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

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2
votes
2answers
205 views

how to understand the generation of cauchy distribution from uniform distribution?

I am learning some basic idea on generating cauchy distribution from uniform random generator $u \in [0, 1]$. I know it was discussed before in How to generate a Cauchy random variable, but during my ...
1
vote
1answer
125 views

Probability Generating Function of a Negative Multinomial Distribution

Derive the probability generating function (pfg) of a negative multinomial distribution with parameters $(k; p_{0}, p_{1}, ..., p_{r})$ where the k-th occurrence of the event with the probability ...
0
votes
0answers
17 views

limit of characteristic function Normal

I need to find the limit of the following characteristic function as $s \rightarrow\infty$ $\frac{e^{-it\frac{s}{\sqrt{s^2+s}}}}{(1-(e^{-it\frac{1}{\sqrt{s^2+s}}}-1)s)}$ The top part seems to reduce ...
0
votes
3answers
1k views

Probability of balls drawn with replacement

We have two bags, Bag A has 40 red balls and 15 blue balls, Bag B has 40 blue balls and 10 red balls. One of these bags is selected at random and from it five balls are drawn at random, replacing each ...
-2
votes
1answer
73 views

Proving the variance of pareto random variable equals (a*lambda)/((a-1)^2*(a-2))

So my PDF for the Pareto distribution is: $$\dfrac{a\lambda^a}{x^{a+1}},\quad x\ge\lambda$$ To find the variance, you need to find the integral of $x^2\dfrac{a\lambda^a}{x^{a+1}}$ and subtract it from ...
0
votes
1answer
31 views

Visualizing a probability measures through a probability density functions

I found a previous question with a very nice answer, but still there is something that is not completely clear to me. We start from a space $(X, \Sigma)$, endowed with a $\sigma$-algebra, and we let ...
2
votes
0answers
31 views

Distribution of some linear combination of Normal RVs

I would like to ask for help concerning this question lifted from the book An Introduction to the Theory of Statistics by Mood, Graybill, and Boes (2nd ed.). Let $X_1$ and $X_2$ be independent ...
0
votes
3answers
451 views

Finding a constant $ z $ such that $ P(Z \leq z) = 0.95 $ when $ Z \sim \text{N}(0,1) $.

This is for a homework assignment on normal distributions. Question: a) Find a constant $z$ such that $P(Z \leq z) = 0.95$ b) Find a constant $z$ such that $P(Z \geq z) = 0.95$ I'm having trouble ...
0
votes
1answer
24 views

Sum of multiple distributions

Three fish $F_1, F_2, F_3 $ are selected at random from the pond, their weights are independent and identically distributed $ \sim N(20.3, 0.13^2) $ a) Find $ P(\bar{F} \ge 20.1) $ b) Find $ P(F_1 ...
1
vote
2answers
114 views

How to calculate the probability distribution function (PDF)?

Sorry for the dumb question, I've been struggling with understanding the probability distribution function formula, what does "x" and "d" stand for in the formula , and how to use the formula? I've ...
1
vote
1answer
71 views

Compute lower tail probability from upper tail - bivariate normal

Let $X,Y$ be bivariate normal with correlation $\rho$. I'd like to compute $\mathbb{P}(X \leq x, Y \leq y)$, but I have only a function (implemented on a computer) that returns $\mathbb{P}(X \geq x, ...
1
vote
1answer
57 views

Birth-death process: What is the distribution of reached states before reaching an absorbing state?

Intro I am working on a birth-death process. For a given choice of parameter ($n=6$, $Wa=1$, $Wb=0.95$, see below), the transition matrix is $$\left( \begin{array}{ccccccc} 1. & 0.144928 & ...
0
votes
0answers
36 views

what is determinantal process?

Would anyone please explain what does this mean? A random point process $P$ on a discrete base set $Y = \{1,\ldots,N\}$ is a probability measure on the set $2^Y$ of all subsets of $Y$. Let $K$ ...
6
votes
0answers
157 views

Normalizing factor for product of Gaussian densities - interpretation with Bayes theorem

The normalizing factor for the product of two multivariate Gaussian densities, $f(x)$ and $g(x)$ with mean vectors $a$ and $b$ respectively, and covariance matrices $A$ and $B$ respectively, is itself ...
0
votes
1answer
135 views

How to determine the support (bounds) of a cumulative distribution function

Suppose that X is uniformly distributed on [0,2]. Suppose that Y = X$^3$ Find the probability density function for Y and state the support for Y. I know the CDF will be G(y) = P(Y $\le$ y) = P(X$^3$ ...
-4
votes
1answer
119 views

An unbiased estimator for the parameter of exponential distribution

The times between arrivals at a fish shop queue can be described by an exponential distribution $X$ with parameter $\theta$ . If $Y$ is the random variable given by the sample mean on $n$ ...
0
votes
1answer
161 views

