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

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constructing an counter example

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 $\sum_nVar(X_n)=\infty$.
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0answers
14 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!
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1answer
19 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 ...
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1answer
24 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 !
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0answers
18 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?
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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 ...
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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 ...
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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 ...
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1answer
14 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
19 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 ...
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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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0answers
29 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 ...
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1answer
34 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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2answers
67 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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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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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 ...
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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 ...
3
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1answer
42 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| ...
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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?
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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 ...
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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. ...
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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 ...
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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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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 ...
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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 ...
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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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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 ...
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2answers
46 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
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 ...
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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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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 ...
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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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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 ...
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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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0answers
20 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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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, ...
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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 ...
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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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1answer
11 views

maximum-likelihood: a sequence of events described by a Bernoulli distribution

I am having quite some troubles with the following homework: In a city it's measured for the whole year whether it rained or not. A distribution $\textrm{Bernoulli}(r_t|\rho)$ characterizes the ...
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0answers
9 views

What is the normal distribution probability? [on hold]

1) A company is contemplating surveying the passengers on a particular ferry service. Over the years, the average number of passengers per trip on the ferry service has been 60 and the standard ...
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2answers
19 views

Conditional probability in multinomial distribution

Consider a multinomial distribution with $r$ different outcomes, where the $i$th outcome having the probability $p_i$, $i$=1,...,$r$, $\sum_{i=1}^r p_i = 1$. Denote $X_i$ be the number of times the ...
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1answer
33 views

Using the inverse Gaussian integral to find percentiles

I need some help with the following: Let $$R=\mu+\sigma*\epsilon \hspace{1cm} \epsilon \sim N(0,1)$$ I want to argue that $$ \mu + \sigma*\Phi^{-1}(u)$$ are the percentiles of the model when ...
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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 ...
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1answer
54 views

What is the probability the best case occurs? (Comp Sci Type Question)

I'm having trouble figuring out what's the probability the best case occurs? It's my first time bringing together probabilistic knowledge into computer science. The question goes as such. Consider ...
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1answer
36 views

An inverse problem on tail probability

This is a question out of curiosity. Assume that $f(x)$ is a density function for which there is a constant $C>0$ so that $$ \int_t^\infty f(x) dx \le C f(t) $$ holds for large enough $t>0$. My ...
3
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1answer
20 views

Generating random variables with complicated probability distribution functions

I have an interesting question I need to solve, and as much as I try, I cannot wrap my head around it. Given a postive random variable X with p.d.f. ...
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2answers
35 views

Let $X_{1},X_{2}, \dots, \sim Exp(1)$ i.i.d. - Calculate the probability of $P[\max{(X_{1},\dots,X_{n},)} < \log(n) - 5] $ for $ n > e^{5}$

Let $X_{1},X_{2}, \dots, \sim Exp(1)$ i.i.d. - Calculate the probability of $P[\max{(X_{1},\dots,X_{n},)} < \log(n)-5] $ for $ n > e^{5}$ as well as $n \rightarrow \infty $ The correct ...
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1answer
18 views

For exponential random variables $X_i$, how to find $P(t-X_1<X_2\mid t-X_1<X_3)$?

Assume $X_1, X_2, X_3$ are three independent exponential random variables with means $1/A$, $1/B$ and $1/C$ resp. How do we calculate $P(t-X_1<X_2\mid t-X_1<X_3)$? My try: \begin{align} ...
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0answers
24 views

Stat problem! Why is this? [duplicate]

This is a statistics problem. although this is not a problem which needs an answer, I want to know the reason Why this is right. Can you guys help me ? Thanks in advance!