a,b)$ distributed. What is the Pdf of $W=XY$ ? Thanks !
7
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0answers
241 views

How do you compute numerically the Earth mover's distance (EMD)?

I was trying to compute numerically (write a program) that calculated the EMD for two probability distribution $p_X$ and $q_X$. However, I had a hard time finding an outline of how to exactly compute ...
-1
votes
0answers
15 views

Stat problem help me! [on hold]

Hello! I got a problem when I was solving stat problem. I solved by(c), but after that, I found it hard to solve. Can you guys help me or give me a hint? Thank you anyway!
1
vote
1answer
21 views

The distribution of sample proportion for given population proportion and sample size

If the population proportion is 0.90 and a sample of size 64 is taken, what is the probability that the sample proportion is more than 0.89? (4dp) work: $n=64$, $\hat p=0.89$, so $X=n \hat p ...
0
votes
0answers
41 views

An Integral and its limit

Consider the following integral, $$K(\alpha)=\int_\mathbb{R}\log^2(g/f)(g/f)^\alpha f \, \mathrm{d}\mu$$ where $\alpha\geq 0$, $\,\mu$ is some measure and $f,g$ are some distinct continuous ...
0
votes
0answers
22 views

approximation by binomial [on hold]

Consider a finite region $\Lambda \subset \mathbb{R^2}$ (e.g. a square), and a model in which $N$ indistinguishable particles are placed randomly and uniformly in $\Lambda$. Let $\Delta \subset ...
0
votes
1answer
15 views

How to compute $P(X\leq Y)$ and $E(X^2 Y)$ with given probabilities [on hold]

Given $P(X=1, Y=0) = 0.1; P(X=1, Y=1) = 0.1; P(X=1, Y=2) = 0$ and $P(X=2, Y=0) = 0.2; P(X=2, Y=1) = 0.4; P(X=2, Y=2) = 0.2$ How do I compute $P(X \leq Y)$ and $E(X^2 \cdot Y)$?
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0answers
20 views

Ratio between normal distributed and gamma distributed variables

Let $X \sim N(0,1)$ and $G \sim Gamma(a)$. Why is $\frac{X}{G}$ t-distributed?
1
vote
1answer
18 views

A urine test, the VMA test

Neuroblastoma is a rare, serious, but treatable disease. A urine test, the VMA test, has been developed that gives a positive diagnosis in about 70 % of cases of neuroblastoma. It has been proposed ...
0
votes
0answers
6 views

average dirichlet distribution [on hold]

Is it possible to combining 2 Dirichlet distribution averaging their values? The resulting probability distribution is still a Dirichlet distribution? If not how can I merge 2 similar Dirichlet ...
3
votes
1answer
44 views

$P(X^2+Y^2<1)$ of two independent n(0,1) random variables

Suppose that X and Y are independent n(0,1) random variables. a) Find $P(X^2+Y^2<1)$ Attempt: a) Let $U = X^2 + Y^2$, $V = Y$. Then $X = \sqrt{V^2 -U}$, $Y = V$. $J = \left| ...
1
vote
0answers
21 views

Expectation of absolute value of Brownian motion

I'm working on this problem that I can't seem to figure out. The problem involves a 1-dimensional Brownian motion, $B_t$, where the subscript denotes the time, and it asks me to show that the ...
0
votes
0answers
23 views

Relationship between a distribution function and the truncated distribution function

Let $F(x)$ be a distribution function and $G(x)$ be $F(x)$ truncated on some interval $(a,b)$. I want to show that: $$G(x)=\frac{F(x)-F(a)}{F(b)-F(a)}, a<x \leq b$$ I want to do this by using ...
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votes
1answer
35 views

some question about expected value

Let $X$ be a non negative random variable. We Know that $E(X)=0$. Is that correct that $X=0$ for some $X$. And more general: Is there a point in the probability space for which E[X]≤X and a ...
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votes
0answers
32 views

CDF of two random variables [on hold]

Suppose Y and Z are independent N(0, 1) random variables and suppose X = |Z|. Consider the random point (X, Y). (a) Derive the CDF FD(d) = P(D ≤ d) of the distance from the origin D =√X2 + Y2. Sketch ...
0
votes
2answers
86 views

How to find $P(X>x)$ when the density is known but the integral does not seem to converge

