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

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19 views

Distribution function of Sum of IID Exponentiation Variables of Variable amount

So I'm trying to determine the distribution function of a random variable, S, give: $N \sim Geo(\frac{1}{1+\lambda}) $ $S_i \sim Exp(\mu), \forall i\in [0,N]$ $S = \Sigma^{N}_{i=0}S_i$ $S = ...
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1answer
28 views

Find the conditional pmf of $Y$ given $X = 0$

Let $X$ and $Y$ have the joint pmf defined by $f(0, 0) = f(1, 2) = 0.3$, $f(0, 1) = f(1, 1) =0.2$ $(a)$ Tabulate the conditional pmf of $Y$ given $X=0$ $(b)$ Tabulate the conditional pmf of $X$ ...
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0answers
28 views

joint pdf for two independent uniform distribution

Suppose that $𝑋_1$ and $𝑋_2$ are independent and follow a uniform distribution over $[0, 1]$. Let $π‘Œ_1 = 𝑋_1 + 𝑋_2$, and $π‘Œ_2 = 𝑋_2 βˆ’ 𝑋_1$. a) Find the joint pdf $𝑓_{π‘Œ_1,π‘Œ_2} (𝑦_1, 𝑦_2)$ ...
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1answer
27 views

Finding the Probability of a Normal Distribution

The mean IQ scores of 30 primary school students is 108.56 and the Standard deviation is 12.33. Assume that IQ scores for primary school students that have been kept for 50 years illustrate a normal ...
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1answer
41 views

Finding a joint probability mass function

I have to find the joint probability mass function (pmf) of (X,Y) for the following problem: Roll a die repeatedly until a five or six appears, and let X be the number of rolls before a five or six ...
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2answers
31 views

Method for separating 'randomness' and 'non-randomness'

Let's assume I have a random two signals: Sin(t) R(t) Sin(t) is of course the trignometric function, but R(t) is a random process. So let's now assume I ...
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1answer
87 views

$N = Poisson(\lambda)$, $\{U_i\}$ iid $\implies (N_1, N_2) = Po(\lambda p_1)$x $Po(\lambda p_2)$

Let $\{N\}\cup\{U_i\}$ be independent random variables. $N = $ Poisson$(\lambda)$ $\{U_i\}$ iid, taking values in $\{1,2\}$, $\mathbb{P}[U_i = 1] = p_1$ and $\mathbb{P}[U_i = 2] = p_2$, $p_1 + p_2 ...
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1answer
25 views

Cumulative distribution function (CDF) strictly less than

Suppose a distribution function for the random variable $X$ is given by $$F(x)=\left\{ \begin{array}{11} \hfill 0 \hfill & x \lt 0\\ \hfill \dfrac{x}{2} \hfill & 0 \leq x \lt 1\\ \hfill ...
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0answers
19 views

Generate Correlated Normals

I want to generate normals $X,Y,Z$ with the correlation matrix $R$ but with means $0, 1, 2$ and variances $4, 16, 25$ respectively. How can I do this? Is it possible to apply Cholesky?
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1answer
34 views

How do can i solve the integral, finding cdf [closed]

Let $X$ be an exponential random variable with mean 1 and Y a uniform random variable between $0$ and $1$. Assume X and Y are independent and let $Z =e^{X/2}$ Find the joint cumulative ...
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0answers
14 views

Generating 'bursty' traffic using probability distributions beyond Poisson

I'm trying to develop a more realistic vehicle generation model for populating a traffic microsimulator. I'm trying to model a real-world intersection from which I have historical flow data. Currently ...
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1answer
30 views

Find $g(x|y=\frac{1}{2})$, the conditional pdf of $X$ given $Y = \frac{1}{2}$ (Need confirmation)

Let X and Y be continuous random variables having the joint pdf $$f(x,y) = 8xy , 0\leq{y}\leq{x}\leq{1}$$ I found that the marginal pdf of Y is $f_2(y) = 4y - 4y^3$. Does $g(x|y=\frac{1}{2}) = ...
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1answer
32 views

Probability. Find the CDF of $Y = X^2 $

Let $X$ have the uniform distribution $U(βˆ’1, 3)$. Find the CDF of $Y = X^2$. I thought this would be simply $$G(y)= \int_{-\sqrt{(y)}}^{\sqrt{(y)}} \frac{1}{4} dx$$ where $0\leq{y}<9$. Which is ...
3
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1answer
26 views

Is a subsequence of an exchangeable sequence exchangeable?

