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

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2
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0answers
61 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
votes
1answer
14 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
vote
1answer
44 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
votes
0answers
36 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
14 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
19 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
vote
1answer
26 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 ...
-1
votes
1answer
32 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
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
96 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
votes
0answers
33 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 ...
1
vote
3answers
75 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
votes
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
votes
1answer
25 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
votes
0answers
66 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
25 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
votes
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
vote
1answer
31 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
votes
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
vote
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
votes
1answer
25 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
votes
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 ...
1
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0answers
22 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
votes
2answers
37 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
28 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 ...
0
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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 ...
-1
votes
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
50 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
vote
1answer
27 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 ...
0
votes
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
18 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
votes
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
34 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
votes
2answers
26 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 ...
0
votes
0answers
23 views

Distribution estimate from a sample or just a probability theory task

Here is the task: I have a sample of replies to my e-mail from my mail box. A sample is taken over a period of 90 days, 1000 e-mails and replies if any. (We only consider a pair of {my original ...
1
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1answer
18 views

Is this form of distribution considered Bimodal or Uniform?

Would this form of distribution be considered Bimodal or Uniform? I have been searching through distribution images and the Bimodal distributions generally appear to refer to a pair of Normal ...
0
votes
2answers
31 views

Trouble deriving sum of squared normals is Exponential with mean $2$

Box-Muller method hinges on the fact that $R = Z_1^2 + Z_2^2$ is Exponential with mean 2, where $Z_1, Z_2$ are independent standard normals. I want to derive this fact but am getting stuck. I proceed ...
0
votes
1answer
16 views

Kernel density estimation including measureed uncertainty

I am trying to plot the distribution of a measured variable from a scientific experiment, in this case a velocity. After making a simple histogram, I have been reading this, ...
0
votes
1answer
29 views

How do you transform probabilities from the form P[X=x] to P[X =< x]

I'm working on a problem that requires you to use a binomial distribution to solve the problem. Now we want to determine x such that P[X > x] =< 0.01 or, ...
1
vote
0answers
33 views

What is meant by Stable Law?

I am reading a very complex paper consider a set of random variables $\left\{ X_{i}\right\} _{i=1}^{\infty }$ whose common distribution $F_{X}$ belong to the domain of attraction of an $\alpha ...
1
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1answer
30 views

Variables defined as floor and fraction part from exponentially distributed random variable

Random variable $X$ has exponential distribution with parameter $\lambda>0$. Let $T=[X]$, $R=\{X\}$ where $[x]$ is floor from number $x\in\mathbb{R}$ and $\{x\}$ is it's fraction part. What is the ...
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vote
0answers
26 views

Proof for the distribution of a two-sample t-test with unequal population variances.

I am having trouble finding documentation showing a proof, or at least some outline for it, illustrating how to derive the distribution and degrees of freedom of the test statistic for a two-sample ...
0
votes
0answers
6 views

Maximum singular value of a Gaussian tensor

Suppose there is an $ n \times n \times n $ tensor where each entry is a normal random variable. Are there results known about the distribution of its maximum singular value, or it's expected value? ...
1
vote
1answer
11 views

Binomial distribution for number of family members

The question below is a part of a more complex question in probability theory. I can't get my head around why is my solution incorrect... So here it goes. Distribution of number of boys in a family ...
3
votes
1answer
29 views

Convergence of expected values as random variables converge almost surely

Let I have a sequence of random variables $X_n$ that converges to random variable $X$ almost surely as $n\to\infty$. How can I proof that $\lim_{n\to\infty}\mathcal{E}[X_n]=\mathcal{E}[X]$ where ...