I am trying to evaluate $$P(X>x) = \int_x^{\infty } t^{\kappa } \exp{\left(-\rho t^{\alpha\kappa + 1}\right)} \, dt$$ where $\kappa$, $\rho$ and $\alpha$ are all constants. I have tried some ...
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votes
0answers
25 views

function of a random variable problem 2 [on hold]

Let $U$ be a continuous random variable with uniform distribution over $[0,1]$. Define $X$ by $$X=\operatorname{Int}\left(\frac{\ln(1-U)}{\ln(1-p)}\right)+1$$ where $\operatorname{Int}(x)$ is the ...
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votes
1answer
31 views

Cumulative distribution function [closed]

The delay of the train in minutes is given by the CDF $$F(x) =\begin{cases} \dfrac{(x+5)}{30} &;\ -5<x<0, \\[15pt] \dfrac{2}{3} + \dfrac{x}{180} &;\ 0<x<60. \end{cases} $$ ...
0
votes
2answers
28 views

Ratio of Gamma random variables

If $X_i$, $i=1,2$ are independent gamma$(\alpha_i,1)$ random variables, find the distribution of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$. Attempt: Let $Y_1 = \frac{X_1}{X_1+X_2}$ and ...
1
vote
2answers
106 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 ...
1
vote
1answer
19 views

Finding distribution of random variable if X is exponential $(1)$

Let X be an exponential (1) random variable, and define Y to be the integer part of X+1, that is $\hspace{15mm}Y=i+1$ if and only if $\hspace{5mm}i \leq X \leq i+1, i = 0,1,2,...$. Find the ...
0
votes
1answer
22 views

Uniformity of the difference between two random variables

What can I say about the distribution of two random variables $A$ and $B$ such that $A-B$ is uniformly distributed?
1
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1answer
39 views

Pdf of the product of an exponential rv and a $f_Y=Ka^{-K}y^{K-1}$ distributed rv …

Let $X$ and $Y$ are 2 independent random variables, where $X$ has an exponential distribution with parameter $1$ and $Y$ has the following Pdf: $f_Y=Ka^{-K}y^{K-1}, 0 \le y \le a $. Someone claims ...
0
votes
2answers
47 views

Poisson random variables and Binomial Theorem

I'm working on a problem from Casella and Berger's Statistical Inference. X is distributed as Poisson$(\theta)$ and Y is distributed as Poisson$(\lambda)$, with X and Y being independent. We let U = X ...
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0answers
21 views

Correlation and First Order Stochastic Dominance

Suppose we have a random variable $X \sim [0,1]$ with a continuous distribution $F_X(x)$. Suppose $I \in \left\{0,1\right\}$ is a discrete random variable with $\text{Prob}(I=1 \ | \ X=x)$ strictly ...
2
votes
0answers
53 views

Autocorrelation function of random process

Let $X_t$ be a wide sense stationary random process indexed by $t\in\mathbb{R}$ with finite mean and variance. (http://en.wikipedia.org/wiki/Stationary_process) Q1) Is the autocorrelation function ...
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0answers
10 views

Exponential Generalised Beta distribution type 2

I am doing some toy examples with EGB2. I would like to simulate variates and then estimate back parameters I used for the simulation. I can achieve the former by using gamlss.dist package in R. ...
0
votes
1answer
11 views

Percentages in Normal Distribution

A statistics problem involves: Lengths of a certain type of carrot have a normal distribution with mean 14.2 cm and standard deviation 3.6 cm. (i) 8% of carrots are shorter than c cm. Find the value ...
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votes
1answer
24 views

A drug treatment [on hold]

A certain drug treatment cures 90 % of cases of hookworm in children. Suppose that 20 children suffering from hookworm are to be treated, and that the children can be regarded as a random sample from ...
0
votes
1answer
26 views

Explicit CDF associated to Gamma PDF [on hold]

Thanks in advance for the help with this! I'm struggling to follow the solution in the book for this problem. Any help is greatly appreciated. Let the distribution function of X for x>0 be: $$F(x) = ...
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vote
0answers
8 views

Creating random integers with distribution schema

I need to create an array that includes 0..5 integers. I'm able to create them randomly. But I need to create them according to below distributions. How can I get below distributions? Ps: I'm using ...
0
votes
1answer
12 views

