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

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0
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1answer
600 views

Binomial Distribution with mean and variance

let x1 x2 x3 x4 be random sample from the population that satisfies an binomial distribution with n = 3 and p = 1/4 a) Find the mean and variance for Sum Y = x1 + x2 + x3 + x4. b) Find the mean and ...
0
votes
1answer
151 views

can chernoff bounds be used for proving upper bounds as well as lower bounds

I have a hw problem where it is asked to show theta(n) using chernoff bounds. I am able to prove for O(n) but not in the reverse way.Is it possible to prove both bounds using chernoff?
1
vote
2answers
139 views

Exponential variables

Suppose we have two exponential random variables $X_1$ and $X_2$ with parameters $\lambda_1$ and $\lambda_2$. Would the sum of them have any recognized distribution? If they have the same parameter ...
1
vote
1answer
86 views

A simple inequality in probability

I need to prove this seemingly simple inequality. If $X$ and $Y$ are iid discrete random variables, how does one prove that $$2P(|X-Y|=0)\ge P(|X-Y|=x)$$ where $x$ is any other positive integer. Is ...
3
votes
3answers
5k views

Finding a CDF given a PDF

The PDF for $Y$ is $$f_Y(y) = \begin{cases} 0 & |y|> 1 \\ 1-|y| & |y|\leq 1 \end{cases}$$ How do I find the corresponding CDF $F_Y(y)$? I integrated the above piecewise ...
3
votes
1answer
146 views

Estimating number drawn from one distribution based on sum of that number and number drawn from another distribution

I have been working on this for several days and have been unable to come up with an answer. The problem is very simple to state, but it seems difficult to solve. A computer draws a number $x$ at ...
3
votes
1answer
1k views

How does a function acting on a random variable change the probability density function of that random variable?

Given a random variable $X$ with probability density function $P(X)$, and given a transformation function $f(x)$, how does one determine the new resultant probability density function: $P(f(X))$? For ...
0
votes
1answer
83 views

Vector of normal distributes random variables

If I have $n$ random variables $X^n=(X_t^{(n)})_{t\ge 0}$, all $X^i$ normal distributed and they are independent. Now I define new processes: $$Z_t:=X^{(1)}_t+\dots+X^{(n)}_t$$ Since $X^{(i)}$ are ...
1
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1answer
136 views
2
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3answers
118 views

A probability distribution and random variable

Assume we have $X_1,...,X_n$ independent poisson random variables. What is the cdf or pdf of $ \sum_{i=1}^{n} X_i$ ??
2
votes
2answers
747 views

Computation of the probability density function for $(X,Y) = \sqrt{2 R} ( \cos(\theta), \sin(\theta))$

Let $R$ be a almost surely non-negative continuous random variable with absolutely continuous measure, and $\Theta$ be an independent random variable, uniformly distributed on the interval $[0, 2 ...
0
votes
1answer
278 views

Finding probability distribution of a random variable from the others

Let $X_1,...,X_n$ be independent exponential distribution.and now i want to calculate the density function of $\sum_{i=1}^{n} X_i$,i tried to find its distributing function,$F(T(X)\le x) $ but then i ...
1
vote
1answer
391 views

PDF/CDF and expected value of a function

How can I compute the PDF/CDF and expected value of the following function: $$ \frac{\alpha}{r^2} $$ where $r$ is generated as follows: draw $x$ and $y$ from a uniform distribution in the range ...
2
votes
1answer
324 views

Understanding the Kesten Multiplicative Process

I read in D. Sornette's Critical Phenomena in Natural Sciences about the Kesten Multiplicative process: $$X_{n+1} = a_n X_n + b_n$$ Where $a_n$ and $b_n$ are stochastic variables drawn from the pdfs ...
1
vote
1answer
182 views

The Nature of Probability Mass/Density Functions

Consider a certain random variable and all its possible probability mass functions (or probability density functions). What structure does this space have? For example, it can be endowed with a ...
3
votes
1answer
1k views

coin flips and markov chain

Consider the case of an infinite (or finite $n$) string of coin tosses, and let $q$ and $1-q$ be the probabilities that the coin comes up tails and heads, respectively. (For simplicity, we can take ...
2
votes
0answers
108 views

expectation of rademacher chaos

Let $\sum_{i=1}^n\sum_{j=n+1}^m\epsilon_i\epsilon_jb_ib_j$ be Rademacher Chaos of degree two (here $b_k\in R$ and $\epsilon_k$ are Rademacher random variables) and such tat ...
1
vote
1answer
194 views