Consider a finite sequence of random variables $X_1,...,X_n$ (1) SUFF COND: Suppose $X_1,...,X_n$ are exchangeable, meaning that the joint probability distribution of $X_1,...,X_n$ is equivalent to ...
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1answer
75 views

Distribution of max number of common balls

I have $n$ different balls numbered $\{1,\dots, i, \dots, n\}$. I choose $n$ balls, uniformly at random, with replacement. Let $X_i$ denote the number of times ball $i$ has been chosen. I would like ...
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1answer
13 views

Continuously go from a lognormal distribution to a power law

Do you know any phenomena that are described by a continuous mappings between a lognormal and a power law distribution? Of course, one could give a simple linear combination of the two ...
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0answers
7 views

Variance of truncated multivariate Gaussian

Let $X \in R^n$ be distributed as the standard multivariate Gaussian i.e. $\mathcal{N}(0,I)$. I want to understand the covariance of the distribution conditioned on certain sets. Let $P_S$ be the ...
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0answers
16 views

How to determine parameters of a normal distribution from a limited range of points?

In an experiment my data points are almost normally distributed with meanvalue != 0. My problem is I can only detect positive points (located on the right side of y ...
2
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1answer
96 views

Exchangeability and independence of random variables

I have a question on the relation between exchangeability and independence between random variables. Consider the random vectors $$u_1:= \begin{pmatrix} \epsilon_{1}\\ \epsilon_2\\ \epsilon_3 ...
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0answers
25 views

How can I calculate definite integral of chi-squared pdf with one degree of freedom

enter image description here I need a calculating process of the above definite integral please help me.. (sorry for my poor English)
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1answer
56 views

Questions on probability law

I'm trying to prove/disprove the following true or false statements, and I want to know if they are correct For every measurable function $g:\mathbb{R}\to \mathbb{R}$, $\mathbb{E}[g(X)]$ is ...
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1answer
20 views

Weibull distribution: from mean and variance to shape and scale factor

I need to sample values from a Weibull distribution whose mean and variance are provided (respectively 62 and 4275). I am running a Matlab code, therefore if I want to use wblrnd(shape,scale) I need ...
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0answers
19 views

deriving the profit function given probability distributions

I can't seem to get much further in deriving the profit function for part (c). I've attached the question and my attempt, but I'm not sure on what to do next, or if I've done something completely ...
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1answer
44 views

probability functions

The total time, measured in units of $100$ hours, that a teenager runs her hair dryer over a period of one year is a continuous random variable $X$ that has the density function $$f(x)= \begin{cases} ...
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0answers
32 views

Odditiies in a StackExchange reputation distribution

As we know from for eg M.SE reputation distribution the reputation distribution is a neat power function, that's to be expected. However, on Travel StackExchange (possibly elsewhere, didn't research ...
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0answers
10 views

what is the spatial distribution of waiting time?

Suppose rimu trees are spread in the territory of some area according to a time homogeneous Poisson process. Suppose a rimu tree is at point x, what is the distribution function of the distance to its ...
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1answer
31 views

Consider a random variable X having the following PDF…

So I have to calculate the value of $c$ that makes this a legit PDF but I only know how to do it when it only has one function (set equal $1$ , integrate, solve for $c$). How do I calculate it for ...
2
votes
2answers
40 views

Is covariance preserved under transformation?

Let $X_1,X_2$ be normally distributed random variables with $\rho = 0.5$, mean equal to $0$ and variance equal to $1$. Let $U_i = \Phi(X_i)$ where $\Phi$ is the marginal distribution of $X_1,X_2$. We ...
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0answers
22 views

Hypothesis Testing on Renewal Processes

We have time $[0,T]$ to observe a renewal point process, where the inter-renewal timings are i.i.d, and then decide whether the observation is according to a renewal process in which the pdf of ...
2
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2answers
33 views

Simple Probability - Enumeration and Geometric Distributions

I am not sure as to why this particular practice problem does not use a geometric distribution. A prize is randomly placed in one of ten boxes, numbered from 1 to 10. You search for the prize asking ...
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1answer
33 views

slot machine can display the numbers 1, 2 or 3.

A slot machine can display the numbers 1, 2 or 3. 1 has probability 0.1, 2 has probability 0.3, and 3 has probability 0.6. The machine is run 10 times, and the numbers that show are added. At ...
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3answers
51 views

100 lottery tickets are distributed, only 2 of them have a prize.

100 lottery tickets are distributed, only 2 of them have a prize. Rupert Murdoch buys n of these tickets (n of course is some number between 0 and 100). What is the probability that Murdoch wins ...
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0answers
33 views

Estimating distribution from two distributions

I have been doing a survey on Family Incomes in India. The income of male and females are denoted by x and y. x and y are strictly positive. Per chance, individual values of y were deleted. I only ...
4
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1answer
54 views

Question on proving tight sequences.

I was just wondering how you would go about showing that a sequence of random variables is a tight sequence. For example suppose $X_{n}$ is distributed Exponentially($\lambda_n$) how would I show that ...
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0answers
19 views

Finding the survival and distribution function of a system.

