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

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Probability density of transformed random variable

Let $X$ be a random variable whose probability density function is $f(x) = xe^{x-2}$, if $1 < x < 2$ and $0$ elsewhere. Let $F(x)$ be the cumulative distribution function of $X$. Find the ...
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9 views

Evaluating the spectral density of generated noise through the autocovariance

Arguably more of a question for the signal processing page, but I feel it could also belong here. I'm working on generating noise signals $X(t)$ (with $t \in \left[0,T\right]$ with step size $\delta ...
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2answers
15 views

Relation between expectation and sample points

Suppose that the expectation of a random variable $X$ is $5$. Which of the following statements is true? There is a sample point at which $X$ has the value $5$. There is a sample point at which $X$ ...
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0answers
12 views

Convert a landau function to a gauus function

Assume the Landau Distribution $$p(x) = \dfrac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}\left(x+e^{-x}\right)}$$ What I would like to do is "convert" it to a gauss function $$g(x) = ...
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0answers
39 views

Calculate the result of combining three multivariate Gaussian distributions

A Bayesian derivation of the Kalman Filter was provided by Ho and Lee (1964); this paper is available as a free pdf here. As part of their derivation, they substituted three multivariate Gaussian ...
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11 views

Methodology for probability dist of function or random variables

Is there a methodology that allows us to derive a distribution of functions of random variables? How would someone approach this problem? What are the key ingredients? For example in many electrical ...
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1answer
32 views

Joint distribution of multivariate normal distribution

So the question asks: Let $X = (X_1, ... ,X_{2n})$~ $ N (0, ∑)$ (multivariate normal distribution with mean vector $(0,..., 0)$ and covariance matrix $∑$ ), where $n≥ 1$. Find the joint distribution ...
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2answers
32 views

Error of Stirling’s approximation for Binomial with central limit theorem

So the question asks: Let $X_n$~Bin(2n,1/2),use Stirling’s approximation for $n!$ to show $P [X_n = n]$~ $1/√(πn)$ as $n→ ∞$, and show the error in the estimate for $P [X_n ≤ n]$, given by the central ...
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1answer
31 views

Finding marginal pdf from joint pdf

$$f_{X,Y}(x,y)~=~\begin{cases}6xy & 0\le x \le 1 ~,~ 0\le y\le \sqrt{x} \\[1ex] 0 & \text{otherwise} \end{cases}$$ Find $F_Y{(y)}$ Now, at first I thought this was a straightforward question ...
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0answers
20 views

Probability that $t^2 + 2\sqrt{x}t + y = 0 $

X and Y are independent X~Geom(P) Y~Exp($\lambda $) Compute the probability that $t^2 + 2\sqrt{x}t + y = 0 $ Steps I've taken so far: Found where the determinant of the quadratic is $\geq 0$. ...
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1answer
51 views

Joint Probability Distributions (Alice and Bob go on a date!)

Alice and Bob have their first date tonight, and they agreed to meet at a restaurant at a certain time. NOTATION: $U(a,b)$ is the uniform distribution between $a$ and $b$, and $W(c)$ is the ...
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1answer
30 views

probability distribution question .1

on average a bowler takes a wicket every eight overs. what is the probability that he will bowl ten overs without succeeding in a getting a wicket?
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2answers
29 views

If $X\sim U(0,1)$ and $Y\sim U(0,X)$ what is the density (distribution) function $f_Y(y)$?

If $X\sim U(0,1)$ and $Y\sim U(0,X)$ what is the density (distribution) function $f_Y(y)$? I know the answer and I also found it on this site (link bellow). However, I just can't get the intuition ...
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1answer
32 views

Independence of some derived random variables

Positive random variables $X_1, X_2,$ and $X_3$ have joint probability density give by $$f_{\lower{0.5ex}{X_1,X_2,X_3}}\!(x_1,x_2,x_3)= \begin{cases}48~x_1~x_2~x_3 & : \textsf{if ...
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1answer
32 views

Giving a method for generating random numbers with a cumulative distribution function

So let's say I have a cumulative distribution function: $$F(x) = \frac{1}{2} (x + x^2) \space for \space 0 \lt x \lt 1$$ How do I find a method for generating random numbers from this function?
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1answer
27 views

($N_t$) is Poisson process with $\lambda = 1$. Calculate $E(N_2\mid N_1)$ and $E(N_1\mid N_2)$

($N_t$) is a Poisson Process with constant rate $\lambda = 1$. $1)$ Calculate $E(N_2\mid N_1)$: So this is how far I've gotten: Let $N_2 = N_1 + (N_2 - N_1)$ $E(N_2\mid N_1) = E(N_1\mid N_1) ...
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1answer
15 views

Proving the Best Test for a Beta Distribution.

