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

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apply the law of total expectation

I'm a little bit confused about applying the law of total expectation. Suppose $v_1,v_2,v_3$ are three random variables drawn independently from the same distribution $\mathrm{uniform}(0,1)$, which ...
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15 views

Probability Density following affine transformation

Suppose $X$ is a random variable in $R^n$ and $Y=a^{T}X+b ∊ R$. If $f_X$ is the density of $X$, then what (and how!) can I obtain $f_Y$ the density of $Y$? It is assumed that $a\neq 0$. I saw the ...
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17 views

What is the Cumulative Distribution Function of $a/x^b$?

I was just wondering what the CDF of $$\frac{a}{x^b}$$ would be? $a$ and $b$ are positive constants and $x \ge 0$. It's puzzling me even though it seems trivial. Thanks for your help.
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3answers
44 views

Adding two discrete distributions

I am taking a probability course and I am having trouble adding two discrete distributions. The two distributions given are: $X$ has a discrete uniform distribution on the integers $0,1, ... ,9$. ...
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2answers
19 views

Name of the probability distribution

If $X\sim N(0,1)$, then the density function of random variable $X^3$ is as follows: $$f(y)=\frac{1}{3\sqrt{2\pi}}\left | y \right |^{-\frac{2}{3}}e^{-\frac{1}{2}\left | y \right |^{\frac{2}{3}}}$$ ...
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7 views

Wireless networks on two sequential office floors: Random partitions of a finite interval via a point process on a line

Construct a Poisson point process of density one on a line of length $L$. Allow each point in the process to "see" part of the line to their left, and part of the line the their right (such that the ...
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1answer
12 views

Finding out the percentage points.( F - Distribution).

How to find the values of these $x_1$ and $x_2$ , given , $P(x_1<F_{7,7}<x_2) = 0.90$ , using the F-Distribution tables.. Can anyone provide me a hint for this ?
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1answer
21 views

Combined arrival rate

Let us suppose a scenario with two clients, $a$ and $b$, each one generating load at rate $\lambda_a$ and $\lambda_b$, respectively. The server receives the requests from both clients. What will be ...
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15 views

Minimizing $\ell_2$ error in a Bayesian setting

Let $(P_1,P_2,P_3)$ be drawn according to the Dirichlet distribution with parameters $(\beta,\beta,\beta)$, with $\beta>0,$ i.e. on $\{p_i\geq 0, p_1+p_2+p_3=1\},$ $$f_{P_1,P_2,P_3}(p_1,p_2,p_3) ...
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1answer
11 views

Derivation of t(n-1) distribution

While trying to prove that $\displaystyle \frac{\bar{X}\,-\,\mu}{S/\sqrt{n}}\sim t_{n-1}$ I came across a manipulation that I can not seem to understand the reasoning behind it. Why does ...
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15 views

Why distribution of multiple recursive random number generators is uniform?

I was reading the article of L'Ecuyer on random number generation. The title of this article is "Uniform Random Number Generation". One of the proposed PRNGs there, is multiple recursive random ...
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1answer
23 views

Conditional probability distribution with geometric random variables

Let X and Y are independent random variables following geometric distribution with parameter p. Find the distribution of X given that X + Y = n. I made it this expression... $$P\{X ...
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1answer
33 views

Writing the expected value of a random variable in terms of its cumulative distribution function

My professor said that an alternative expression for the expected value of a random variable can be written as: $$ E[X] = \int_{0}^{\infty} (1-F_X(x)) \, dx - \int_{-\infty}^0 F_X(x) \, dx $$ No ...
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0answers
63 views

Polynomial Interpolation When part of $y_i$'s are Shuffled

Hypothesis: Let $\vec{x}=[x_1,...,x_n]$ be elements of field $\mathbb{Z}_p$, where $p$ is a large prime. $x_i \neq x_j$, $x_i \in \mathbb{Z}_p$. Note $x_i$ values are NOT picked uniformly random and ...
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1answer
18 views

the probability that a chi square distribution smaller than its degree of freedom

Suppose $X$ is a $\chi_k^2$-distributed random variable, then is there any explicit form for the probability $$\mathbb{P} (X < k)?$$ In particular, I'm interested in the asymptotic value of ...
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6 views

Finding the Normalization constant for a wave function

I don't really even know where to begin on this. It doesn't look like a Gaussian. $$ Ψ(x) = Ae^{|x|/a} $$ We are supposed to find the normalization constant 'A' to begin with. I know that I need to ...
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1answer
17 views

Expectation of max absolute value of a Gaussian vector

Let $X$ be a joint Gaussian vector of dimension $k$ with zero mean and covariance matrix $K$ (where $K$ may not be diagonal). I am interested in sharp estimates on $$\mathbb{E}\max_{i=1,2,\ldots,k} ...
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2answers
20 views

Find the following distribution?