2 User Queuing Model Probability Problem

Consider two users who arrive to a system with exponential arrival times with parameters $\lambda_a$ and $\lambda_b$. Once they arrive, the users stay in the system for an exponentially distributed ...
2
votes
0answers
57 views

Random variables and the topology of weak convergence

To see what's going on, I am trying to translate the idea of topology of weak convergence on a random variable setting, just to get some concrete intuition. This is what I have got so far (where the ...
2
votes
2answers
61 views

Sums of independent random variables (more than two) [closed]

I read that the convolution of two iid random variables is $$(f * g) (z) = \int f(z-y) g(y) dy$$ What is the general formula for more than two RVs? For example, for three RVs.
1
vote
1answer
40 views

continuous probability density functions

Continuous distributions assign probability 0 to individual values. But, according to DeGroot, it doesn't mean that it is impossible for the random variable to take individual values. So, why not make ...
5
votes
1answer
63 views

Distribution of occurrences of “pairs of heads” in $N$ coin tosses

Let's say we toss a weighted coin $N$ times, each with probability $p$ of landing heads up. What's the distribution of the number of times we'll see $k$ pairs of heads? For example, HTHHHTHH would ...
-1
votes
2answers
849 views

Binomial Random Variable and Bernoulli trials problem

Let X be a Binomial random variable defined as the sum of 6 independent Bernoulli trials. The probability of a Bernoulli taking the value 1 is given by p. Suppose that prior to the 6 Bernoulli trials, ...
1
vote
2answers
60 views

Probability problem without the usual information

A random variable $\xi$ has the normal distribution with expectation value of $25$. The probability of such random variable being in the interval $(10,15)$ is $0.07$. The question now is how can one ...
1
vote
1answer
63 views

Random number generator from a piecewise PDF

I'm trying to create a random number generator on the interval $(a,c)$ given a probability density function defined as: $$f(x) = \left\{ \begin{array}{lr} \dfrac{C}{x} &, x \in (a,b)\\ ...
2
votes
2answers
69 views

Is there a meaningful way to approximate a discrete random variable?

Is there a meaningful way to find a continuos approximation of a discrete random variable? Thoughts for the $L^2$ case If $X \in L^2$, then we may want to consider the subspace $V = C^1 \cap L^2$ ...
1
vote
1answer
57 views

How do we approximate sum of random variables?

Suppose we have independent, identically distributed random variables $X_n \notin L^1$. I would like to approximate, in some way, the distribution of their sum $\sum X_n$ .The problem is that these ...
2
votes
0answers
23 views

A bound on $\mathbb{E}[\max_{i} e^{pX_i}]$

Suppose $X_1, \dots, X_n$ are independent, nonnegative random variables satisfying $\mathbb{P}(X_i > t) \le a e^{-bt^2}$ for some constants $a, b >0$. Can one show that for each $p>0$ there ...
0
votes
1answer
18 views

$(X,Y)$ normal, find $P(\mathrm{sign}(X) \neq \mathrm{sign}(Y)$

Suppose $(X,Y)$ is a bivariate normal with mean $0$ and $E[X^2]=E[Y^2]=1, E[XY]= \delta$. I think that $P(\mathrm{sign}(X) \neq \mathrm{sign}(Y)) = \frac{1}{\pi}\arccos\delta$ and am wondering what ...
1
vote
1answer
27 views

If $X$ has density, when has $X\cdot I_A$ a density?

Let $(\Omega, \mathcal F, P)$ be a probability space, and $X$ be a random variable with some density function $f_X$. If $A \in \mathcal F$, then the indicator function $I_A$ has, as a discrete random ...
-1
votes
1answer
75 views

Why is the area under the pdf for the Von Mises distribution not one?

I've been playing with the Von Mises distribution for a project I'm doing in python and I'm confused about it. I'm drawing the pdf, which is defined by wikipedia here as $p(x|\mu, k) = \frac{\exp{(k ...
1
vote
1answer
147 views

Probability confusing question

I saw this in my probability class past exam papers I saw the answer key but I still can't fully understand. I wish somebody can walk through this with me :) A company takes out an insurance policy ...
1
vote
1answer
67 views

Calculating the distribution of a compound random variable

Given $X\sim U(1, 0)$ and $Y\sim Exp(1)$, determine the density function of $Z:=\frac{X}{Y}$. Now, without looking up how to do it I tried to figure it out myself. The value of the density function ...
3
votes
0answers
221 views

Joint pdf of N > 1 i.i.d. random variables isotropic if and only if they are centered gaussian?