I am having problems trying to solve the following problem: Let $X = (X_{1}, X_{2}, ..., X_{n})$ be a random sample, where $X_{1}$ has pdf given by $f(X_{1};\theta) = \theta x_{1}^{\theta -1} ...
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1answer
29 views

Suppose that in a region the chance that someone having a health insurance coverage is 80% and that a sample of five people is selected at random [closed]

Suppose that in a region the chance that someone having a health insurance coverage is 80% and that a sample of five people is selected at random for a survey concerning health insurance. Let $X$ ...
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1answer
44 views

The birthday-cake problem: a variation on the birthday problem

Suppose we randomly select 100 people and record each of their birthdays. (Assume a year of 365 days, and an equal chance of being born on any of them.) Suppose further that over the course of a year, ...
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1answer
22 views

Finding density function of a conditioned function

Let $(X, Y)$ be a point taken uniformly at random on the unit square $[−1, 1]^2$. Let X|E be X conditioned on the event $E$ = ${\{X + Y \ge 0\}}$. Find the p.d.f. of X|E. Now, the ...
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2answers
51 views

Poisson Processes with Gamma Arrivals

I think the title is the best description I can give of my problem (but I'm not 100% sure - the problem set-up has me very confused). So, given a sequence of i.i.d. Gamma RV having parameters 3, ...
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1answer
19 views

Given that we have a Coxian-2 distribution, what is P(X<0.5)?

I am working on a probability question that asks me to compute the variance and the expected value given that we have a coxian 2 distribution with $\mu_1=5$, $\mu_2=6.1$ and $\alpha=0.2$. Once this ...
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0answers
24 views

continuous variable $X$ such that $\Pr(X=x) \neq 0$, for some value $x$.

I'm interested in cases where you have a probability function $\Pr$ defined over the values of the continuous real valued variable $X$, where some particular value or values have non-zero, non-unitary ...
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0answers
19 views

Gaussian distribution with Gamma variance

I am using a hierarchical Bayesian model. In one part of it, I have a normal distribution with mean zero and a variance sampled from a Gamma distribution for some hyper-parameters $a_0$ and $a_1$: ...
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1answer
58 views

Can I assume that random variables with exponential distribution are positive?

Let $(Y_n)$ be i.i.d random variables following exponential distribution with parameter $1$. Let $X_n=\min(Y_1,\dotsc, Y_n)$ Prove that $ X_n \xrightarrow{P} 0$ It's easy to prove that ...
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0answers
27 views

BM hitting times with exponential killing process

Assume a BM in 3d domain (infinite) with a small absorbing subdomain (cube, sphere, ect), centered at point $p_s=(x_s,y_s,z_s)$. BM starts at point $p_0=(x_0,y_0,z_0)$ and when it riches the subdomain ...
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21 views

I like to know that am i correctly find pdf of some random variable

Let suppose there is two random variable $X$ and $Y$. $$X:N(0,\sigma_{1}^{2})$$ $$Y:N(0,\sigma_{2}^{2})$$ And we define new random variable $Z$ like below ...
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9 views

Is there any way to fit multivariate distribution on set of differently distributed random variables

Let us suppose I have 4 random variable $X_1,X_2,X_3X_4$ which are normal, binomial, normal, Poisson distributed respectively. Now I want to find a single multivariate distribution of ...
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17 views

Neyman-Pearson Tests and Types of Error

For the family of Neyman Pearson tests, show that the larger the size of the test ($\alpha=E_{0}\phi(X)$), then the smaller the type II error ($\beta=1-E_{1}\phi(X)$). Moreover, if $1-\beta$ is the ...
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39 views

Find pdf of $Z = X + Y$.

Suppose $X, Y$ are independent random variables with probability density functions $f_X (t) = f_Y (t) = \frac12 e^{-|t|}, t ∈ R. $ Find the pdf $ f_Z (t) $ of $ Z = X + Y .$ Hint: Consider the cases ...
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2answers
55 views

Find f(z) where Z= X+Y

Let $f_{X,Y}(x,y) = \frac{1}{8}$ for $-2<x<2$ and $0<y<2$. Find $f(z)$ where $Z = X+Y.$ Should I find the marginal of X and Y first, then $$f(z) = \int_{-\infty}^{\infty} f_X(x) ...
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0answers
15 views

PDF of correlated lognormal RVs

Let's say that I have two correlated lognormal RV's, $X$ and $Y$ with $E(X) = 20, SD(X) = 10, E(Y) =15$, and $SD(Y) =5$, where $X$ and $Y$ have correlation coefficient $\rho=.6$. How can I find the ...
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10 views

Find SDE coefficients such that solution follow given distribution

Consider a cdf F and suppsoe $F^{-1}$ is also known. Denote by W a standard Brownian Motion. 1) Find function G such that random variable $G(W_1)$ has cdf F. Answer: $G(x) = F^{-1}(Ф(x))$, where Ф ...
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13 views

Variance converging to zero implies weak convergence to delta measure?

