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

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5
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1answer
132 views

Probability of seeing to a certain distance in a forest and related problems

I was walking in a forest one day and saw trees all around me. I begun wondering about how far do I see in the forest on average. I was also reminded to the "proof" that the age of the universe is ...
1
vote
1answer
382 views

What does the error rate mean in Naive Bayes.

Can anyone explain what the Bayes error rate is in Naive Bayes, for instance in matlab: ...
3
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2answers
114 views

Estimate the density function of a distribution based on binomial distributions.

Let's consider a set of nodes $V$, and let some nodes be colored with one color choosen between two possible colors; denote the color $\alpha$ and $\beta$, with respectively $I>0$ and $K>0$ ...
3
votes
0answers
76 views

Hellinger Distance between Laplace Distributions

I'm looking for a closed form for the Hellinger distance between two (generalized) Laplace distribution with the same covariance (but different means). Any suggestions? Thanks in advance, Federico
0
votes
1answer
142 views

Probability - Dice rolling question [duplicate]

Possible Duplicate: Probability that the sum of all values of 5 pairs of dice will be between 30 and 40 Roll 10 dice. What is the probability the average is between 3 and 4? I know E[x] = ...
4
votes
1answer
204 views

“General” non centered Chi distribution (having correlated random variables)?

Let $\mathbf{X} = [X_0, X_1]^t \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ with $\boldsymbol{\mu} = [\mu_0, \mu_1]^t \in \mathbb{R}^2$ and $ \boldsymbol{\Sigma} = \begin{bmatrix} ...
0
votes
2answers
53 views

Why does the distribution of the probability change this way?

Bob wants to test a possibly biased coin's probability $p$ of getting heads. His prior prediction is that $p \sim \mathrm{Uniform}(0, 1)$. After getting all heads on 2 flips of the coin, why should ...
0
votes
1answer
177 views

A poll carried a survey to examine the approval rate of a policy in a country. Which is most appropriate?

Suppose a polling company carried a sample survey to examine the approval rate of a newly implemented public policy in a country. 1,600 citizens aged 18 and above were randomly selected in this ...
2
votes
1answer
56 views

Poisson distribution huge comfusion

I have this problem... although I tried a few failed attempts but.. The number of boats that being repaired at a repair campus , during a week, is a Poisson distribution with k parameter . If k is ...
2
votes
1answer
200 views

Binomial vs Poisson Distribution

On a problem in my textbook it asks to give an exact distribution of the a random variable and then in the next part it says to give a simple approximation for the same distribution. I know that the ...
0
votes
0answers
571 views

Binary symmetric channel capacity or mutual information inequality

I proved that I(X,Y) <= 1 - H(p) to the following way: How can I prove if I start in that way I(X,Y) = H(X) - H(X|Y), I ...
0
votes
1answer
298 views

Distance between random variables

Let $\Delta(X,Y)$ denote the (variational) distance between two discrete random variables $X$ and $Y$. $\Delta(X,Y)$ is given by: $\Delta(X,Y)=\frac{1}{2}\sum_{v \in V} |\Pr[X=v] - \Pr[Y=v]|$. $V$ is ...
3
votes
3answers
186 views

Using Binomial Distribution to evaluate this probability distribution

I am trying to find the value of a skewed distribution but can't make sense of what to plug in to evaluate the answer. This is the given: $$ \text{Let X be Binomial(n, p). } \text{Using that, ...
2
votes
1answer
318 views

Help understanding the Feynman-Kac formula

From wikipedia: Suppose we wish to find the expected value of the function $e^{-\int_0^t V(x(\tau)) d\tau}$ in the case where $x(\tau)$ is some realization of a diffusion process starting at $x(0) = ...
3
votes
2answers
1k views

Probability of one normdist being greater than another [duplicate]

I have two independant normally distributed random variables. X ~ N(657, 3) Y ~ N(661, 2) P(x > y) = ? How do I calculate the probability of X being greater ...
0
votes
1answer
314 views

covariance of two linear combinations of a bivariate normal distribution

$X$ and $Y$ are jointly normal, with the mean vector and covariance matrix given by: $$\mu= \begin{pmatrix} 1 \\ 2 \\ \end{pmatrix} \Sigma= \begin{pmatrix} 2 & 0.4 \\ 0.4 & 1 \\ ...
1
vote
0answers
172 views

To obtain the closed-form expression of CDF and PDF from the recurrence relation

Now I have a question, in which I need to find the probability mass function and the cumulative distribution function. But now I only have the recurrence relation. Here is the details: Assume ...
1
vote
1answer
582 views

Approximating a sum of two binomial distributions

A club basketball team will play a 60-game season. Thirty-two of these games are against class A teams, and 28 are against class B teams. The outcomes of all the games are independent. The team will ...
1
vote
1answer
76 views

biasedness/unbiasedness of an MLE.