Are centered Gaussian densities given by $$f_X(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-x^2/(2 \sigma^2)}$$ the unique densities such that the joint pdf of $N > 1$ independent and identically ...
0
votes
1answer
27 views

“Distance” of iid gaussian variables [duplicate]

Take two i.i.d. Gaussian R.V.s $X$ are $Y$ both of which are $~N(0,a\sigma)$. Define a new R.V. $D = \sqrt{X^2 + Y^2}$. What's the expected value $E(D)$? In researching this I'm seeing references ...
1
vote
1answer
74 views

Joint distribution function from marginals

Is it possible to obtain joint distribution function when only the marginal distribution functions of random variables are given and, the random variables are not independent? If possible, it would ...
0
votes
1answer
42 views

Conditional probability distribution notation versus conditional probabilities of a single sample space?

When writing conditional marginal probabilities, the following seems to be the notation: $$p_{i|Y=y_{j}} = P(X=x_{i}|Y=y_{j}) = \frac{P(X=x_{i},Y=y_{j})}{P(Y=y_{j})}=\frac{p_{ij}}{p_{+j}}$$ This is ...
0
votes
1answer
456 views

Unbiased estimator for geometric distribution parameter p

I believe that the MLE of parameter $p$ in the geometric distribution, $\hat p = 1/(\bar x +1)$, is an unbiased estimator for $p$ and would like to prove it. So far, I have: $E[\bar x + 1] = E[\bar ...
2
votes
2answers
75 views

Expected value with negative exponent

I am trying to solve identify the expected value of a statistic that involves a fraction. I have simplified the expression to: $E[\frac{1}{1+ \sum_i x_i}] = E[\frac{1}{1+ T}]$ However, I am not sure ...
0
votes
1answer
62 views

Approximating a joint pdf using normal density of two independent variables

I know that given these two random variables (which correspond to the $x$ and $y$ coordinates of a random walk after $n$ steps, their joint probability density function can be $approximated$ by a ...
0
votes
1answer
37 views

Binomial-Poisson limit

I want to show that if $Z_n$ has the binomial distribution with parameters $n$ and $\lambda/n$ with $\lambda$ fixed, then $Z_n $ converges in distribution to the Poisson distribution, parameter ...
1
vote
1answer
22 views

Probability function and distribution - taking out fish from a pool

In a pool of fish there are 4 fish of type A, 3 fish of type B, 2 fish of type C, 1 fish of type D. We take out fish without returning them until we get fish of type C for the first time. ...
0
votes
1answer
34 views

Binomial distribution tail inequality

Let $X \sim \mathrm{Bin}(n,p)$ does there exist $l$ ideally $l=f(n)$ such that $P(X<l)=o(1)$ in the limit $n\rightarrow \infty$? I'd be looking for the largest possible $l$.
0
votes
1answer
41 views

A Continuous random variable X has probability density function $f(x)=ae^{-ax}$

A Continuous random variable X has probability density function $f(x)=ae^{-ax}$ where I found $a=0.5ln2$ I Found that the mean of this distribution occurs at X=2. Now, I was then asked what is: ...
0
votes
1answer
51 views

Poisson Distribution to Calculate plane crashes

The number of passenger planes that crash every day follows the Poisson distribution with parameter p. The number of crashes each day is independent. What is the probability of exactly 3 planes ...
0
votes
1answer
44 views

How to get the value of 'scaled' binomial distribution?

People kindly told me that there is not a equivalent popular distribution for $aX$ when $X$ is distributed as Binomial, but it is just a 'scaled' distribution. Here, $a$ is a positive constant. ...
0
votes
1answer
38 views

Poisson Probability (Shopkeeper Sales)

SOLUTIONS: (A) 0.1804 (B) 0.0166 (C) 0.3233 Mean = 2/7*5 (a) x = 3 (b) x > 5 I'm still unsure how to approach each question, because I still get the wrong answers.
0
votes
1answer
64 views

pdf: What is the distribution of aX when X ~ Binomial / Gaussian

Question When $X$ is distributed as binomial or Gaussian, is $aX$ equivalent to some famous distribution? Here, $a$ is a real and positive number. Background I know a general formula giving $aX$'s ...
0
votes
1answer
37 views

Joint Probability with many values

Consider I have the following tree structure which provides the relation between various entities. Associated with this, I have the following table with data. ...
1
vote
1answer
46 views

Is there any simple formula for this probability distribution of random walk?

Assume $\{S_n\}_{n\geq 0}$ transits as follows: $S_0=0$, for $k\geq 1$, $P(S_{n+1}=k+1|S_n=k)=\alpha$, $P(S_{n+1}=k|S_n=k)=\beta$ and $P(S_{n+1}=k-1|S_n=k)=1-\alpha-\beta$, where ...
1
vote
1answer
48 views

Expected value of Bernoulli with probability of success Gaussian distributed

I have a circle with centre $(0,0)$. I am generating Matlab code to include $N$ neurons in a neural network. The probability of including individual neurons in a network decays exponentially with ...