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

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30 views

Which probability distribution should I use to get unconditional PDF?

Here is my question. I want to get the unconditional PDF of a random variable t with PDF $f(t\mid\tau_1,\tau_2,\dots,\tau_n)$. Depending on the relation between $\tau_1,\tau_2,\dots,\tau_n$, I have ...
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1answer
36 views

Distribution of Coin Tosses

A fair coin is tossed repeatedly until the first time we see H T (in this order, though not necessarily consecutively). Calculate the distribution of the number of tosses, i.e. give a ...
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1answer
378 views

Mean and covariance of Wiener process

Let $x(t), x(0)=0$ be a Wiener process with the parameters $a$ and $\sigma.$ Prove that its mean equals $a \cdot t$ and its covariance $R(t,s)$ is equal $R(t,s)=\sigma \min(t,s)$
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1answer
399 views

Sample from multivariate normal distribution with given positive-semidefinite covariance matrix

I want to draw a random vector from a multivariate normal distribution with given covariance matrix $Σ$. I'm following this algorithm: A widely used method for drawing a random vector $x$ from ...
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1answer
34 views

I want to show$X : ((0,1],\mathcal B,\lambda)\to \mathbb R$ is random variable

Let $F:\mathbb R \to [0,1]$ be a distribution function of a probability measure $P$ $(i.e.,F(x)=P((-\infty,x])) $. Then show that There is a random variable $X : ((0,1],\mathcal B,\lambda)\to \mathbb ...
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1answer
143 views

Bernoulli distribution vs the probability mass function

What is the difference between the two? They seem to mean the same thing to me. The probability mass function can be used to find the probability of getting a tail from a coin flip: X{1 ...
1
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1answer
93 views

How to find the pdf of the minimum of absolute differences of Uniform distributions.

Let $X_1$,$X_2$ and $X_3$ are independent random variables that are uniformly distributed over $(0;b), b>0$. What is the probability density function of z=min($Y_1$,$Y_2)$, where $Y_1=|X_1-X_2|$ ...
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0answers
121 views

replacing dependent random variables with independent random variables.

I have $x$ and $y$ independent sub-Gaussian random variables and the quadratic form: $z= x^2+xy$ Let $x'$ be an independent and identically distributed copy of $x$. If I use the replacement $z'= ...
2
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2answers
65 views

how to show $P_{\hat X}=P_{X}$.where $P_{X}$ is distribution.

Let $X$ be a random variable on the probability space $(\Omega,\mathcal B,P)$, with distribution $P_{X}$. Consider the random variable $\hat X$ on the probability space $(\mathbb R,\mathcal B_{\mathbb ...
0
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1answer
175 views

Finding joint pmfs from marginal pmfs

Let a, b > 0. The random variables X and Y are independent and their densities are : f(x) = 1/gamma(a)*x^(x-1)*e^-x, x>= 0 f(y) = 1/gamma(b)*y^(b-1)*e^-y, y>= 0 Let U=X+Y and V=X/X+Y Find the ...
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1answer
42 views

Finding PDF of random variables

Let $X$ and $Y$ be random variable with joint pdf: $$f(x,y)= \begin{cases} \frac{1}{4} e^{-\frac{1}{2}(x+y)} & x \geq 0, y \geq 0 \\ 0 & \text{ otherwise} \end{cases} $$ $U= ...
2
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0answers
63 views

Differential Equation for CDF

Consider the following differential equation $$F(cx) = F(x) + x F'(x)$$ for $c>1$. Does this differential equation belong to a some well known class? Is there a way to find all the solutions ...
3
votes
3answers
179 views

p.d.f of function of two random variables

I am trying to find the p.d.f (but will calculate the c.d.f first) of $Z = Y - {(X - 1)}^2$ knowing that $(X, Y)$ is distributed uniformly on $[0, 2] \times [0, 1]$. So, $$f_{X, Y}(x, y) = ...
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1answer
59 views

Derivation Expected value of l-step ahead forcasting term

I have some trouble solving the following problem: Given is the causal AR(1) model $X$: $X_t = qX_{t-1}+Y_t$ Where $Y_t$ is distibuted $IID(0,\sigma^2)$ Now I'm trying to figure out how to derive: ...
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4answers
757 views

The joint density of the max and min of two independent exponentials

Let $X=\min(S,T)$ and $Y=\max(S,T)$ for independent exponential variables $S$ and $T$. Find the joint density of $X$ and $Y$. Are $X$ and $Y$ independent? How would you suggest I approach this?
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0answers
46 views

Distributions of a group of i.i.d Gaussians after Gram-Schmidt Orthogonalization

If I have a collection of i.i.d standard Gaussian random vectors, say $\{\mathbf{x}_i\}_{i=1}^{\lambda}\sim \boldsymbol{\mathcal{N}}(\mathbf{0},\mathbf{I}_n),\lambda < n$, then I orthogonalize them ...
4
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2answers
78 views