To show whether an MLE I just found is biased/unbiased, would I need to find the expectation of the answer? Plus would I do this by integrating $\text{MLE} \cdot \text{pdf}$. My MLE is $ ...
0
votes
1answer
69 views

CDF for random variable $X(\omega) = 2(1-|2\omega - 1|)$

I don't know how to calculate this cdf, the modulus is very annoying, because the cdf definition is $P(X< x)$ in my case $P(\omega < x)$. But in the modulus equality I get this $P(-\omega < ...
2
votes
2answers
1k views

How do I read this distribution function: $\min(X,Y)$?

I'm confused on what the $\min$ means. For example if I need to find the distribution function of $\min(X,Y)$ what am I looking for exactly? Am I looking for the distribution of the minimum value of ...
1
vote
2answers
73 views

Stats - Likelihood function

Let $X_1, X_2, \ldots , X_n$ be a random sample from a distribution with the following pdf $$f(x|\theta) = \begin{cases} 1/(\theta_2−\theta_1), &\quad\text {for}\quad \theta_1 \leq x\leq ...
0
votes
1answer
1k views

How to calculate sum of negative binomial distribution ? it E[X] Var[x] and P(X = n |1 st event occurs on the 5 th try)

How to calculate sum of negative binomial distribution ? it $ E[X], Var[x]$ and $P\{X = n |\text {first event occurs on the 5-th try}\}$ $$P(X=k)= \binom{k-1}{r-1}p^r(1-p)^{k-r}$$
2
votes
2answers
142 views

Why is $\frac1\pi$ the answer for this Cauchy Distribution?

So I have this problem here, and I know what the correct answer is but I'm not sure of why it's the right answer. It says: Let $X$ be a random variable with the density function $$f(x)= \frac{1}{\pi ...
0
votes
1answer
87 views

How do I show that $P(N=n) = \frac{x}{n} P(X=x)$

Suppose that independent Bernoulli trials with parameter $p$ are performed successively. Let $N$ be the number of trials needed to get $x$ successes, and $X$ be the number of successes in the first ...
1
vote
1answer
130 views

Probability question on conditional prob

Suppose a-priori chance of getting malaria is 10%. A positive blood test indicates a 80% chance of actually having the disease; but 5% of time healthy people also test positive. Suppose you test ...
1
vote
2answers
83 views

Find the distribution of $X$ given that $P(X=k)=\frac 23 (k+1)P(X=k+1)$

A discrete random variable $X$ of values in $\mathbb N$ verifies the property that $$P(X=k)=\cfrac 23 (k+1)P(X=k+1)$$ What is the distribution of $X$? I found that $$P(X\ge 0)=\sum_{k=0}^\infty ...
0
votes
2answers
65 views

If $\text P(X>k+1)=\cfrac 12 \text P(X>k)$, what is the distribution of $X$?

If $\text P(X>k+1)=\cfrac 12 \text P(X>k)$, $k\in \mathbb N^*$ what is the distribution of $X$? So I did; $$\text P(X=k+1)= \text P(X>k)- \text P(X>k+1)=\cfrac 12 \text P(X>k)$$ ...
0
votes
1answer
150 views

Product of two independent stochastic variables $XY$

I have two independent variables $X\sim \mathcal B(n,p)$, Binomial and $Y\sim \mathcal P(\lambda)$, Poisson. How would I go about finding the distribution of $Z=XY$ and the couple $(Z,S)$, where ...
1
vote
2answers
283 views

Proof of $\sum_{k=0}^n k \text{Pr}(X=k) = \sum^{n-1}_{k=0} \text{Pr}(X>k) -n \text{Pr}(X>n)$

$X$ is a random variable defined in $\mathbb N$. How can I prove that for all $n\in \mathbb N$? $ \text E(X) =\sum_{k=0}^n k \text{Pr}(X=k) = \sum^{n-1}_{k=0} \text{Pr}(X>k) -n ...
1
vote
1answer
941 views

How to prove/show that some distribution is symmetric at 0

I have to prove that the Cauchy distribution is symmetric at 0. However, I'm not entirely sure how to do this. I'm given the problem: Suppose that a particle is fired from the origin in the $(x,y)$ ...
1
vote
0answers
125 views

generating a binomial distribution

I'm trying to sample from a data set using a binomial distribution with parameters p and n. Implementation-wise, I follow these steps I generate an array containing the values of the cumulative ...
0
votes
2answers
234 views

M/GI/1 service time distribution

I want to compute the distribution of the waiting time and the number of jobs for M/GI/1 where the service time is Heavy-Tailed or more specifically Pareto. I found this paper ...
3
votes
1answer
100 views