Binomial probabilities

Okay, so here is probably the easiest question ever on this website. A question on binomial distribution. In a city, the percentage of left-handed women is 16% and the percentage of left-handed men ...
0
votes
1answer
34 views

Estimating how much two probability distributions differ

I have two probability distributions A and B. First I would like to estimate how much they differ. In this regard I use as metric the Jensen–Shannon distance (i.e. the square root of Jensen–Shannon ...
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votes
0answers
7 views

Deriving the multivariate t-distribution from the normal mixture representation

I'm trying to derive multivariate t-distribution from its representation as a normal variance mixture distribution by following the calculations in Appendix 4 of ...
1
vote
1answer
18 views

How to visualize probability distributions in terms of sets - joint and marginal?

Let there be two sets, $\mathcal{X},\mathcal{Y}$, both finite, and they represent the set of values that the discrete random variables, $X,Y$ can take. $\mathcal{P}_{Y|X}$ be all possible ...
3
votes
2answers
46 views

Random variable $X^2$ determined by moments

Let $X$ be a real random variable, with standard normal distribution. Is the distribution of $X^2$ determined by its moments? In general, if $n \in \mathbb N$, is the distribution of $X^n$ ...
1
vote
1answer
304 views

Solving Probability Density Function for continuous random variable

The probability density of a random variable $x$ is $$f(x)=a\ \cdotp x^2\ \cdotp \mathrm{e}^{−kx}\ (k>0,\ 0\leq x\leq \infty)$$ Then, the coefficient $a$ equals $$(i)\frac{k^3}{2}\ \ \ \ (ii)\ k^3 ...
1
vote
2answers
38 views

Find $E[N]$, where $N = \min\{n>0: X_n = X_0\}$

Let $X_i$, $i\geq 0$ be independent and identically distributed random variables with probability mass function $$ p(j) = P\{X_i=j\},\; j=1,...,m,\;\sum^{m}_{j=1}P(j)=1 $$ Find $E[N]$, where ...
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1answer
26 views

Product of two distribution functions.

Let F and G be two distribution functions, does the product FG still a distribution function?
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0answers
7 views

Uniform conditional on maximum distribution

If $U_1,U_2,\dots,U_n$ are i.i.d. $U(0,1)$ and $U_{(n)}=max(U_1,U_2,\dots,U_n)$, I want to show that $U_n|U_{(n)}$~$U(0,U_{(n)})$. I know that the pdf of $U_{(n)}$ at $t$ is $nt^{n-1}$. I did the ...
0
votes
1answer
885 views

Finding joint cdf and pdf of independent random variables

Let $X$ and $Y$ be independent random variables. Each has an exponential distribution with parameter $\lambda$. Define two new random variables by $W = \min({X,Y}) $ $Z = \max({X,Y})$ Find the ...
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0answers
21 views

Deriving the Pareto Distribution from an Exponential Distribution [on hold]

Let $T$ be an exponential random variable with hazard rate $a>1$. Consider a random variable defined by the condition $X = b (e^t - 1)$." I need to find the density of $x$.The answer is ...
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votes
0answers
35 views

Normal distrubition [on hold]

Let Xi denote the weight of a randomly selected prepackaged one-kilogram bag of potatoes. Of course, one-kilogram bags of potatoes won’t weigh exactly one kilogram. Actually, history suggests that Xi ...
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votes
1answer
21 views

Get unknown value in discrete random variable

Let $X$ be a discrete random variable (i) Assume that the PMF of $X$ is given by $$\operatorname{Pr}(X=x)=\begin{cases}kx^{2} & x \in \{-4,-2,0,2,4\} \\ 0 & x\not\in \{-4, -2, 0, 2, ...
2
votes
1answer
33 views

Expected Value on code

I'm trying to figure out the expected number of times this algorithm will print. I'm stuck on how to go about doing so. I used an indicator variable to keep track of the number of print statements ...
0
votes
3answers
56 views

Is it even possible to find the variance of this moment generating function?

This is my moment generating function: $M_x(t) = \frac{6e^t}{t^2} + \frac{6}{t^2} + \frac{12e^t}{t} - \frac{12e^t}{t^3} + \frac{12}{t^3}$. I have to find the mean the variance of it. After taking ...