Tagged Questions

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

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votes
1answer
70 views

What is the distribution of $\frac{(X_1 + X_2)^2}{(X_1 - X_2)^2}$ [closed]

If $X_1, X_2 $ is a random sample of size 2 from an $N(0,1)$ population then $\frac{(X_1 + X_2)^2}{(X_1 - X_2)^2} $ follows ?? A) $X^2 _ {(2)}$ B) $F_{2,2}$ C) $F_{2,1}$ D) $F_{1,1}$ Plz help ...
0
votes
1answer
28 views

Finding a CDF from the PDF of another Random Variable

Given a function $$ Y=ae^x $$ With the distribution: $$ f_X(x)=be^{-bx} \,\,\,\, x\geq0 $$ Show that the cumulative distribution function is: $$ F_Y(y)=1-(y/a)^{-b} \,\,\, y\geq c $$ My approach ...
1
vote
1answer
11 views

If $f(x)$ belong to set of probability distributions, then what can be deduced about $\frac{1}{a} f(\frac{1}{a}\cdot x)$

My question is in the context of probability distributions, whose Fourier transforms (characteristic function) almost always exit. If $f(x)$ be some function such that $ \int_{-\infty}^\infty f(x) \, ...
0
votes
1answer
23 views

Listing the values of particular probability distributions

Three balls are selected at random from a bag containing 2 red , 3 green , and 4 blue balls. Define the random variables R = the number of red balls drawn, and G = the number of green balls drawn. ...
-1
votes
1answer
43 views

random variable and joint probability

A hamburger chain's game card has ten squares, each of which has a covering that can be rubbed off to reveal what is pictured beneath. Seven squares show different foods, two square show the same ...
-2
votes
1answer
23 views

Continuous Random Variables and Probability Distributions [closed]

Let X denote the amount of time a book on two-hour reserve is actually checked out, and suppose the cdc is: $F(x) = (x^2)/4$ (when $0\leq x<2$), $1$ (when $2\leq x$) find $E(X)$ The answer is ...
0
votes
0answers
13 views

Estimate number of data points necessary to generate a normal distribution

I've written a program that generates random normally-distributed variables using the Box-Muller transform. My question is if I can find any formula that relates the number of data points that I ...
0
votes
0answers
12 views

Question regarding Balanced Incomplete Block Design

Question: Consider a BIB design with a treatments, b blocks and c < a number of plots in each block where a,b,c ≥ 2. Let $n_{ij} = 1$ if an observation is made on the ith treatment in the jth ...
0
votes
1answer
25 views

calculating exponential distributions for products going bad

half of our products (follows an exponential distribution) gone bad during a a week. calculate how long does it take for 1/3 of the products to go bad? my answer (I put this in calculator and get no ...
0
votes
2answers
31 views

Is zero random variable a continuous random variable?

In my probability textbook I got stuck on the following problem: cosntruct a positive (it means $\geq 0$) real function $f(x)$ (not necessarily continuous, of course) such that it is constantly $0$ ...
0
votes
0answers
33 views

Creating a random network (graph) with a $\textbf{random}$ number of vertices and given degree distribution

I was trying to find an answer to my question on google scholar, however I didn't find anything that is close to what I am looking for. I would be very grateful for your help. There is a theory of ...
0
votes
1answer
15 views

Bounds of integral in Power function

Here is the question: Let $X_1,X_2$ be iid uniform $(\theta,\theta+1)$. For testing $H_0:\theta=0$ versus $H_1: \theta>0$, we have two competing tests: $\hspace{15mm}\phi_1(X_1):$Reject $H_0$ if ...
1
vote
1answer
34 views

MLE of $n(\theta,a\theta)$ family

Question: A special case of a normal family is one in which the mean and variance are related, the $n(\theta,a\theta)$ family. If we are interested in testing this relationship, regardless of the ...
1
vote
1answer
18 views

The probability that a device doesn't work during specific time interval

Assume both the time to failure and time to repair are exponentially distributed. The failure rate is $\lambda$ and the repair rate is $\mu$. The repair starts immediately after the failure occurs. We ...
1
vote
1answer
43 views

Probabilities and Conditional Expectation

Good day! Please check my answers. Here is the problem: Let $ X,Y, Z$ have joint pdf $f(x,y,z) = \frac23 (x+y+z), 0<x<1, 0<y<1,0<z<1,$ zero elsewhere. Find the marginal probability ...
1
vote
2answers
47 views

Should I avoid distribution functions in probability?

I'm reading Erhan Çınlar's book on Probability and Stochastics, and in Chapter 2, he says that distributions are used extensively in elementary probability theory in order to avoid measures. And ...
0
votes
1answer
25 views

What is application of gamma distribution on pure math or probability theory?

