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

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2
votes
2answers
59 views

How to prove something at Uniform distribution…

$X\sim U (0,1)$. The point $X$ divides $[0,1]$ to two parts. $Y=\frac{\text{The big part}}{\text{The small part}}$. ($Y$ is the ratio... $Y\ge1$). What is the density function of $Y$? I'd like to ...
0
votes
1answer
35 views

Solving this random variable problem

This is an earlier problem Proving this random variable problem but generalised, maybe you want to take a look at that one first? $X_1,X_2,X_3,\ldots$ are IID random variable taking values in ...
1
vote
2answers
55 views

Proving this random variable problem

$X_1,X_2,X_3,\ldots$ are IID random variable taking values in $(-1,\infty)$. Also $t\in(0,1)$. Define random variables $Y_1,Y_2,Y_3,\ldots$ recursively like $$Y_1 = (1+tX_1)$$ $$Y_n = ...
1
vote
0answers
28 views

Probability equations

Sorry, I wasn't sure what to name this one. I have the following assigned problem, and I am not entirely sure how to attack it: Let $A_1 = B_1^2$ and $A_2 = B_1 B_2$, where $B_1$ and $B_2$ are two ...
2
votes
1answer
51 views

Uniform distribtion: clarification of $f_X(x)$

I have $Y=2(X-1)^2 -1$ where $X$ is uniform distributed on $[0,2]$ I want to find the pdf of $Y$ and expected value of $Y$. My question is just: Does $X$ have pdf $f_X(x)= \frac{1}{2}$?
1
vote
1answer
37 views

Expectation of Continuous variable.

Given the probability density function $$ f(x) = \begin{cases} \frac{cx}{3}, & 0 \leq x < 3, \\ c, & 3 \leq x \leq 4, \\ 0 & \text{ otherwise} \end{cases} $$ I have found $c$ to be ...
0
votes
2answers
34 views

Chi Square Test for one variable

I got a question about the use of Chi Square test. Let's assume I am conducting a survey. And I have a question: "Have you ever heard of the Internet"? The possible answers are: "Yes", "No", "Not ...
1
vote
2answers
50 views

probability distribution of a random selection from one of two bernoulli random variables

Say I have two bernoulli random variables, $X$ and $Y$, and that I want to randomly select from either one of them with some probability $p$. In other words: $X$ ~ $Be(p_X)$ $Y$ ~ $Be(p_Y)$ Then I ...
1
vote
0answers
46 views

What is the expectation of $X^2$ where X has a truncated normal distribution?

Suppose that $X\sim N\left(a,\mbox{ }\sigma^{2}\right)$, what is $E\left\{ \left[1\left\{ X>b\right\} \exp\left(X\right)\right]^{2}\right\}$? $b$ is a constant.
2
votes
1answer
133 views

P.d.f. of $XY$, where $X, Y$ are independent uniformly distributed over $[0,1]$ [duplicate]

I tried to change the variables: Let $U=XY$ and $V=Y$; so then the Jacobian is $1/v$. So joint pdf $g(u,v) = f(x,y)\cdot (1/v) = 1/v$ Would you then integrate over $v$ from $0$ to $1$ to get the ...
0
votes
1answer
45 views

Limit of Poisson Distribution

Just for fun, I'm looking at the concentration of the Poisson Distribution near it's mean. For $\lambda=10$, there is a 36% probability of being within 10% of the mean. For $\lambda=100$, that ...
1
vote
1answer
31 views

Simple distribution problem

I have $F_X(x)= \frac{1}{2} + \frac{1}{\pi}arctan(x)$ And I know $Y = aX+b$ Does that make $F_Y(y) = \frac{1}{2} + \frac{1}{\pi}arctan(\frac{y-b}{a})$ Pretty sure $F_Y(y) = F_X(g^{-1}(x)$ where we ...
3
votes
2answers
530 views

How to find nth moment?

I'm quite new to the field so please bare with me. Problem: Let ξ be a random variable distributed according to a log-normal distribution with parameters μ and $σ^2$, i.e. log(ξ) is normally ...
1
vote
1answer
62 views

Which Queue to Join at the Super Market

Last night I started wonder about the fastest way to take a shopping trip with my university flat mates and was wonder about how we should queue for the check out. I have a feeling that queue theory ...
1
vote
0answers
28 views

Maximum possible distance between two vectors sampled from n-variate Gaussian

What would be the probability distribution of the distance between two vectors sampled from n-variate Gaussian distribution? Thanks.
2
votes
5answers
276 views

Find the distribution of $X_1^2 + X_2^2$? [duplicate]

Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ which means (mean = 0, variance = $\sigma^2$) random variables. What is the distribution of $X_1^2 + X_2^2$? My approach is that $X_1\sim N(0, ...
0
votes
1answer
19 views

Is the statement: $p\left(\left.y\right|h^{-1}\left(\varphi\right)\right)=p\left(\left.y\right|\varphi\right)$ correct?

