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

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2
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30 views

Is it possible to construct any random variable on the Euclidean Probability space?

Let $(\Omega,\mathscr A,P)$ be an arbitrary probability space, and let $X:\Omega\to\mathbb R$ be a random variable. Then, one can generate a random variable $Y$ from the probability space ...
1
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1answer
24 views

Distribution of the sample variance of n iid exponential variables

I have to check some properties of an estimator, but I can't find its distribution. Let $X_1,...,X_n $ be independent identically distributed exponential variables with parameter $ \theta $, i.e. ...
0
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1answer
22 views

Poisson distribution equation

This is probably a very simple and silly question to ask, but I just don't understand the steps for b). I don't quite understand where the negative (-) sign came from? Could somebody please shed some ...
2
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3answers
123 views

Some issues concerning joint random variables

Let the joint random variable $P[x;y]$ be $P[x;y] = c[2x^2 + y^2], x=-1;0;1, y=1;2;3;4$ $=0$ $elsewhere$ So I had to find the value of $c$ that makes $P[x;y]$ a joint discrete random variable. I ...
0
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2answers
47 views

Given a CDF, find P(-.5<X<.5)

Given the following CDF: \begin{equation*} F(x)= \left\{ \begin{array}{lr} 0 & x<-1, \\ \frac{x+2}{4} & -1 \leq x < 1 \\ 1 & x \leq 1 \end{array} \right. \end{equation*} Compute ...
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0answers
17 views

applicability of Poisson distribution

Do all random variables dealing with number of random, independent events in a continuous interval follow a Poisson distribution?
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2answers
39 views

Finding the value of c for which two probabilities are equal

May I please get help with this question? The amount of a certain chemical in a type A cell is normally distributed with mean of 10 and a standard deviation of 1, while the amount in a type B cell is ...
0
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0answers
28 views

Computing the expectation of following interesting random variable

Every package of some cereal includes a plastic animal. There are $N$ different types of animals, and each package is equally likely to contain any type. Your children make you buy one package of the ...
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0answers
28 views

Distribution of the sum of many lognormal random numbers from same distribution

In my application I have to sum up a lot (between 1000 and 2000) lognormally distributed random numbers and use their sum. All random numbers that I sum up follow the same distribution. The current ...
1
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1answer
64 views
+50

Optimization with probability densities - Lagrange multipliers

This question is concerned with the paper "A Lower Bound for a Probability Moment of any Absolutely Continuous Distribution with Finite Variance" by Sigeiti Moriguti appeared in Ann. Math. Statist. ...
2
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0answers
8 views

Hypergeometric RV - what is the sample/population?

An instructor who taught two sections of engineering statistics last term, the first with 20 students and the second with 30, decided to assign a term project. After all projects had been turned in, ...
3
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1answer
13 views

$L^p$ integrability of products of Gaussian variables

Gaussian variables have moments of all orders, so by Hölder's inequality the product of two Gaussian variables $\xi$ and $\eta$ has finite $L^1$-norm: $$ \|\xi \cdot \eta\|_1 \leq \|\xi\|_2 \cdot ...
1
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1answer
42 views

What is the correct equation for conditional relative entropy and why

I was trying to understand the concept of conditional relative entropy. As in: $$D(P(X\mid Y) ||Q(X\mid Y))= E [\log\frac{P(X\mid Y)}{Q(X\mid Y)}]$$ I would have thought that its equations would ...
4
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0answers
30 views

Memoryless property and geometric distribution

Suppose $X$ is a random variable taking values in $\mathbb N_0$ with the memoryless property,i.e., for each pair of number $s,t \in \mathbb N$, $$P(X\geq s+t\mid X>t)=P(X\geq s)$$ Show that a ...
0
votes
1answer
13 views

Geometric Random Variable of a coin toss which is tossed 1 time.