What is application of gamma distribution on pure math or probability theory? i saw it on several probability textbook as a definition, but it seems to me mathematician couldn't derived it if it is ...
0
votes
2answers
22 views

calculating the expected value of random variable, which is net income

there is 1000 lots in the lottery. you can win 1 unit of 100£, 10 units of 50£ and 15 units of 20£. One lot costs 1£. calculate the expected value of random variable, which is net income. here is my ...
2
votes
1answer
36 views

What could be the mathematical model behind “beginner's luck” (followed by losses) in gambling?

I recall a documentary in which a slot machine had trial runs and at first, the desired "bingo" outcome came out more often, but later waned into losses. A scientist plotted the graph, a discrete ...
0
votes
1answer
18 views

Is truncating a discrete probability mass function possible?

I have random variable X, and probability distribution: $P[X = A] = .4$ $P[X = B] = .3$ $P[X = C] = .2$ $P[X = D] = .1$ I want to create a conditional probability with event F. Where F is the ...
0
votes
1answer
29 views

pleas help me with probability distribution [closed]

The Gumbel distribution is used to model maxima of certain random variables, for example, Normal random variables. The CDF of a standardized Gumbel random variable $X$, is given by ...
0
votes
1answer
37 views

Help with probability distribution [closed]

Let $X \sim \exp(1)$ and $Y =\sinh(X)$. Find the PDF of $Y$. Hint: $\sinh^{-1}(x) = \ln\big(x+\sqrt{1+x^2}\big)$. Question is showed above. How to find PDF of $Y$?
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votes
1answer
68 views

Is $\theta_1-\theta_2$ independent of $\theta_1-\theta_3$ given all are uniform random variables between $[-\pi,\pi]$

I have three random variables $\theta_1, \theta_2, \theta_3$ all are i.i.d uniformly over $[-\pi,\pi]$. These in reality represent angles in my problem that I am trying to solve. I have a linear ...
3
votes
0answers
476 views

Mean Absolute Deviation for a Stable Distribution as a Function of the Tail Exponent

Consider the standard Lévy-Stable (or Alpha Stable) distribution $S(\alpha,\beta, \mu, \sigma)$ where $\alpha$ is the tail exponent, $1 \leq \alpha \leq 2 $. Picking the symmetric case with $0$ mean ...
5
votes
1answer
101 views

Determining a consistent estimator/asymptotic relative efficiency

Question: Let $X_1,\ldots,X_n$ be i.i.d. as $N(0,\sigma^2)$. a) Show that $\delta_1 = k \sum_{i=1}^n |X_i|/n$ is a consistent estimator of $\sigma$ if and only if $ k = \sqrt{\pi/2}$. b) Determine ...
0
votes
0answers
17 views

Normalize or standardize dataset

I have the following data-set: -user: 1, words:300, edits:200 -user: 2, words:200, edits:180 -user: 3, words:250, edits:230 -user: 4, words:150, edits:170 Can I normalize or standardize these ...
1
vote
0answers
16 views

Uniform Convergence and exponential inequality to demonstrate Stirling's formula

If $0<R\leq\sqrt n$ and $Z_{k} \sim exp(1)$ are independent and identically distributed as prove that $$P\left[\left|\dfrac{Z_{1}+Z_{2}+\cdots +Z_{n} -n}{\sqrt n }\right|\leq R\right] \geq 1- ...
-1
votes
1answer
52 views

A cumulative distribution function is discontinuous at some points. Then Random Variable X is discrete?

Here is the distribution So, is the random variable X discrete or continuous? Note: first condition is if x<0 Edit: I have never come across this kind of distributions before. Extensive ...
-1
votes
0answers
35 views

Uniform Convergence and exponential inequality to demonstrate Stirling's formula.

If $0<R\leq\sqrt n$ and $Z_{k} \sim exp(1)$ are independent and identically distributed as prove that $$P\left[\left|\dfrac{Z_{1}+Z_{2}+\cdots +Z_{n} -n}{\sqrt n }\right|\leq R\right] \geq 1- ...
0
votes
1answer
33 views

When computing the CDF from a PDF, why is the integral bound a different variable? $F(x) =\int_{-\infty}^x f(t)\,dt$

Right now, I know the variable in the integral must be x, otherwise, the final result does not match the published CDFs of popular distributions. But I don't know why conceptually this is. If the CDF ...
0
votes
1answer
28 views

How to use Chi-Square for a rolling die

I'm watching this video. The guy said that this kind of test allows us to check whether the experiment occurs by chance or something wrong with the way the experiment is being done. I have done very ...
0
votes
0answers
21 views

What sort of Bernouilli trials are taking care of the counted failures here?