I have the following question: Suppose that I have a sequence of random variables $X_n$ such that all moments exists and are finite. I have that $E[X_n]\to a$, where $a$ is a finite number and also ...
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1answer
33 views

Uniformly Most Powerful Test for a Uniform Sample

Let $X_{1}, \dots, X_{n}$ be a sample from $U(0,\theta), \theta > 0$ (uniform distribution). Show that the test: $\phi_{1}(x_{1},\dots,x_{n})=\begin{cases} 1 &\mbox{if } ...
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2answers
32 views

Find the probability density function of Z=X+Y

Suppose X,Y are independent random variables with probability density functions (pdf) $$fX(t) = fY(t) = \frac{1}{2}e^{-|t|}$$ Find the pdf $fZ(t)$ of $Z = X + Y$. Hint: Consider the cases $t<0$ ...
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1answer
20 views

PDF for a continuous random X, where $a$ is a constant such that $1< a \leqslant 5$

I have a pretty decent length question - so if you don't feel like answering each part in its entirety... the first few steps might be all I need. :) The information given Consider the probability ...
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3answers
190 views

How to win Matt Parker's jackpot - finding the median of the following distribution

In a recent video the legendary Matt Parker claimed he kept flipping a two-sided (fair) coin untill he scored a sequence of ten consecutive 'switch flips', i.e. letting $T$ denote a tail and $H$ a ...
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25 views

equality in distribution for infinite sequences [closed]

Let $Y_1, Y_2, \ldots $ be an infinite sequence of random variables taking values on a countable set $V$ and defined on $(\Omega, \mathcal{F})$ . Let $P_1$ and $P_2$ be two probability measures on ...
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How to justify that we should assume the variances are equal or not?

The following data set gives the recorded birth weights of 50 infants who displayed severe idiopathic respiratory distress syndrome(SIRDS) D indicate died while S indicate Survived. The analysis data ...
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18 views

Distribution of the sum of two (dependent?) random variables

There are two random variables $X$ and $Y$, each of which can take on the values $0$ or $1$. Furthermore: $P(X=0,Y=0)=p$, $P(X=0)=1/2$ and $P(Y=0)=1/2$. So these two shouldn't be independent in ...
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1answer
35 views

How can I find expection value?

Let suppose that $X$ is gaussian RV whose mean value is $0$ and variance is $\sigma^2$ and $Y=10^\frac{X}{10}$ . What I like to know is $E[Y]$. First of all I search pdf of $Y$ like belows. ...
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ratio distribution of two dependent Poisson variables

Consider a homogeneous Poisson point process $\Phi$ in a disc with radius $R$, and the density of $\Phi$ is $\lambda$ per unit area. Denote the total number of points in the whole disc as N, and ...
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1answer
58 views

Compute $P(X>2U)$ for $X$ exponential and $U$ uniform on $(0,1)$

I am trying to calculate the $P(X>2U)$, where $X-exp(\lambda), U-Uniform(0,1)$. But my answer calculated by hand doesn't matched with the answer calculated using Mathematica. ...
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1answer
23 views

Exercise on bivariate normal distribution [closed]

Let X and Y be normal random variables with mean 0, variance $\sigma^2$ and correlation coefficient $\rho \in (-1,1)$, so that the density is given by $$f(x,y) = \cfrac{1}{2\pi ...
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1answer
20 views

The product of a uniform probability density function and -1

What happens when you multiply a uniform probability density function between -1 and 1 by -1? Does the new uniform distribution become -1/2 between -1 and 1? I am asking because I am trying to find ...
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1answer
45 views

Probability - A drunk man walking along the axis

A drunken man walks along the X-axis, where the probability that he goes right is $p$ and the probability he goes left is $(1-p)$. What is the probability that the drunken man will be at $+1$ after ...
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1answer
30 views

Prove that $\text{lim}_{\Delta t} \rightarrow 0$ of the transition PDF of a std Weiner process is 0

The transition probability density function of the standard Wiener process is: $$ f(x_2,t_2|x_1,t_1) = \frac{1}{\sqrt{2 \pi (t_2-t_1)}}e^{-\frac{(x_2-x_1)^2}{2(t_2-t_1)^2}} $$ I know that if Markov ...
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1answer
26 views

Exercise on stopping times

Let $(Y_n)_{n \geq 1} $ be a sequence of independent r.v.'s s.t. $$P(Y_n=y) = {n \choose k } \left(\frac1n\right)^y \left(1-\frac1n\right)^{n-y}\quad {\rm if }\;y \in \{0,1,\dots,n\}$$ How to show ...
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1answer
21 views

Intuitive explanation of distribution of ratio of independent random variables

Case 1: I have two independent exponentially distributed random variables $X$ and $Y$. Intuitively, it makes sense that the sum of those variables is essentially exponentially distributed, but is that ...