I have been given the following problem: The probability density function of a random variable X is given by: $f(x;θ) = \dfrac{2(θ−x)}{θ^2}$, if $0< x<θ$, $0$ otherwise* Find the ...
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2answers
33 views

Infinite sequence of exponentially distributed random variables

Consider an infinite sequence of exponentially distributed random variables, $X_k$, where$ k \in \{1, \ldots, n\}$ with $\lambda = 1$. I am trying to evaluate: $$\lim_{n\to\infty} \frac{\max_{1 \leq ...
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0answers
11 views

Probability That a Polynomial has Specific Root when we use Permutation Polynomial

To some extent similar question was asked here: Polynomial Interpolation and Security Imagine we have $\vec{x}=(x_1,...x_n)$ and two polynomials $P_1$ and $P_2$. Degree of $P_1$ is fixed $n-2$, ...
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3answers
68 views

Can someone give me real world example of uniform distribution [0,1] of a continuous random variable.

Can someone give me real world example of uniform distribution [0,1] of a continuous random variable, because I could not make out one.
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1answer
22 views

When randomly distributing n points amongst m people, what are the odds that one certain person will get a certain amount of points?

I'm mostly curious about how to find this in general, but the actual problem is with 20 points and 5 people. I know probability problems are very counterintuitive, and thus I was unsure after ...
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28 views

Probability, expected frequency and resultant distribution skewed or not?

A population consisting of a certain proportion of defective items has mean $\mu = 2$. If a sample of 4 items is examined and repeated 200 times, obtain a) probability of an item being defective, ...
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19 views

Probability that a Polynomial Has Specific Root When $y_i$'s are Not Random.

Imagine we have $\vec{x}=(x_1,...x_n)$ and two polynomials $P_1$ and $P_2$. Degree of $P_1$ is fixed $n-1$, but degree of $P_2$ can be at most $n-1$. $P_1$ has root $\beta$, where $\beta \leftarrow ...
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1answer
18 views

Notation: Codomain of a probability density function

I need some help with the correct notation for the codomain of a probability density function. Consider the following problem. Let $$ F : V \to (0,1), \, x \mapsto \int\limits_{\inf V}^{x} f(t) \, ...
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23 views

How many poker hands until statistically significant winner

How many poker hands do I have to play to determine a statistically significant winner? What is the best approach to get a 95% confidence interval? To give some more context: I have been building a ...
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21 views

Probability Help with three events [on hold]

When a piece of information (a bit) is transmitted over a communications channel, it may be wrongly communicated. One method of improving reliability is to transmit the same piece of information an ...
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3answers
28 views

Dice roll - Geometric Distribution Question

I am having a hard time understanding the concept of a negative binomial distribution. For example the question: How many times do you expect to roll a six-sided die before landing on the number ...
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2answers
42 views

What is the distribution of a binomial variable where the number of trials is itself random?

We do the following experiment: Select a random element $k$ from $\{1,\dots,n\}$. Toss $k$ fair coins. Define $X$ = the number of heads. What is the distribution of $X$? Given $k$, the variable ...
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5 views

Fitting power law to existing integral

I have empirical data - people from cities - a certain number of people for a certain number of cities. I know the exact number of cities, as well as the exact number of total people - e.g. the ...
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14 views

Lognormal approximation of the sum of successive values of a lognormal process

I would like to use a lognormal process to approximate the successive values of another lognromal process. Let $X_t$ be a lognormal process. I would like to approximate $$ Y_t := \sum_{t=0}^T X_t $$ ...
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0answers
25 views

A model to describe probability to win at certain skill ranges?

Let's say we have a list of all the chess players in the world, and we want to predict the likelihood of success if any player goes up against any other player. (Hypothetical example) I'm assuming ...
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2answers
13 views

Can a geometric random variable have a finite sample space? [closed]

Can it be finite? I think it has to have an infinite sample space (according to my lecture notes)
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1answer
74 views

Probabilities of errors in three independent transmissions

i have been working through some old exam papers and have gotten stuck on this last one. can anyone help? When a piece of information (a bit) is transmitted over a communications channel, it may be ...
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2answers
39 views

95% Confidence Interval Problem for a random sample

The sample mean of a random sample of $25$ observations is $9.6$ and the sample variance is $22.4$. Derive a $95$ confidence interval for the population mean. I calculated the following: Confidence ...
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17 views

Higher order terms in Taylor expansion tend to infinity faster.

