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

learn more… | top users | synonyms

0
votes
1answer
40 views

limit of probability distribution

Let $(F_n)$ and $F$ be one-dimensional cumulative distribution functions, with $F_n\to F$ in distribution. This means for all continuouity points $x$ of $F$ $\lim_{n\to\infty} F_n(x)=F(x)$ holds. For ...
0
votes
1answer
38 views

Discrete Distribution question

Can anybody help out or give me a hint please? Thanks a lot!
2
votes
1answer
48 views

Finding the joint density of $Z=X+Y$ where $X\in U(0,1), Y\in U(0,\alpha)$

I'm trying to find the joint density of $Z=X+Y$ where $X\in U(0,1), Y\in U(0,\alpha)$ Here $U$ is the uniform distribution. The method I use i to introduce an auxilary variable $W=X$ and then use ...
1
vote
0answers
63 views

Convergence in distribution for $\frac{Y}{\sqrt{\lambda}}$

Given a sequence of independent r.v's $\{X_n\}_{n\geq 1}$ such that $P(X_n=x)=\frac{1}{2}$ if $x=-1$ and/or $x=1$ Let $N\in Po(\lambda)$ be independent of $\{X_n\}_{n\geq 1}$ and we set that ...
0
votes
1answer
202 views

Mean and median of a linear piecewise pdf

I am studying for a test and I was wondering what is the mean and median of any linear piecewise pdf and why. Also just so you know p.d.f is probability density function.
0
votes
1answer
43 views

Why do different probability distributions have different restrictions on their parameters?

Is it correct that the parameters of the following distributions must be taken from the intervals given below? Bernoulli. $p$ from $[0, 1]$ Binomial. $n$ positive integer, $p$ from $[0, 1]$ ...
1
vote
2answers
153 views

Density of the sum of $n$ uniform(0,1) distributed random variables

I am working on the following problem: Let $X_1, X_2, \ldots, X_n, \ldots$ be iid. random variables, each of Uniform$(0,1)$ distribution. Denote by $f_n(x)$ the density of the random variable $S_n ...
0
votes
1answer
27 views

How can I show that the minimum is first sucess distribution?

Given the random variables $X_1, X_2, \ldots X_n \ldots$ with probability mass function: $$p_X(-1)=\frac{1}{4},\, p_X(0)=\frac{1}{2},\, p_X(1)=\frac{1}{4}.$$ We introduce a new random variable $N=$ ...
1
vote
1answer
268 views

On the mgf of the Logistic Distribution

So the Logistic Distribution pdf (w/ mean = 0 and shape parameter = 1) looks like this: $$f_X(x)=\frac{e^{x}}{({1+e^{x}})^2}\;\;, \;\;-\infty<x<\infty$$ Now, I am interested in getting its ...
0
votes
0answers
34 views

From continuous distribution to discrete and vice-versa.

Assume a random variable $a$ that can be either 0 or 1 with probability $\mu(0)=\mu_0$ and $\mu(1)=\mu_1$ respectively (without lost of generality, $\mu_0<\mu_1$). Assume a second variable $X=(a_1, ...
3
votes
1answer
258 views

Distribution function of the sum of poisson and uniform random variable.

Merry Christmas to everybody. I am working on the following problem. Let $X$ and $Y$ be independent Poisson($\lambda$), respectively Uniform$(0,1)$ random variables. Find the distribution function of ...
0
votes
2answers
56 views

discrete math probability question

To find an object X, one needs to search linearly through two lists: L1 of 50 elements and L2 of 40 elements. All the elements from both lists are distinct. The probability of X∈L1 is 40%, of X∈L2 is ...
1
vote
1answer
51 views

locally linearize a CDF

I have a sequence of discrete CDF's that converge to continuous CDF. Assume we call it $F_n(x)$. If say at some point, say $R$, $F_n$ is differentiable, then we can write $F_n(R+\xi) \approx ...
0
votes
1answer
37 views

If $X_{n} \xrightarrow{d} X$ , then show that $X_{n}^2 \xrightarrow{d}X^2$.

Let $X_{n} \xrightarrow{d} X$ , then show that $X_{n}^2 \xrightarrow{d}X^2$ converges in distribution to $X$.
1
vote
1answer
109 views

Normal approximation for binomial distribution isn't giving correct result, z score comes out 0

I'm trying to use the normal distribution to calculate approximate values for the (cumulative) binomial distribution with large values (since it's impractical to evaluate the factorials). I'm very ...
0
votes
1answer
23 views

Question about normal distribution statistics

I am doing a project about descision analysis, but I don't understand one crucial step in my book. F(x) is the standard normal cumulative distribution function, and we have: $$F(x) = 0.744$$ From ...
0
votes
0answers
22 views

Find x, when F(x) is a cdf with given mean and std

I have a small question. For a problem in desicion analysis course, I come to the equation F(x) = 0.8 where X is distributed with mean 50 and standard deviation 10. How can I find x? (my probability ...
8
votes
1answer
202 views

What is the intuition behind the generalized confidence interval?