Say I have a likelihood function $p\left(\left.y\right|\theta\right)$ and I make the reparameterization $\varphi=h\left(\theta\right)$ using the bijective function $h$ with inverse $h^{-1}$. Then it ...
1
vote
1answer
123 views

Is a log-normal distribution uniquely determined by its moments or not?

Wikipedia states that A log-normal distribution is not uniquely determined by its moments $\text{E}[X^k]$ for $k\ge 1$, that is, there exists some other distribution with the same moments for all ...
3
votes
1answer
115 views

Circular distribution of circles

Suppose you have $n$ objects , distributed randomly, in a circular manner of radius $r$. Each objects is of area $A$. So my question is if you draw line everywhere from the center to the surface of ...
1
vote
1answer
50 views

Speed Networking for Small Group (10 people)

I'm hosting a speed networking session where there are 10 people at 5 tables (2 people per table.) Each table is hosted by a different person. There will be four rotations. It does not matter if ...
3
votes
0answers
107 views

determine type of probability distribution

let us consider following model $$y(t)=A_1 \sin(\omega_1 t+\phi_1) + A_2 \sin(\omega_2 t+\phi_2) + A_3 \sin(\omega_3 t+\phi_3)+ \ldots +A_p \sin(\omega_p t+\phi_p)+z(t)$$ we have three parameter ...
2
votes
1answer
26 views

Discrete distribution with the minimum variance

Consider a discrete random variable $X \in \{x_1, x_2, \ldots, x_n\}$, where $n < +\infty$ and $x_1 < x_2 < \ldots < x_n$. Let pose $p_i = \text{Pr}(X = x_i)$, with $\sum_{i=1}^N p_i = ...
1
vote
2answers
53 views

Find vector of expected values ​​and covariance matrix

For vector (X,Y) with density $f(x,y)=C exp \{ -4x^2-6xy-9y^2 \}$ find constans C, vector of expected values ​​and covariance matrix. How to do this kind of exercises?
1
vote
1answer
34 views

Probability to remove complete buckets from histogram

I have some data and its distribution as a histogram. Let's say for example there are the following 20 data items: 3 times a A 5 times a B 4 times a C 4 times a D 3 times a E 1 times a F Now I ...
0
votes
1answer
84 views

Standard deviation with multiple means and deviations

The amounts of a certain mineral that can be produced in a day from mines $1$, $2$, and $3$ are independent normal random variables with means equal to $80$, $90$, and $75$ pounds, respectively, ...
1
vote
1answer
25 views

Exponential deviation with two $x$ values

I recently got interested in this topic of standard deviation. My TA did not have any time to go over this topic so I was trying to teach myself it recently. My TA said if he had more time he would ...
0
votes
2answers
41 views

A probability question regarding two independent uniform distrbutions.

I am thinking about this question: $X_1$ and $X_2$ are independent $Unif(0,1)$ random variables. (a) Derive the pdf of $\overline{X}=\frac{X_1+X_2}{2}$. (b) Calculate $E(\frac{X_1}{\overline{X}})$. ...
5
votes
1answer
257 views

Continuous probability distribution with no first moment but the characteristic function is differentiable

I am looking for an example of a continuous distribution function where the first moment does not exist but the characteristic function is differentiable everywhere. Cauchy distributions do not ...
0
votes
1answer
18 views

Calculate the Cumulative density.

$U$ is a random variable in the range of $(0,3)$. The random variable $W$ is the output of the clipper described by $W=g(U)=U$ for $U\le 1$ and $1$ for $U>1$ find the cdf of $FW(w)$ ...
3
votes
3answers
113 views

How does one 'correct' a table that doesn't add up to $100\%$?

I have a table consisting of a number of whole percentages $x_i$ between $0\%$ and $100\%$. However, they don't add up to $100\%$ (rather they add up to $101\%$). But they 'should'. Assuming that any ...
1
vote
2answers
54 views

What's wrong with this random variable proof?

Let $X$ be a Binomial random variable $\sim B(p, n)$. Show that for $\lambda > 0$ and $\epsilon > 0$, $P(X - np > n\epsilon) \le \mathbb{E}\{\displaystyle e^{\lambda(X - np - ...
0
votes
1answer
26 views

definition of distribution function of random variable

please help me to understand fully following definition : i am using this book http://www.math.harvard.edu/~knill/books/KnillProbability.pdf page 79,i can't understand some part,in spite of this ...
1
vote
0answers
51 views

Warren's proof for Benford's Law

Warren has a little proof of Benford's law in Hacker's Delight. To quote: Let $f(x)$ for $1 \leq x < 10$ be the probability density function for the leading digits of the set of numbers with ...
0
votes
1answer
58 views

Puzzle for Applying the Definition to a t distribution

The coeffcient of variation (CV) for a sample of values $Y_1,\ldots, Y_n$ is defined by $$ CV = S/ \bar{Y}.$$ Let $Y_1,\ldots, Y_n$ be a random sample of size $10$ from a normal distribution with mean ...
1
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0answers
38 views

From joint to conditional probability distribution?

