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

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5
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1answer
442 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 $F$, ...
1
vote
1answer
141 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 $?. ...
0
votes
2answers
115 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 ...
0
votes
1answer
1k views

Joint cdf and pdf of the max and min of independent exponential RVs

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 ...
1
vote
2answers
96 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 ...
1
vote
0answers
30 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 ...
1
vote
0answers
72 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 $P(XY<0)=\frac{\beta}{\pi}$. ...
-1
votes
1answer
47 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?
1
vote
0answers
62 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
vote
2answers
1k 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
votes
4answers
814 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
vote
1answer
38 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
159 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 ...
0
votes
1answer
84 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
vote
1answer
800 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 ...
2
votes
1answer
514 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 $\mathcal{CW} _m(n_2,\Sigma_2)$ ...
1
vote
1answer
43 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 ...
1
vote
1answer
61 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
votes
0answers
96 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. ...
0
votes
2answers
43 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
votes
2answers
382 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 ...
1
vote
1answer
35 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 ...
0
votes
1answer
134 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 ...
0
votes
1answer
134 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 ...
0
votes
1answer
159 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
votes
1answer
62 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
votes
1answer
44 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
votes
1answer
211 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
vote
1answer
45 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
171 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
vote
2answers
92 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 ...
0
votes
1answer
46 views

Urn probability function

Suppose I have an urn with an infinite number of balls which can be either red or white. I do not know what the proportion of each colour is, but I do know it's a fixed proportion. After drawing $N$ ...
1
vote
1answer
68 views

Drawing without replacement, a special case

I have the following problem: you pick a set $x$ containing $|x|$ elements from a bag, containing $k$ marbles of $m$ possible types. Once a marble of a certain color is drawn, however, all other ...
1
vote
0answers
141 views

Simplify the expectation of the maximum of two random variables

My aim is to simplify the maximum of two expressions each of which are a function of exponentially distributed random variables Given: positive constants $a,b,c,d$. Independent random variables $x,y ...
12
votes
3answers
589 views

Formula for picking time closest to (but after) target

Let's say you have an arbitrary length of time. You are playing a game in which you want to push a button during this time span after a light comes on. If you do so, you win ($+1$), if not, you lose ...
1
vote
0answers
183 views

Unbiased estimators of theta

Suppose $\hat\theta_1$ and theta $\hat\theta_2$ are both uncorrelated and unbiased estimators of $\theta$, and that $\text{var}\hat\theta_1=2\cdot \text{var}(\hat\theta_2)$. a) Show that for any ...
1
vote
0answers
661 views

What is the expected time you have to wait until the first bus comes?

 three buses, bus A, B, and C come to a bus stop every hour. The time at which each bus arrives at the stop is distributed as a uniform random variable, i.e., TA,TB,TC ∼ Unif[0,1] hours. The ...
1
vote
3answers
51 views

Can we compute $ \mathbf{Pr}[x_{1} < X < x_{2}] $ if we know the cumulative distribution function $ F $?

Assume that we have a cumulative distribution function $ F $. How can we calculate the quantity $ \mathbf{Pr}[x_{1} < X < x_{2}] $? I know the answer for $ \mathbf{Pr}[x_{1} < X \leq x_{2}] ...
0
votes
1answer
84 views

Is monotonicity condition not required in this short derivation?

For given density functions $p_1(x)$ and $p_0(x)$ ($x\in\mathbb{R}$) the following equation is to be satisfied: $$(1-\epsilon_1)\{P_1[p_1/p_0>c] +cP_0[p_1/p_0\leq c]\}=1$$ where $c\in\mathbb{R}^+$ ...
1
vote
1answer
27 views

Determine $f(y_1, y_2)$ precisely.

Context of the problem: Continuous bivariate random variable $(Y_1, Y_2)$ has the uniform density $f(y_1, y_2)$ on support S = $(y_1, y_2) \leq 1-y_1^2, y_1 \leq 0, y_2 \leq 0$. Thus, $f(y_1, y_2)$ ...
1
vote
1answer
254 views

Probability Theory: On weak convergence of distribution functions.

On the top of page 97 in Durrett's text on Probability, it says that a sequence of distribution functions $F_n$ converges weakly to some limit function $F$ if it converges to $F$ for all continuity ...
0
votes
1answer
89 views

Find marginal distribution for Pareto prior

I have the following problem: The prior distribution for $\theta$ is distributed $\pi(\theta) = \frac{aP^a}{\theta^{a+1}}$, $\theta >P$ The likelihood for X is uniformly distributed, i.e. ...
0
votes
1answer
45 views

How is this cumulative distribution formed?

I have a probability density function that equals $f(x)=\begin{cases}.1\quad \text{for $0\le x\lt 2$}\\.2\quad \text{for $2\le x \lt 4.5$}\\.3\quad \text{for $4.5\le x\lt5.5$}\\ 0\quad ...
1
vote
1answer
105 views

Variance of this probability density

I have the function $\rho(x) = \frac{sin^2(x)}{x^2}$ and I want to calculate its variance on $\mathbb{R}$. Does anybody know how to do this? Cause afaik the integral does not converge.
0
votes
2answers
50 views

In PDF, why are these two interval notations the same?

0 <= x <= 10 0 < x < 10 When finding the probabilty for each of those ranges, why are they attributed to be the same answer? In other words, why does adding the equals symbol make it ...
0
votes
1answer
144 views

What is the probability that a customer waits for lesser than 3 minutes?

The rate of service is exponential and the service rate is 12 customers served per hour. The arrival of customers is in a Poisson distribution at the rate of 30 per hour. There are 3 servers and the ...
0
votes
1answer
321 views

Random variable and Poisson distrubtion

Given that $X$ is a random variable having a Poisson distribution, compute the following: (a)When $μ=0.5$, $P(X>3)$, My attempt: $1-e^{-.5}-e^{-.5}-e^{-.5}\cdot0.5-\frac{e^{-.5}\cdot0.5^3}{3}$ ...
1
vote
0answers
23 views

Distribution properties of the arithmetic mean of a subordinated Gaussian Process

Let $(X_i)_{i\in\mathbb{N}}$ be a stationary standard gaussian sequence and $G$ be a function such that $\operatorname{E}\left[G(X_1)\right]=0$ and $\operatorname{Var}(G(X_1))<\infty$. Further let ...
1
vote
1answer
505 views

Maximum likelihood estimator of minimum function with exponential RV and a random number

I'm having some problems with the following assignment: Let $X_1, X_2, ...,, X_n$ be samples from a exponential distribution with parameter $\lambda$, and let $c_1, c_2, ..., c_n$ be a sequence of ...
1
vote
0answers
104 views

what is relative entropy between to random binary string with length of $L_1$ & $L_2$?

I want calculate relative entropy between two strings of binary such as: $L_1:11000100011101001$ $L_2:00101110110111001$ It is primarily when the lengths of two strings is same and in general when ...