We have a random variable $X\sim Gamma(3,c)$, so that means $f(x)=\frac{c^3}{\Gamma(3)}x^2e^{-cx} ; \ x>0$, with $c$ being appropriately selected scale parameter. We also have $P(U_2 \leq x)=x^2$ ...
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1answer
20 views

Density of a distribution given by a Gaussian copula and a set of marginals

Suppose the distribution of an $n$-dimensional random vector $X$ is characterized by a Gaussian copula $C_R$ with correlation matrix $R$ and a set of marginal $\{(F_{X_i}, f_{X_i})\}_{i=1}^n$ (pairs ...
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0answers
32 views

Limit of monotone decreasing function on generalised inverse.

Consider a right-continuous, monotone decreasing, non-negative function $\bar F(x)$ (its the tail of a probability distribution, but that doesn't matter). Now let \begin{equation} I_{n}=\{x : \bar ...
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2answers
95 views

Finding the distribution of a function of random variables using the definition (without the convolution theorem)

I'm trying to find $f_Z(z)$ with $Z=2X-Y$, for $X$ and $Y$ with joint density function $f_{XY}(x,y)$: $$ \begin{cases} x/8 & 1 \le x \le 3 \land -1 \le y \le 1 \\ 0 & \text{elsewhere} ...
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1answer
14 views

Does this distribution relation have a name: $\mathbb E(X^n)=b^n \mathbb E(Z^n)$ [closed]

More precisely, if $X$ and $Z$ are distributions so that $X=bZ$, then $\mathbb E(X^n)=b^n \mathbb E(Z^n)$. I found it on this site and would like to know "where it comes from".
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1answer
28 views

Convergence of sequence of random variables 2

If I know $\lim\limits_{n \to \infty} \mathbb{P}(X_n<c-\gamma)=0$ for all $\gamma>0$, how can I prove supremum of all reals $\alpha$ for which $\lim\limits_{n \to \infty} \mathbb{P}(X_n\leq ...
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0answers
26 views

Is the CDF of a mixture distribution uniformly distributed?

It is well-known that if $Y = F(X)$, such that $F$ is a continuous and a strictly increasing cumulative distribution function with a well-defined quantile function $F^{-1}$, then $Y \sim U(0,1)$. Now, ...
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1answer
30 views

Even moments of distribution given probability density function

Given the probability density function $f(x)$, and the $𝔼[X] = \frac{2}{\sqrt{\pi\lambda}} $, how best should I go about deducing the even moments of this distribution? $f(x) = ...
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1answer
40 views

How to find the Probability of $X \gt 1$ for $p(X)=2e^{(-2X)}$ [closed]

A PDF is given by formula $p(X)=2e^{(-2X)}$, $x>0$. Determine P($X \gt 1$)? we input 2 and got $=.03663$ but we do know as $X$ goes to infinity $p(X)$ will be $2$.
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0answers
22 views

Distribution of convex combination of Bernoulli random variables

Suppose $Y_1,Y_2, \ldots $ are i.i.d Bernoulli$(p)$. What is the distribution of $$\sum_{i=1}^{\infty} \frac{Y_i}{2^i}$$ I could deal case for $p=\frac{1}{2}$ using characteristic functions but for ...
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0answers
12 views

how to calculate cumulative distribution function inverted exponential in this pic

how to calculate cumulative distribution function inverted exponential in this pic enter image description here
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3answers
28 views

Binomial Random Walk

For the random walk with step sizes: $S_i = \begin{cases} &+1 &\text{probability} &p, \\ &-2 &\text{probability} &q=1-p \end{cases}$ Let $T_n = \sum_{i=1}^mS_i$ be the ...
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6answers
385 views

Die that never rolls the same number consecutively

Suppose we have a "magic" die $[1-6]$ that never rolls the same number consecutively. That means you will never find the same number repeated in a row. Now let's suppose that we roll this die $1000$ ...
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2answers
28 views

A coin is tossed 6 times. What is the probability that the no. of heads in the first 3 throws

A coin is tossed 6 times. What is the probability that the no. of heads in the first 3 throws is the same as the number number in the last three throws? To be honest, I don't know how to tackle this ...
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1answer
55 views

Simulate two centered normal random variables with given variances and given covariance

How can I, by the central limit theorem, simulate two random variables $Z_{1}$ and $Z_{2}$ such that $$Z_{1}\sim N(0,\sigma^{2})\qquad Z_{2}\sim ...
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1answer
22 views

Distribution of $aXa^T$ for normal distributed vector $a$

Let $a$ be $1\times n$ random vector with entries chosen independently from normal distribution with zero mean and unit variance. What is the distribution of $aXa^T$ for a given $n\times n$ matrix ...