The geometric random variable is defined (in this example) as the number of tosses needed for a head (a fair coin) to come up for the first time. $P_x(k)$ = $(1 - p)^{k-1}$ * $p$ So I calculated the ...
0
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1answer
21 views

Poisson Process question (joint PMF and expectation)

Stuck on this question, would really appreciate any help. Thanks!
-1
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1answer
18 views

cumulative density function exercise [closed]

$F(x)=\begin{cases} 0 \ \text{for} \ x<0 \\ sin(x) \ \text{for} \ 0< x<π/2 \\ 1 \ \text{for} \ x>π/2 \end{cases}$ Find the probability as the result of experiment that the random ...
1
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1answer
22 views

How is this Negative Binomial Random variable used to solve this problem?

I was looking at the solution to this problem below and I don't understand how they used a negative binomial R.V. to solve the problem. A research study is concerned with the side effects of a new ...
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1answer
27 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
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1answer
23 views

The probability of a certain number of trials before a defective object is selected

A box contains 20 items of which 4 are defective. Joshua draws one item after another with replacement until he gets a defective one. What is the probability that the number of trials ...
4
votes
3answers
63 views

Deriving Mean and Variance of Laplace Distribution

It has been a long time since I have used calculus, and I am trying to understand how the mean and variance of the Laplace distribution with pdf $$f(x|\mu,\sigma) = \dfrac{1}{2 ...
3
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0answers
32 views

Distributions question. I'm getting the wrong answer?

An oil exploration firm is to drill $10$ wells, with each well having probability $0.1$ of successfully producing oil. It costs the firm ${10}$ million dollars to drill each well. A successful well ...
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3answers
71 views

What kind of distribution is this and how do I calculate the expected value

Jack buys four items from a firm; the four are randomly selected from a large lot known to contain 10% defectives. Let $Y$ denote the number of defectives among the four that Jack has bought. ...
0
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0answers
28 views

Is there an error in this question - binomial distribution

The median time a customer waits to be served at a large retail company is 20 minutes. On a day when 6 customers pitch up, what is the probability that more than half will have to wait more than 20 ...
0
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1answer
22 views

Ratios and probability mass function

It was given that $p_{2,3} = 2$ and $p_{0,1} = 12$ $p_{k, k+1} = \frac{P(X=k+1)}{P(X=k)}, k=0,1,2,...,n = (\frac{n-k}{k+1})(\frac{1-\theta}{\theta})$ The question was: Find $P(X \geq 2)$. Answer: ...
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0answers
20 views

Binomial Distribution Problem

Hello can someone please help me to answer this question it, it a binomial distribution question: An email message advertises the chance to win a prize if the reader follows a link to an online ...
5
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0answers
61 views

Probability that a five is seen before any of the even numbers are seen

A fair die is repeatedly tossed. What is the probability that a five is seen before any of the even numbers are seen? I have my own solution below and just want someone to verify it. According ...
0
votes
1answer
22 views

The joint distribution of two linear combinations of independent standard normal variables

If $W$ and $V$ are standard independent normal variables find the joint distribution of $3W+2V$ and $2W-3V$. Since $W$ and $V$ are standard independent normal variables then would it just be the ...
0
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1answer
19 views

Binomial distribution question regarding one after another selection

Say you have a manufacturer who manufactures a product and the process historically averages 5% defective products. Now suppose the products are randomly selected and inspected for defects one after ...
0
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1answer
8 views

Subset Probability to Element Probability (part II)

Asking in conjuction with the previous question: Subset Probability to Element Probability If John selects any sized-subset (from 1 element to N elements), which is the probability of selecting ...
3
votes
1answer
53 views

Calculus Question: Improper integral $\displaystyle\int_{-\infty}^{\infty} x^{2}e^{x-e^{2x}}dx$

I am curious about evaluation of the following integral $$\int_{-\infty}^{\infty} x^{2}e^{x-e^{2x}}dx$$ Is it possible to evaluate it? This not my homework but I will share my attempt. I tried ...
1
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1answer
29 views

tricky integrating ranges x1-x2

So, we know the sum of n i.i.d. exponential(lambda) is gamma(n,lambda). But I am looking at a problem with X1-X2. So I get the joint dist of z=x1-x2 and w=x2. Then I integrate out w on range 0 to ...
0
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0answers
18 views

Unable to understand Steps in derivation expectation maximization

In Paper, System Identification using Symbolic Chaotic Sequence, Authored by A. Kurian and H. Leung download link under section II B, can somebody please explain ...
0
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0answers
13 views