Suppose $g$ is a smooth bounded and symmetric probability density function (pdf). Let $\{(X_1,Y_1), ..., (X_N,Y_N)\}$ be a random sample from the joint pdf $t(x,y)$. Further assume $a\to 0$ and $Na ...
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1answer
21 views

Is $a+r \cdot b$ an uniformly random value when $a,b$ are fixed and $r$ is random value?

Imagine we have two fixed values $a,b \in \mathbb{Z}_p$ and a uniformly random value $r\leftarrow \mathbb{Z}_p$, for large prime number $p$. Question: Is $v=a+b\cdot r$ an uniformly random value in ...
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1answer
45 views

A coin probability question

Let $p$, $q$ be values in $[0,1]$ and $\alpha \in [0,1]$. Assume $\alpha$ and $q$ known, and that $p$ is unknown parameter we would like to estimate. A coin is tossed n times, resulting in the ...
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0answers
9 views

Estimating Johnson Distribution Parameters by Quantile

In this paper: http://www.researchgate.net/publication/31291960_Quantile_Estimators_of_Johnson_Curve_Parameters the four parameters for the Johnson distribution ...
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2answers
17 views

Probability Distributions (Tree Diagram)

Satish picks a card at random from an ordinary pack. If the card is ace, he stops; if not, he continues to pick cards at random, without replacement, until either an ace is picked, or four cards have ...
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2answers
25 views

Using Normal Distributions to find Proportion

The height of a randomly selected woman from a population is normal with $\mu=165cm$ and $\sigma=7cm$. The heights f the men in this population are normal with $\mu=178cm$ and $\sigma = 8cm$. I am ...
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1answer
30 views

Functions of a random variable

Assume that $Y$ ~ $Exp(Ω)$. Find the cdf and pdf of $Z$ = |$Y$ - $δ$|. In order to solve this question so far, for $Y$, I am thinking about using the pdf equation for the exponential distribution i.e. ...
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3answers
34 views

Finding the Marginal Distribution of Two Continuous Random Variables

The continuous random variables $X$ and $Y$ have the joint probability density function: $$f(x, y)= \begin{cases} \dfrac{3}{2}y^2, & \text{ where } 0\leq x \leq 2 \text{ and } 0 \leq y ...
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1answer
28 views

Poisson Approximation of Binomial

I have to prove the Poisson approximation of the Binomial distribution using generating functions and have outlined my proof here. Given, \begin{align} & \lim_{n\to \infty} np_n = \lambda \\ ...
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51 views

Need simplified formula of probability equation

I have RV $x$ which is function of independent continuous RVs $x_1$ and $x_2$. After some manipulations, I came up with an expression for the outage probability of $x$ as $$P_\text{out}(y)=\Pr(x\leq ...
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1answer
59 views

Breaking probability theory by having a different number of random variables depending on a conditioning random variable.

I suspect I'm breaking probability theory but I don't know how or why. How does one handle working with conditional probabilities where one can have a different number of random variables depending on ...
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0answers
42 views

Sufficient condition for convergence in distribution in the plane

I'm trying to show convergence in distribution for a sequence $X_n$ of random variables in the plane. Here's what I know. I have a sort of squeeze theorem for the probability of the r.v.s being in a ...
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2answers
44 views

Is $P(n) = \frac{a n }{b}$ or $\frac{(a+1) n}{b + 1}$?

I investigated Some random data and I was a bit confused. Could be Mathematical coincidence but i'm not sure. Consider the integers $1,2,3,...,a$ Randomly Pick $b$ dinstinct element out of them. ...
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1answer
19 views

If $x_1, \ldots, x_n$ have probability distribution function $F(x)$, then the maximum has probability distribution function $F(x)^n$

A random sample $x_1,x_2,.....,x_n$ is taken from a population , which has the probability distribution function $F(x)$ and the density function $f(x)$ . The values in the sample are arranged in ...
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1answer
31 views

Understanding the matrix normal distribution

A random $n \times p$ matrix $X$ is distributed according to a matrix valued normal distribution iff $\mathrm{vec}(X) \sim \mathcal{N}_{np}(\mu, V \otimes U)$, where $\mu \in \mathbb{R}^{np}$ is a ...