Multinomial distribution: probability that at least one variable takes a certain value

Let $(X_1,\ldots,X_M)\sim \operatorname{Mult}(N;p_1,\ldots,p_M)$ follow a multinomial distribution. What is the probability that at least one of the variables takes a certain value, i.e. ...
0
votes
1answer
229 views

A problem of bivariate normal distribution

suppose $(X_1,X_2)\sim\mathcal{N}(0,0,1,1,\rho)$. find distribution $$\mathbb{Y}=(X_1^2-2\rho X_1X_2+X_2^2)$$
0
votes
1answer
61 views

the expectation of a Normal r.v conditioned by possible observation

I got stuck with this problem, Suppose there is a Normal r.v $X \sim \mathcal{N}(\mu, \sigma^2)$, where $\sigma^2$ is known and $\mu$ is unknown and will be updated using Bayesian inference. We give ...
2
votes
0answers
95 views

The expectation after Bayesian inference of a Normal r.v

I'm confusing myself with this question. Suppose there is a Normal r.v $X \sim \mathcal{N}(\mu, \sigma^2)$. We known the variance $\sigma^2$ however don't know the mean $\mu$, and choose to use ...
1
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0answers
318 views

Generating an asymetric triangular distribution

I am trying to generate an asymetric triangular distribution; a is lower limit, b is higher limit and c is mode. I found this following way to generate a random variable $X$ with triangular ...
1
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2answers
3k views

Determining p-th Quantile From Probability Density Function

I'm not sure how to derive the $p$-th quantile. I know that it is point which divides the distribution of $X$ into two parts, but I'm not sure what I'm supposed to do here. If the random variable ...
4
votes
1answer
2k views

If $X$ is a Poisson distribution with mean $\lambda$ how is $X^2$ distributed?

If $X$ is a Poisson distribution with mean $\lambda$ how is $X^2$ distributed? Any explanation would be very appreciated.
-1
votes
1answer
192 views

Probability and Odds when Gambling!

Here is a 'lunch break' problem from a rather old publication. Devise one set of rules for a dice game, where any number of players and one representative of the bank (mandatory), with one die ...
0
votes
2answers
104 views

Finding a distribution of a random variable generated using a Monte Carlo method

I would greatly appreciate if somebody could confirm or negate my result to the following problem. I am especially not sure about "putting it all together" step. Generate $U_1,U_2,U_3 \sim ...
0
votes
1answer
229 views

Finding a CDF from a PDF

If we have a probability density function given by $f(y)=\frac{a}{y^2}$ where $0<a\leq y$, how do we find F(y)?
13
votes
1answer
258 views

Help with a Bollobás proof - Switching between random graph models

I'm trying to make my way through Bollobás' book 'Models of Random Graphs', and unfortunately I've come entirely unstuck on one of his typical 2-line "and of course, this is entirely trivial"-style ...
3
votes
2answers
1k views

Characteristic function of Cauchy distribution.

When computing the characteristic function of Cauchy distribution, we applied the Cauchy Integration theorem: $$ \int_{C_{R}}\frac{e^{i\alpha z}}{z^{2}+1}dz=\int_{-R}^{R}\frac{e^{i\alpha ...
1
vote
1answer
193 views

What kind of distribution is that?

Could you tell me what kind of distribution is that? I can also provide the original data if needed.
0
votes
1answer
725 views

pdf of a quotient of uniform random variables

Suppose $x_1, x_2$ are IDD random variables uniformly distributed on the interval $(0,1)$. What is the pdf of the quotient $x_2 / x_1$?
0
votes
2answers
228 views

An identity for exponentially distributed random variables

Let $T_k \sim E(q_k)$ be a countable family of independent exponentially distributed random variables. In a book by Norris (http://www.statslab.cam.ac.uk/~james/Markov/), at pg. 72, I found the ...
3
votes
1answer
4k views

coin tossed until two consecutive heads or tails appear

A fair coin is tossed repeatedly and independently until two consecutive heads or two consecutive tails appear. What is the PMF of the number of tosses? edit: in italic
3
votes
2answers
86 views

Unique continued fraction

If $x$ is a uniformly random number in $[0,1]$, what distribution should the $n$-th term in its continued fraction expansion follow? What is the expected vale of $a_n$ in $[a_0;a_1,a_2,\dots]$? Here ...
0
votes
3answers
6k views

Discrete Probability Problem: determining probability mass function and cumulative distribution function