What is the intuition behind the generalized confidence interval? My best description on GCI that it is the way to derive a formula to calcuate the area of the center region in a asymetry distribution ...
1
vote
1answer
119 views

Multivariate Gaussian decomposition

I've seen around the claim that an $n$-dimensional Gaussian random variable (say, having unit covariance) can be decomposed into the product of two independent random variables. $$U=ZS$$ where $Z$ is ...
0
votes
1answer
33 views

Given a distribution to generate a set of numbers, what is probability of generating two consecutive numbers whose difference is greater than k?

Suppose I am generating a set of numbers {$x_1$, $x_2$, $x_3$ ... $x_n$} from a given probability distribution $f(x)$. Is it possible to calculate the probability of finding $x_{i+1}-x_i \geq k$, ...
1
vote
2answers
39 views

Integration of the product of probability densities

Does a probability density $f(x|\alpha)$ multiplied by another probability density $g(\alpha)$ , where of course both integrate to one, also integrate to one if we integrate with respect to $\alpha$? ...
0
votes
1answer
35 views

Generate random numbers from a family of PDFs

For a part of a simulation task, I need to generate (lots of) random numbers from the distribution $$P(E_k | N) = \frac{1}{E_k}\left(\frac{E_k}{k_BT}\right)^{N-1}\frac{1}{(N-2)!} e^{-E_k/k_BT} $$ ...
5
votes
0answers
94 views

Distribution for ratio of dependent quadratic forms.

Random vector $\mathbf{x}_{0}$ $\sim$ $\mathcal{N}\left(\boldsymbol{\mu}, \mathbf{\Sigma} \right)$ is a sum of two orthogonal random vectors: $\mathbf{x}_{0}$ = $\mathbf{x}_{1}$ + ...
3
votes
1answer
67 views

How to find $P(X=r)$ from probability generating function of $X$?

I have a probability generating function $$G_X(s) = \frac{p+ps}{1-s+p+ps}$$ and I need to find $P(X=r)$. How do I get this from the probability generating function? I was thinking about finding ...
0
votes
1answer
54 views

Scaling a probability distribution function

I have the following PDF that gives the probability of a certain annual wage being drawn: $f(w)=0$ if $w<20000$ $\frac{w-20000}{50000^2}$ if $w \in [20000,70000]$ $\frac{120000-w}{50000^2}$ if ...
1
vote
0answers
74 views

Expectation of a function of a Binomial random variable

Is it possible to obtain the explicit form of the following expectation, $E\left[\left(\frac{X}{1+X}\right)^n\right]$, where $X$ is a random variable following Binomial distribution with parameters ...
1
vote
1answer
305 views

How to recover the probability mass function from probability generating function?

Would someone please provide me an example of where we take a p.g.f and use it to derive the p.m.f. ? I understand that you were have to take the derivatives of the pmf, which is understandable ...
0
votes
1answer
189 views

How do I find $P(X+Y = k)$ for a geometric distribution?

If $X$ and $Y$ are independent identically distributed random variables where $P(X=k) = P(Y=k) = pq^{k-1}$ where $q = 1-p$. How do you find $P(X+Y=k)$? Is it acceptable to say that $$P(X+Y=k) = ...
0
votes
2answers
404 views

Compute Cov(X,Y) while X is the number of 1's and Y is the number of 2's in n dice rolls

Let $X$ be the number of 1's and $Y$ be the number of 2's that occur in $n$ rolls of a fair die. Compute $Cov(X,Y)$. What's wrong with my solution? Here it is: $Cov(X,Y)=E[XY]-E[X]E[Y]$ Compute ...
1
vote
1answer
113 views

Uniformly distributed points over the surface of the standard simplex

I would like to generate points that are uniformly distributed over the SURFACE of a standard $k$-simplex ($k$ dimensions, $k+1$ vertices). One way to efficiently generate points that are uniformly ...
0
votes
2answers
272 views

PDF of X/Y when X, Y are uniformly distributed

The question is as follows: Let $X$ and $Y$ be random variables uniformly distributed on $[0, 1]$. Find the PDF of $Z = Y/X$. I approached it in the following manner: $P(Z < z) = P(Y/X < z) = ...
0
votes
1answer
45 views

density function of $\frac{1}{\left(X+Y\right)^{2}+1}$.