Expectation of $\frac{1}{1+X}$ for Gamma

I am trying to evaluate the following integral: $$ \int_{0}^{\infty} \frac{1}{c+x} \frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x} dx $$ I have tried simple transformation and ...
1
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1answer
41 views

Mean and Variance of probability distributions

I know how to calculate mean and variance of some given numbers but I have trouble computing them for probability distributions especially when it is a continuous probability distribution. For ...
2
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1answer
118 views

Stationary distribution of a vector-autoregressive process

Given a $K\times K$ real matrix $\mathbf{\Phi}$ and given a sequence $\boldsymbol\varepsilon_t$ of multivariate normal variables $\boldsymbol\varepsilon_t\sim ...
5
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1answer
336 views

Empirical Distribution Function Understanding

I'm studying this topic by myself and I'm pretty sure there will be some big misunderstanding on my part, so please be patient with me. Given the sample $X_1,\ldots, X_n$, iid with distribution $F$, ...
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1answer
94 views

Numeric approximation for fitting a Gamma Distribution with a single parameter

Given a series of $N$ observations $\left(x_1, \ldots, x_N\right)$ that follow a Gamma distribution with a single parameter, $ \text{Gamma}(k, k)$, what is the maximum likelihood estimate of $ k $?. ...
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2answers
96 views

Theoretical impossibility? Deviation from normality with a sample greater than 300?

Huge thanks in advance! I've been lead to believe that the following is a theoretical impossibility: a population larger than 300 records without an approximation of a normal distribution. The ...
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1answer
1k views

Finding joint cdf and pdf of independent random variables

Let $X$ and $Y$ be independent random variables. Each has an exponential distribution with parameter $\lambda$. Define two new random variables by $W = \min({X,Y}) $ $Z = \max({X,Y})$ Find the ...
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2answers
91 views

independent chi squares mean independent non central chi square?

Let $Y$ be a multivariate normal random vector with covariance $\Sigma$. Let $A_0,A_1$ be matrices such that $$A_0\Sigma A_1=0.$$ It is known that in this case $Y'A_0Y$ and $Y'A_1Y$ are independent ...
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0answers
27 views

Anti-derivative of a function involving exponentially distributed variable

Suppose a random variable $x$ with p.d.f $f(x) = \lambda e^{-\lambda x}$ such that $\lambda$ is the parameter of $f$. Given a function $ g(x) = (a + bx )e^{- \frac{\lambda x}{a}} $ where $a,b \in ...
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0answers
65 views

For $(X,Y)$ bivariate normal, show that $P(XY<0)=\frac{\beta}{\pi}$

Let $(X,Y)$ be bivariate normal with mean 0 and correlation coefficient $\rho$. Let $\beta$ be such that $ \cos \beta = \rho$, $(0\leq \beta \leq \pi)$, and show that $P(XY<0)=\frac{\beta}{\pi}$. ...
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1answer
44 views

Prove $\limsup X_n = 1$ has probability 0

If $X_n$ are i.i.d. random variables U[0,1], is it true that $$ \{\omega : \limsup X_n(\omega) =1\} $$ has probability 0? How would you prove that?
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0answers
54 views

Conditions for non-decreasing conditional expectation

Let $X$, $Y$ and $Z$ be three real random variables. I would like to know if assuming Regression Dependence * , in the sense that $\Pr[Z\leq z |Y=y]$ is non-increasing in $y$, is sufficient or ...
1
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2answers
877 views

Proof that negative binomial distribution is a distribution function?

In my textbook, a clear proof that the Geometric Distribution is a distribution function is given, namely $$\sum_{n=1}^{\infty} \Pr(X=n)=p\sum_{n=1}^{\infty} (1-p)^{n-1} = \frac{p}{1-(1-p))}=1.$$ ...
2
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4answers
465 views

How do you find the second moment of the beta distribution?

I'm required to show $ E(Y^2) = \dfrac{\alpha(\alpha + 1)}{(\alpha + \beta + 1)(\alpha + \beta)} $ for the beta distribution using the definition of expectation. Now so far I have $ \int\limits_0^1 ...
1
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1answer
36 views

Expected value on non-normalised PDFs

Suppose the following is known: $$\int{g(x)dx}>\int{g'(x)dx}$$ Considering that $g=kf$ and $g'=k'f'$ where $f$ and $f$ are probability distributions on $X\in[0,1]$. Is the following true: ...
2
votes
1answer
154 views

Relationship between cdfs

Suppose we have random variable $X$ and two observers. For observer one cdf of $X$ is $F(.)$ and for observer two, it is $G(.)$. For some particular value of $x$ I am looking conditions/relations for ...
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1answer
75 views

Find the constant term of a generating function

I have a generating function for $S_1$: $g_1(z)=\frac{1}{4}(z+z^{-1}+2)$, and I want to know the distribution of $S_n=\sum_nS_1$. According to the convolution stuff, ...
1
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1answer
444 views

