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

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2
votes
0answers
14 views

Modelling commodity price uncertainty with brownian motion - time period impacts

background I have two separate models of a metals resources company. Each model produces a series of accounting and cashflows forecast for different assets, and consolidates these to a overall ...
0
votes
1answer
20 views

Poisson sampling

Suppose I have a pdf $f(S)$. $f(S)$ describes the size of firms in the economy. Also define the Bernoulli variable $X_{f} \in \{0,1\}$ where $P(X_{f}=1)=g(S_{f})$ and $P(X_{f}=0)=1-g(S_{f})$. $S_{f}$ ...
1
vote
1answer
15 views

Random variable with density function that is scaled geometric mean of density functions of two independent normally distributed random variables

Given two independent normally distributed random variables A and B: $$A \sim \mathcal{N(\mu_A, \Sigma_A)}$$$$B \sim \mathcal{N(\mu_B, \Sigma_B)}$$ is there a way to find normally distributed random ...
11
votes
1answer
78 views

Distribution of $\sum\limits_{i=1}^{N}X_{i}$ conditionally on $\sum\limits_{i=1}^{N}X_{i}^{2}$ for i.i.d. standard normal $X_i$s

Assume that the random variables $X_{i}$ are i.i.d $\mathcal{N}\left(0,1\right)$, then: $$S_N=\sum_{i=1}^{N}X_{i}\sim\mathcal{N}\left(0,N\right)\qquad\qquad ...
3
votes
0answers
56 views

Max and sum of random variables

I have a set of independent random variables $\{A_1, A_2, B_1, B_2\}$. All of them have the same distribution function $F(x)$. I want to find distribution function of a variable $C$, where $C=max(A_1 ...
0
votes
1answer
22 views

Understanding function's notation

I have been given a question on the following pdf: Suppose the random variable, X, follows a uniform distribution on the interval (0, θ). The pdf of X is $f(x;θ)$ = $1/θ$, $if$ $0≤x≤θ$, $θ>0$, ...
0
votes
1answer
7 views

Walking through the reduction of a cumulative probability function to a polynomial

Setup Define $P(p)$ as follows: $$ P(p) = \sum_{N_1-\phi \cdot N_2 \geq \theta} {n_1 \choose N_1} {n_2 \choose N_2} p^{N_1 + N_2}q^{n_1 + n_2 - N_1 - N_2}. $$ Here, $$ q = 1 - p. $$ The sum is ...
-1
votes
1answer
64 views
2
votes
1answer
26 views

transformation of uniformly distributed random variable f(x)=1/2pi into Y=cosx

Let X be a uniformly distributed function over $[-\pi􀀀;\pi]$. That is $ f(x)=\left\{\begin{matrix} \frac{1}{2 \pi} & -\pi\leq x\leq \pi \\ 0 & otherwise \end{matrix}\right.\\ $ Find the ...
-1
votes
2answers
43 views

PDF of $Y=\min(0,X)$ when PDF of $X$ is $\frac34(1-x^2)$ on $(-1,1)$

Let $X$ be a random variable with density $f(x) = (3/4) (1-x^2).$ Range is $-1 < x < 1.$ I have to find probability distribution of $Y = \min(0,X).$ I know that distribution function could be ...
1
vote
1answer
25 views

Change of variable using dirac delta function

How do I intuitively understand the following result to find the probability density function $P_Y(y)$ given $P_X(x)$ after change of variables $y=f(x)$ or several variables. How to derive this from ...
0
votes
0answers
12 views

Conditions for Mellin inversion

Under which conditions is the function $$ g(s)=a^{c(s-1)}\Gamma(s),\qquad a>0,c\in \mathbb{R} $$ the Mellin transform of a probability density function $f$? If $c=-1$, then $f$ is the exponential ...
0
votes
2answers
41 views

T distribution with n degrees degrees of freedom

I would like to prove that $\displaystyle \frac{\bar{X}\,\sqrt{n}\,}{\hat σ^2}\sim t_{n}$. Note that x~N(0,$σ^2$) and they are iid. Could someone explain why $\displaystyle ...
1
vote
1answer
32 views

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 ...
0
votes
0answers
29 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 ...
0
votes
0answers
36 views

What is the Cumulative Distribution Function of $a/x^b$? [on hold]

I was just wondering what the CDF of $$\frac{a}{x^b}$$ would be? $a$ and $b$ are positive constants and $b \gt 1$ ($1.22$ to be exact). $x \in [0, \infty)$ theoretically but in practice once $x$ has ...
2
votes
3answers
61 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 $0,1, ... ,9$. ...
0
votes
2answers
28 views

Name of the probability distribution

If $X\sim N(0,1)$, then the density function of random variable $X^3$ is as follows: $$f(y)=\frac{1}{3\sqrt{2\pi}}\left | y \right |^{-\frac{2}{3}}e^{-\frac{1}{2}\left | y \right |^{\frac{2}{3}}}$$ ...
2
votes
0answers
14 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 ...
1
vote
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 ?
0
votes
1answer
30 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 ...
0
votes
1answer
14 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 ...
0
votes
1answer
22 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 ...
1
vote
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 ...
1
vote
1answer
34 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 ...
1
vote
0answers
73 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 ...
0
votes
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 ...
0
votes
0answers
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 ...
2
votes
1answer
19 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} ...
1
vote
2answers
26 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 ...
2
votes
2answers
34 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 ...
0
votes
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 $n-2$, ...
0
votes
3answers
77 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.
1
vote
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 ...
0
votes
0answers
34 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, ...
0
votes
0answers
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 ...
0
votes
1answer
19 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) \, ...
0
votes
0answers
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 ...
-4
votes
0answers
21 views

Probability Help with three events [closed]

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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 $$ Y_t := \sum_{t=0}^T X_t $$ ...
1
vote
0answers
28 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 ...
0
votes
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)
0
votes
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 ...
0
votes
2answers
40 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 ...
1
vote
0answers
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 ...
2
votes
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 ...
1
vote
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 ...