X and Y are independent continuous random variables with the same density function. Find the density function of $\frac{1}{\left(X+Y\right)^{2}+1}$. I have tried getting the Jacobian where T maps ...
2
votes
1answer
38 views

What is $\Pr[T_a < T_b]$ for independent gamma RVs with same shape

Given independent gamma random variables $T_a, T_b$ with shape $k$ and rates $\lambda_a, \lambda_b$, what is $\Pr[T_a < T_b]$? Estimates are welcome! This question is motivated by the fact that, ...
1
vote
1answer
45 views

Poisson distribution proof

Looking over an exam and I have no idea how to finish when proving this: Prove that for a Poisson r.v. X, if the parameter $\lambda$ is not fixed and is itself an exponential r.v. with parameter 1, ...
1
vote
1answer
31 views

Derivin the Moment Generating Function

Question: Suppose f(x) = 1/10 for 0 < x < 10 and f(x) = 0 elsewhere, compute mx(s) for all s in the Reals This is what I have so far: mx(s) = E(e^(sX)) = integral (from 0 to 10) of ...
2
votes
1answer
75 views

Geometric Probability Distribution, Expected Values

Question: Let $X $~ Geometric $(\theta)$, and let $Y = \min(X, 100)$. Compute (a) $E(Y)$ and (b) $E(Y-X)$ I know that the Geometric distribution is $(1-\theta)^{k-1}\theta$ and I also know how to ...
0
votes
1answer
277 views

Conditional uniform distribution

I had this question in a quiz, and now that I am reviewing it, I am not sure if why my TA gave me the marks because I am pretty sure I am wrong. Let the r.v. $Y$ follow uniform distribution $U(1,2)$ ...
0
votes
1answer
75 views

Covariance problem.

The experiment is a three hat experiment with the following probabilities: $\frac15$ for $(1,2,3)$, $(1,3,2)$, $(2,1,3)$ and $(3,2,1)$, and $\frac1{10}$ probability for $(2,3,1)$ and $(3,1,2)$. Find ...
0
votes
0answers
21 views

Find the Cumulative Distribution Function of F(X) [duplicate]

Supposing that F is the Cumulative Distribution function of some c.r.v. X and that some inverse function F^-1 exists such that F^−1[F(x)] = x for all x ∈ R and F[F^−1(z)] = z for all z ∈ (0, 1) Put ...
2
votes
2answers
46 views

Compounding a binomial distribution where the number of observation is binomial?

I am looking for a definition of a binomial distribution where the number of observation is itself binomial. That is: $X \sim binom(N, q)$ When $N \sim binom(n, p)$. Is this a known distribution? And ...
0
votes
1answer
211 views

Multiple Choice Question Binomial Distribution

Jim didn't study for his math test, and has to guess randomly on 10 multiple choice questions. If each question has 4 choices, what is the probability of gym getting 8 questions correct? I'm ...
1
vote
2answers
1k views

Jointly Gaussian uncorrelated random variables are independent [closed]

Let $X,Y$ be jointly normally distributed and uncorrelated. Why are they independent?
2
votes
2answers
60 views

Poisson Distribution for X > y

I'm curious if there is a faster method for solving the following problem: Michael is observing the occurrence of bicycle accidents, and he has determined that B = the number of accidents in one day ...
1
vote
0answers
152 views

Upper Bound on Supremum of Expected Value

Let $\left( \Omega, F, P\right)$ be a probability space, where $P$ is a probability measure on $\mathbb{X} \subseteq \mathbb{R}^n$, so that $P(\mathbb{X}) = 1$. For all integer $i \geq 1$, consider ...
2
votes
1answer
38 views

if X and Y are Gauss distributed, what's the distribution of X^2-Y^2?

X and Y are independent random variables with identical Gaussian distribution; for simplicity, the variance shall be 1. What's the distribution of Z=X^2+Y^2? With a plus sign, it would be the ...
2
votes
1answer
348 views

Show that the nth order statistic is a consistent estimator of a uniform parameter

We assume that our observations come from a uniform $(0,\theta)$ distribution. Can you please check my work on the following? We can derive the distribution function of the maximum of the sample, ...
0
votes
2answers
82 views

variance and generating function - probability hat problem

Posted this previously but couldn't comment on it with the temp account i created so: We have a $3$ hat experiment where $(1,2,3), (1,3,2), (2,1,3), (3,2,1)$ have a $\frac{1}{5}$ probability and ...
0
votes
1answer
501 views

Question about the Irwin-Hall Distribution (Uniform Sum Distribution)

So I have been reading about the Irwin-Hall distribution online, it is a sum of uniform distributions on $[0,1]$, and it seems very interesting: ...
0
votes
2answers
57 views

What do we know about the pdf of $\bar{X}$

We have $n$ independent random variables $X_i$ all with mean $\theta$ and variance $\sigma^2$. The sample mean is given by $$\bar{X} = \frac{1}{n} \sum\limits_i^n X_i$$ and the means square error is ...
4
votes
2answers
59 views

Limiting case of Binomial(n,p)/n?

Let the random variable $X$ have distribution $X \sim \text{Binomial}(n,p)$. Let $Y = X/n$. What is the limiting distribution of $Y$, as $n \to \infty$? Does it have a simple distribution? Of ...