A question about infinities and distribution functions

Let $\mathcal{P}_i$ be the set of probability density functions to which $f_i$ belongs, $(i=0,1)$. Furthermore assume that $$L(y)=\frac{f_1(y)}{f_0(y)}$$ is an increasing function for any chosen ...
0
votes
0answers
268 views

pmf of X: the number of defects per yard

The number of defects per yard, denoted by X, for a certain fabric is known to have a Poisson distribution with parameter $\lambda$. However, $\lambda$ is not known, and is itself assumed to be random ...
0
votes
1answer
296 views

Time-evolving probability distribution functions with an equation of motion

I came up with this question a while ago and haven't been able to gain any insight on it. You are playing baseball. As a batter with finite vision capabilities, the only information you have about ...
0
votes
1answer
116 views

A confusing conditional probability problem

$n$ vehicles are stopped at random, the probability that a driver who is stopped is a beginner is $p$ while the probability that a driver who is stopped is a professional is $q$. There are drivers ...
0
votes
1answer
114 views

What is the joint probability mass function?

A pond has $r$ red fish, $b$ blue fish, and $g$ green fish. Let $R$ be the number of red fish, $B$ be the number of blue fish, and $G$ be the number of green fish in a random sample size of $N$. What ...
3
votes
2answers
285 views

$L^p$ norm of multivariate standard normal random variable

Given $X_i\sim \mathcal{N}(0,1)$ what is the behaviour of $$ ||X||_{l^p}=(\sum_{i=1}^n|X_i|^p )^{1/p}$$ as $n\rightarrow \infty$? For $p=2$ results about $\chi$-distribution tell us that ...
0
votes
0answers
87 views

Function of random variables

If $X$ is picked from a normalized probability distribution $f(x)$ and $Y$ from $g(x)$, then what is the distribution of $X+Y$ in terms of $f$ and $g$? And that of $XY$? And is there some equation ...
0
votes
1answer
26 views

Taking Limit of a function

I'm reading a proof and I don't understand one step. 1−(1−λ/n)^tn. Now, taking the limit as n→∞, P(Yn/n≤t)⟶1−e^(−λt). How does that work out? In this case, t is ...
2
votes
1answer
121 views

Probability Hyper Geometric Distribution

Can Somone help explain this to me, don't need the answers just a bit of guidance, I'm kind of lost on this one. A consumer advocate claims that 80 percent of cable television subscribers are not ...
0
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1answer
1k views

Cumulative distribution function determine the random variable

I don't know that determine is the right word, but I try to explain. What I need to understand. :) So.. We know's that if a function fit this conditions: Monotonically non-decreasing for each of its ...
0
votes
1answer
2k views

Distribution of minimum and sum of two independent exponential random variables

How can I solve this problem? Is there any formula for this problem Find the distribution of the random variable $Y$ if $Y=\min(X_1,X_2)$ $Y=X_1+X_2$ where $X_1$ and $X_2$ are independent ...
0
votes
1answer
54 views

Change of Variables and independent random variables.

Suppose that we have two IID random variables, $X_1, X_2$, carried by a triple $(\Omega,\mathcal{F},P)$. While solving an exercise I ended to a point that I had to see that, $$ \iint\limits_D x_1 ...
0
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0answers
64 views

Need help with calculating covariances of a random process.

Suppose ${X_n; n \in Z}$ and ${Z_n; n \in Z}$ are mutually independent, i.i.d. Gaussian random process with auto correlations $R_x(k) = {\sigma_x}^2\delta(k)$ and $R_z(k) = {\sigma_z}^2\delta(k)$ ...
0
votes
0answers
59 views

How can I find the distribution of $R$?

We are told that if $X \sim N(0,1)$, then $X^2$ has gamma distribution. Also, if $Y \sim N(0,1)$ and is independent from $X$, then $X^2 + Y^2$ has gammma/chi-squared distribution. Let $(X,Y)$ be a ...
1
vote
1answer
565 views

What is the autocorrelation of a squared Gaussian process?

Suppose $ {X_t; t \in R} $ is a wss, zero mean Gaussian random process with autocorrelation function $ R_X( \tau) ; \tau \in R$ and power spectral density $S_X(\omega); \omega \in R$. If w define ...
4
votes
2answers
143 views

Evolution by death and immigration of Poisson distributed population

This is quite an interesting problem, but I'm not sure how to go about doing it. I know that by using some basic Poisson properties I can figure it out but I'm failing to see how. It goes like this: ...
0
votes
0answers
49 views

Continuous random varible

If a continuous random variable $X$ is normally distributed with a mean of $11$ and a standard deviation of $0.095$, how do we find the probability that $X > 10.9$? Please help.