Three couples and two single individuals have been invited to an investment seminar and have agreed to attend. Suppose the probability that any particular couple or individual arrives late is .4 (a ...
0
votes
1answer
261 views

Gaussian formula for $n$ dimensions

I know this is a very simple one. If this is the formula for the two dimensional Gaussian (no covariance matrix considered - I have one mean and variance for each dimension): $$ A\exp{\left[ ...
4
votes
1answer
196 views

Weak convergence of a triangular array of Bernoulli-RV's

I am looking at the series $$X_{1,1},$$$$X_{2,1}, X_{2,2}$$ $$X_{3,1},X_{3,2},X_{3,3}$$ $$\dots$$ of independent r.v's with $p_n:=P(X_{n,k}=1)=n^{-\frac{1}{4}}$ and ...
10
votes
4answers
537 views

Probability of global epidemic

Consider $\mathbb{Z}^2$ as a graph, where each node has four neighbours. 4 signals are emitted from $(0,0)$ in each of four directions (1 per direction) . A node that receives one signal (or more) at ...
5
votes
1answer
519 views

Range of i.i.d. normal random variables

Let $X_1, \dotsc, X_n$ be i.i.d. standard normal random variables. Define the range $R \in \mathbb{R}_{\geq 0}$ as $R = \max \{X_1, \dotsc, X_n\} - \min \{X_1, \dotsc, X_n \}$. I am looking for a ...
2
votes
1answer
72 views

Hellinger metrics of Weibull probability distributions

The Hellinger distance between two probability distributions $f(x)$ and $g(x)$ is given by: $$H(f,g)=1-\int_\Omega(dx\sqrt{f(x)g(x)})$$ My question is: if $f(x)$ and $g(x)$ are two Weibull ...
2
votes
0answers
250 views

What is the continuous distribution version of multinomial distribution?

I am trying to model a distribution, on the number of occurrences of an event in a 24 hour time span. Right now, I discretize the 24 hour time span into hourly intervals, and each hour is taken as a ...
1
vote
1answer
170 views

Help with a short paper - cumulative binomial probability estimates

I was hoping someone could help me with a brief statement I can't understand in a book. The problem I have is with the final line of the following section of Lemma 2.2 (on the second page): Since ...
6
votes
2answers
713 views

Tail bound on the sum of independent (non-identical) geometric random variables

Suppose $X_1, \ldots , X_k$ are $k$ independent geometric random variables with success probability $p_1, \ldots, p_k$ and let $X = X_1 + \cdots + X_k$. The expected number of trials needed is $$ ...
1
vote
1answer
304 views

Why Binomial Distribution formula includes the “not-happening” probability?

Suppose I have a dice with 6 sides, and I let a random variable $X$ be the number of times I get 3 points when I throw the dice. So I throw the dice for $10$ times, I want to find the probability of ...
0
votes
2answers
153 views

Gaussian Distribution

sorry to bother people with this, but my stats teacher did not make solving gaussian distribution questions, clear AT ALL in his notes, and my exam is coming soon, someone explain to me what is ...
2
votes
1answer
368 views

Combining 1D normal distributions into a 2D distribution

First of all, apologies for my poor terminology - I have a particular problem which I understand in own terms, but I am having difficulty in applying the mathematics in the correct manner. My problem ...
0
votes
0answers
37 views

What distribution is this: $\sum_{x}{c(n)x a(n)^x}=1$?

This arises as a distribution of 'infected' species in some population (stationary distribution in an MC) $$ p(x) = c(n)x a(n)^x $$ where $a(n) >1$ and $c(n)$ is such that $\sum_{x} p(x)=1$. ...
0
votes
2answers
164 views

Defining the distribution for a complicated random variable

I want to come up with at least the expectation, and at best, the cdf, for a variable $Z$ that I think of as the result of a process and am not quite sure how to translate into equations. Let $F(x) = ...
3
votes
1answer
176 views

What's the expected value in this jackpot winning experiment.

If there exists a fair National Lottery, that someone bets £1, Jackpot increases by £1, and there is p chance that he wins a Jackpot. If a Jackpot is won, it is reset to 0. repeats. We can easily ...
2
votes
2answers
112 views

How can I prove this inequation $\Pr\{X+Y<t\} \le \Pr\{X<t\} \Pr\{Y<t\}$

Could you please help me to prove the inequality probability as follows: $\Pr\{X+Y<t\} \le \Pr\{X<t\} \Pr\{Y<t\}$ where $X$ and $Y$ are non-negative independent random variables with common ...