Sum over stochastic processes on the same set of categories

I have a stochastic process consisting of multiple (stochastic) steps, for which I want to know if I can substitute (or at least approximate) it by summing over the deterministic and stochastic parts ...
1
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1answer
26 views

Probability function

Peter and John are designing a game. You take 2 balls (once a time, and without replacement) out 0f 12 balls, where 2 are blue, 3 green, 7 black. Tha gambler needs to pay $10 to be able to play the ...
0
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1answer
24 views

Computing absolute distribution from conditioned probability

for a sum $X:= \sum_{i=1}^n 1_{U_n<U_0}$ of a series of random variables $U_1, ..., U_n$, all of them uniformely distributed on the unit interval as is $U_0$, I computed the following conditional ...
2
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0answers
38 views

Finite discrete approximation to the normal distribution

I wish to derive a finite (that is, which has a finite support) discrete approximation to a normal distribution, with the following considerations: It should have exactly the same mean and variance ...
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0answers
21 views

Derivation of expectation maximization for GMMs

I am referring to these lecture slides on EM estimation of GMM. In particular I have confusion in steps on slide 13 and 14. If we have a $N$ component GMM (defined by parameters $\theta$), likelihood ...
0
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2answers
34 views

Probability of transferring a ball from one bucket to other

There are $n$ buckets. Each of these buckets can hold maximum $k$ balls. The probability that a ball is transferred to $m_{th}$ bucket is denoted as $P(m)$. What is the probability that a ball is ...
0
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0answers
27 views

Expectation in Moment Generating Functions

A moment generating function $M_X(t)$ is defined as $\mathbb{E}(e^{tX})$ where $X$ is a random variable and $t$ is a number. In every calculation of moment generating functions I see the following ...
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0answers
15 views

True or false: If a distribution has a conjugate prior, then it is a member of the exponential family.

I would like to know if it's true that "A distribution has a conjugate prior if and only if it is a member of the exponential family". I know that all members of the exponential family have conjugate ...
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0answers
21 views

Can anyone help with this discrete problem? [duplicate]

Question: How many ways are there to distribute 16 identical pieces of candy to five children such that every child receives at least one piece? Generalize k identical pieces of candy and n children. ...
0
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2answers
48 views

What is the reason for this answer on this coin problem?

Question: How many ways are there to pick a collection of 15 coins from bags of pennies, nickels, dimes, and quarters? (Assume coins of the same denomination are indistinguishable.) I know the answer ...
0
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0answers
28 views

Name of a distribution of subsequent successes of a random event

What I am looking for now is the name of the distribution that describes the situation of exactly $k$ successes in a row of random events of probability $p$. To give an example, if $k = 3$, that ...
1
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1answer
22 views

Cumulative Poisson Distribution Question

For the following question I figured that the expected time between successive arrival is the mean = 1/10 per hour (or 1 per 6 minutes). My question is regarding the second part; does the fact that ...
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0answers
15 views

pdf and random variable transformation?

let y be a multivariate random variable with probability density function $p_y(y)$. Let $x$ be such that $y=f(x)$ where $f$ is a differentiable function (not bijective). What is the density of $x$?
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0answers
11 views

Generating the data from empirical histogram or probability density function.

This question might be redundant of the last question which I asked. Thanks to jameselmore and user159813, I tried to generate the data from my emprical PDF. If I explain again for better ...
0
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0answers
22 views

Relative merits of definitions of “discrete probability distribution”

Some books say that a probability distribution is "discrete" if the set of possible values that a random variable so distributed can assume is either finite or countably infinite. I think a better ...
0
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1answer
29 views

Unbiased sample standard deviation of a custom/unknown probability distribution

Hi i must determine the unbiased sample standard deviation of an unknown probability distribution.I dont have the data of the full population so i must work with a sample. Now according to Wikipedia ...
0
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2answers
22 views

Derivative of Survival Function

I am trying to get through statistical survival analysis - sadly I only have high school math. I have the following equation: $ S(t) = Pr\{T ≥ t\} = 1−F(t) = \int_t^\infty f(x) dx$ $f(x)$ is the ...