K Nearest Neighbor Density Estimation

An intuitive way to estimate the pdf of a distribution $f$ is described here. Given a set of points you find the distance to the $k$th nearest neighbor for a point $x$ that we want to know the value ...
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1answer
314 views

Sum of Wishart matrices

Considering two matrices, $H_1$ and $H_2$, that are independent of each other and follows complex wishart distributions as $\mathcal{CW} _m(n_1,\Sigma_1)$ and $\mathcal{CW} _m(n_2,\Sigma_2)$ ...
0
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1answer
37 views

continuous distribution with change of variable

I'm trying to do this question: If $X$ is a continuous random variable with a mean of 2 and a variance of 4, find the mean and variance of $Y$, where $Y=\log{X}$. I know how to find the expectation ...
0
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1answer
50 views

Combination of Conditional Expectations

Let $(T,S,\theta)$ be random variables in $\mathbb{R}^3$ with joint pdf noted by $f_{T, S ,\theta}(\cdot)$ I want to know if $E[\theta|T\geq t,S\geq s]= \frac{\int_{-\infty}^{\infty} ...
2
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0answers
79 views

How to construct a uniform joint distribution

I have a question that is critical to my work, but I am not sure if it is any possible. Assume that you have two uniform random variables X and Y. The product distribution of Z=XY is not a uniform. ...
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2answers
40 views

probability density function question for logs

I have a question which says the random variable X has a pdf of $f_{X}(x)= \frac{x}{8},\ 0<x<4 $ $f_{X}(x)= 0, \ $ otherwise I have been asked to find the pdf for $Z=log_{e}(X/4)$ Can ...
0
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2answers
299 views

Given two uniformly distributed random variables, find the expected value

X and Y are independent random variables that are both uniformly distributed on the interval [0,1]. Find $$E[Y\,|\,X<Y^2]$$ How would I go about setting this up with the given condition? I am ...
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1answer
34 views

Difference of a likelihood function for a vector and a single value

$p(x\mid C)$ is defined as the probability density of a point $x$ given that it belongs to a class $C.$ But what of $p(\mathbf{x}\mid C)$ where $\mathbf{x}$ is a vector? I'm finding hard to ...
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1answer
132 views

Expectation of Two Variables

The probability of the amount of time taken for a secretary to process a memo independent of others is modeled as an exponential random variable with PDF $ \\ f_{T}(t) = \frac{ 1 }{ 2 }e ^{-\frac{ t ...
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1answer
117 views

Probability Mass Functions ; Limit

Let A and B be two discrete random variables with joint PMF $P _{A,B} (n,m).\\ What\ is \ \lim_{n \rightarrow \infty} P _{A,B} (n,0)$ My idea is that since A is growing to inf, the probability will ...
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1answer
120 views

Covariance of dependent, conditionally independent, variables

I'm trying to find the covariance between two variables that are dependent, but conditionally independent. My two random variables, $X_1$ and $X_2$ are i.d. and their probability density functions ...
0
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1answer
56 views

Distribution of sum of quadratic gaussian matrices

I have two gaussian matrices, $\textbf{Z}_1 \in \mathbb{C}^{M \times N}$ and $\textbf{Z}_2 \in \mathbb{C}^{(T-M) \times N}$ where each entry in $\textbf{Z}_1$ and $\textbf{Z}_2$ is i.i.d. as ...
0
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1answer
40 views

Product of randomly drawn numbers

Here are two code line to run in R: prod(rnorm(100, mean=1, sd=0)) # (1) prod(rnorm(100, mean=1, sd=0.2)) # (2) $prod(..)$ returns the product of a sequence. The sequence it given by ...
0
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1answer
140 views

Limits of integration for a joint PDF

I have $f_{X,Y}(x,y) = \lambda^2e^{-\lambda y}$ for 0 < x < y. If I want to show that this is a joint PDF, I need to do a double integral and show that it is equal to 1. Do I set my integration ...
1
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1answer
44 views

Probability: Random Variables

Let's $T_1$ be a random variable with pdf: $$f(t) = \frac{6+2t}{7}$$ and $T_2 \sim Exp(\frac{1}{3})$ Knowing that $T_1$ and $T_2$ are independent calculate $$P(T_1 + T_2 > 1) $$ During my ...
2
votes
2answers
141 views

Inner Product vs. Integrals with Fourier Series, When to include 1/2pi?

I am confused about when to include a prefactor of $\frac{1}{2\pi}$ when dealing with integrals of functions that are expressed as fourier series. This is what I understand (please correct me if I'm ...
1
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2answers
89 views

What is this mathematics sub-field called?

I would love to answer another question on this site, but I am totally unfamiliar with the required technique. I mean, I don't even know the sub-field's name. The field I am looking for is one that ...