In this question $N$ represents the number of failures before the first success is achieved by iid Bernouilli trials $Y_{1},Y_{2},\dots$ having a chance of $p$ on success. Independent of that there ...
1
vote
1answer
37 views

The smallest integer $n$ for a Poisson distribution

Along a stretch of motorway, breakdowns require the summoning of the breakdown services occur with a frequency of 2.4 per day, on average. Assume the breakdowns occur randomly and that they follow ...
1
vote
0answers
136 views

Does clever noise exist?

This question is about a random noise, which is called "clever" if its distribution function satisfies a certain condition. Let $Y_0$ and $Y_1$ be two continuous random variables on the same ...
0
votes
3answers
19 views

How to compute expected value

How do I solve the expected value of this problem, if I have already calculated the pmf? Let $X$ be a random variable with cumulative distribution function given below: $$F_X(x) = ...
0
votes
2answers
18 views

Bivariate Normal Distributions

Let X and Y have a bivariate normal distribution with parameters μ1 =3, μ2 = 1, σ1^2 = 16, σ2^2 = 25, and ρ = 3/5 . Determine the following probabilities: (a) P(3 < Y < 8). (b) P(3 < Y < ...
1
vote
1answer
12 views

Find distribution of a bernoulli funtion of a unifrom random variable?

I have a uniform random variable $\theta \in [-\pi,+\pi]$. I also have a bernoulli function of this random variable $G(\theta)$, defined as follows, \begin{align} \begin{cases} 1 & \text{if $ - ...
0
votes
1answer
34 views

Dirichlet distribution, sum of Beta distributions

I currently have a problem about Dirichlet distributed Variables. In one of the papers I am currently reading it says: Let $S=(S_1,...,S_m)\sim Dir(\delta\omega_1,..., \delta \omega_m)$, with ...
0
votes
0answers
14 views

problem on ratio of moments

Recently, I encounter the following problem: Let $w$ be a probability density on $[0, 1]$. Let $m_k$ be the $k$-th moment, i.e., $$m_k = \int\limits_0^1 t^k w(t)dt.$$ Under what condition can we have ...
0
votes
0answers
18 views

How do you interpret coefficient of Variation [migrated]

I am trying to understand Coefficient of Variation, and when I try to apply it to the following to samples of data I am unable to understand how to interpret the results. Lets say sample $s_1={0, 5, ...
0
votes
0answers
34 views

conditional expectation of two independent normal random variable

Show that conditional expectation of two independent normal random variable $(X,Y) \approx N(m=(m_1,m_2), \Sigma)$ is equal: $$E[X|Y] = m_1 + \rho \frac{\sigma_X}{\sigma_Y}(Y-m_2)$$ Is there any way ...
2
votes
1answer
24 views

Proving Brownian Motion has Stationary Increments

In Oksendal's 'Stochastic Differential Equations', we define Brownian Motion as follows: Fix $x\in\mathbb{R}^n$ and define for $y\in\mathbb{R}^n$: $$p(t,x,y)=(2\pi ...
-3
votes
1answer
31 views

Probability distribution quesiton [closed]

enter image description here Question is above showed as picture. I am stuck at part c.
0
votes
0answers
15 views

Expectation of truncated negative binomial distribution

This is what I have done, I would like to know if there is a way to simplify it, solve summations,...
0
votes
3answers
42 views

Calculate the PMF, mean and variance of X for x=-1,1

An Urn contains 7 red and 11 white balls. Draw one ball at random from the urn. Let X=1 if a red ball is drawn, and let X=-1 if a white ball is drawn. Give the pmf, mean, and Variance of X. I know ...
0
votes
3answers
32 views

How to find the expected value of a function.

An appliance dealer sells three different models of freezers having 13.5, 15.9, and 19.1 cubic feet of storage space. Let $X$ be the amount of storage space purchased by the next customer to buy a ...
0
votes
1answer
12 views

Probabilities over the Union of Differently Distributed Populations

I'm working with a problem where the sample space is the union of two populations, each normally distributed. Specifically, I'm given that the heights of women are normally distributed, and the ...
0
votes
1answer
21 views

Convergence in distribution - using moment generating function

Let $X_{n1},X_{n2}...X_{nn} $ be independent random variables with a common distribution given as follows: $P(X_{nk} = 0)= 1 - \dfrac{1}{n} - \dfrac{1}{n^2} \quad,P(X_{nk} = 1)= \dfrac{1}{n} \quad ...
0
votes
1answer
35 views

The expected revenue problem

Question : A travel agent company organizes a tour with ticket price $\$50$ and the ticket is non-refundable. The company has a bus with $20$ seats. The company knows that the participant might not ...
1
vote
2answers
38 views

Binomial-like probability problem between two independent groups

Question A study is conducted to monitor the health of two independent groups in a year where there are 10 participants for each group. Each participant will quit from the study with probability 0.2, ...