I found the following expression in a paper but I'm not quite sure where does it come from: let $Y$ and $V$ be two random variables defined on the probability space $(\Omega, \mathcal{F}, ...
1
vote
0answers
131 views

Conditional Probability with Gamma Distribution

I have a stochastic process $X$ that is distributed according to the Gamma distribution: $X$ ~ $\Gamma(\alpha, \beta)$ I would like to compute the following : Probability = $P(X_t > 0.75 | X_{t-1} ...
4
votes
1answer
36 views

Properties of cumulative binomial distribution

Let $F\left(k, n, p\right) = \sum_{i=1}^k\binom{n}{i}p^i\left(1-p\right)^{n-i}$ denote the cumulative binomial distribution function. If $F\left(k, n, p\right)-F\left(k, n, p'\right) \geq ...
1
vote
1answer
11 views

Probabilty of random number distribution

Given a random number generator generating integer numbers in the range 1 to N. What is the probability that a given number appears Q times (not necessarily sequentially, but in any order) in a ...
1
vote
1answer
198 views

Find the bias for the Maximum-likelihood estimator

Let $X_1,...,X_n$ be a random sample from the pdf $$f(x|\theta) = \theta x^{\theta-1} , 0 \leq x \leq 1, \theta >0.$$ I found the Maximum-likelihood estimator of $\theta$ is $$\hat{\theta} = ...
3
votes
2answers
211 views

Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ random variables. What is the distribution of $X_1^2 + X_2^2$?

Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ which means (mean = 0, variance = $\sigma^2$) random variables. What is the distribution of $X_1^2 + X_2^2$? My approach is that $X_1\sim N(0, ...
1
vote
2answers
258 views

Find the pdf of $\prod_{i=1}^n X_i$, where $X_is$ are independent uniform [0,1] random variables.

How do I find the pdf of $\prod_{i=1}^n X_i$, where $X_is$ are independent uniform [0,1] random variables. I know X~U[0,1], -ln(x) is exponential(1). I also know the sum of two or more independent ...
2
votes
1answer
294 views

Central Limit Theorem for uncorrelated (non-independent) but bounded random variables

Given uncorrelated, discrete random variables $X_i$ that are bounded, e.g., they can only take on values $|X_i| \leq 4$, then is there a form of the central limit theorem that one can apply to the ...
1
vote
1answer
32 views

Independence proof

$X,Y\sim\mathscr{E}(1)$ (exp. with parameter $1$) and independent. I'd like to show that $\min\{X,Y\}$ and $|X-Y|$ are independent. Let $Z=\min\{X,Y\}$ and $W=|X-Y|$. The transformation gives a ...
2
votes
1answer
127 views

Expectation of $x^4$ [closed]

Can anyone help me prove that Expected Value of $X^4$ is $3\,($Var$(X))^4$, if the Expected Value of $X$ is zero and Var$(X)$ is the Variance of $X$ $(N(0,\sigma^2))$.
5
votes
1answer
56 views

Probability roots of quadratic lie in unit disc

$A,B\sim\mathscr{U}(0,1)$ and independent. We consider: $$x^2+2Ax+B=0$$ Given that both of the roots of this equation are real, what is the probability that they lie in the unit disc? ...
0
votes
1answer
92 views

How to mathematically prove that we are sampling from same distributions?

The content of this question is about rigorously proving something which is otherwise considered easily correct intuitively. Let's assume we have a multivariate distribution $g(x_1,x_2,...,x_n)$ over ...
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vote
0answers
17 views

Cumulative distribution function of a model similar to the multinominal distribution

I would like to solve a problem similar to the multinominal distribution (http://en.wikipedia.org/wiki/Multinomial_distribution): For k independent trials each of which leads to a success for ...
0
votes
1answer
204 views

Probability Density Function and Proof

Given the Probability Density Function: $f(x)=kx(2-x), 0\leq x\leq 1$ Prove that $k=\frac 3 2$ Looks like it should be a Beta Distribution, but all examples of a beta distribution use the format: ...
1
vote
0answers
18 views

Raffle between different groups composed by different numbers

I've got this issue, I need to prepare a raffle between teams for a cars race. Cars are grouped by teams. Rounds are 1:1, composed by different manches until the cars are done. Total number of cars is ...
1
vote
1answer
38 views

Use a probability function of X to prove the Moment Generating Function

So my question reads: Given the probability function of $X$ as follows: $f(x) = \frac{1}{2} \left(\frac{2}{3}\right)^x$ , $x=1, 2, 3, \dots$ (a) Use the definition $M_x(t)= E(e^{tx})$ to show ...