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

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Distribution of the quotient of two random variables

Let $x$ and $y$ be two random variables with support of $\left[1\hspace{5pt}10\right]$ and $\left[50\hspace{5pt}90\right]$ respectively. The distribution of each of these variables is $p_X(x)$ and ...
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1answer
42 views

Cumulative Distribution Function with New Random Variable

I'm really new to statistics and probability so sorry if this is a really basic question. I just wasn't sure how to do it. I tried looking it up but can't find much information. If I'm given a cdf ...
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32 views

Questions on the theory of Lasso

The linear model ${\bf Y}={X\beta}+\epsilon$, where ${\bf Y}$ is a $n\times 1$ vector, and ${\bf X}$ is $n\times p$ matrix. $n\lt p$ and $rank({\bf X})=n$. $\epsilon\sim N(0, \sigma^2)$. How to prove ...
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1answer
134 views

What is the probability of passing using the coin method?

The problem reads as follows: A multiple choice exam has 20 questions, 4 posible answers for each. A student decides to flip a coin on every question and choose the answer like this: Choice - Toss ...
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47 views

Why is the probability that two independently sampled values have the same value is zero?

In a book about machine learning, it reads, Generally, the probability that $x$ generated independently by a continuous probability distribution $p(x)$ have the same value is zero. Otherwise, ...
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36 views

Definition of Standard Deviation

We note that given a probability distribution function $P$ over a space $U$ the expected value of a function of the elements in U: $$ E(f(x)) = \int_{U} f(x)P(x) $$ We thus consider the mean as the ...
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1answer
1k views

Proof for Standard Deviation Formula for a Binomial Distribution

I understand the concept of standard deviation as the square root of the square of the mean of each sample value - the mean of the sample values. Here is the mathematical representation (I've solved ...
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2answers
40 views

Covariance of normally distributed random variables

If $ X \sim N(0,1) $ and given $ X = x $ then $ Y \sim N(x,1) $ I want to find the $ Cov(X,Y) $ using the relationship stated above. My attempt: $ Cov(X,Y) = E[XY] - E[X]E[Y] \\ E[X] = 0\\ ...
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1answer
52 views

How many pairs of positive constants $a, b$ exist such that $P(a < X < b) = 0.95$, where $X$ has a chi-squared distribution?

$X$ is a chi-squared distribution with $n$ degrees of freedom (sum of the squares of $n$ $N(0,1)$ variables). How many pairs of positive constants $a,b$ exist such that $$ P(a < X < b) = 0.95 ...
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1answer
66 views

On Integrating the joint probability density function of two random variables

Suppose that the joint probability density function of two random variables $x$ and $y$ is given as $p(x,y)$. We know that the probability density function of $x$ can be found by integrating out $y$ ...
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45 views

How can I get a Covariance Matrix from Mean and Variance?

this may be a very basic question. I have the mean and variance for 12 lognormal distributions: ...
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46 views

Tightness of probability measures

Prove: If there is a $\phi(X)\geq0$ such that $\phi(x)\rightarrow \infty$ for $|x|\rightarrow \infty$ and $\sup_n\int\phi(x)dF_n(x)<\infty$ Then $F_n$ is tight. The definition of tightness of ...
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48 views

Cumulative Distribution Function combination

There are n random variables of a Poisson distribution. These random variables are iid. Given the cumulative distribution function of each of these iid to be the same, how can we find the cumulative ...
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1answer
65 views

Calculating joint cumulative distribution function

given the joint probability density function $f_{XY}(x;y)= \begin{cases} 1, & \text{if (x; y) $\in[0; 1]\times[0; 1]$} \\ 0, & \text{elsewhere} \end{cases}$ I want to calculate the joint ...
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1answer
32 views

How to show that the random variables given are independent?

Prove that a sum of random variables say $\sum_{i=1}^{\infty }a_{i}X_{i}Y_{i} $ is independent of the sequence $\left\{ Y_{i}\right\} _{i=1}^{\infty }$ where both the random variables $\left\{ ...
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1answer
27 views

Let k be a natural number. What is the probability of it being equal to a random number in group?

We have 6 groups: A takes values from [0, 1, 2, 3, 4, 5, 6] B takes values from [0, 1, 2, 3, 4, 5] C takes values from [0, 1, 2, 3, 4] D takes values from [0, 1, 2, 3] E takes values from [0, 1, ...
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1answer
44 views

What is the expected value of the random variable with the following pdf

Let $X$ be a random variable with pdf $$f(x \mid \sigma) = \dfrac{1}{2\sigma}\exp\left(-\dfrac{|x|}{\sigma}\right)\text{, } x \in (-\infty, \infty)\text{, }\sigma > 0\text{.}$$ Here are my steps: ...
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1answer
28 views

What is the name for an ODE with an integral as a side condition?

My question: I have to find a function $y: \mathbb R \rightarrow \mathbb R$ fulfilling $$y^\prime(t) = f(t, y(t)),\ \int_{-\infty}^\infty y(t) dt = c$$ with a given $c \in \mathbb R$. What is the ...
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2answers
26 views

Prove that a sequence of maps is a sequence of i.i.d. r.v.

I need an help with the following exercise. I have a probability space $([0,1], \mathcal B, dx)$, where we denote with $\mathcal B$ the borelian sets in $[0,1]$ and $dx$ is the Lebesgue measure. We ...
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32 views

Probabilistic modelling of PSO algorithm's first iteration, mean of distribution of minimum

The particle swarm optimization algorithm (PSO) consists of a set of $I$ particles, each having a velocity $v_i$ and position $x_i$. The algorithm keeps track of the best encountered position for each ...
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1answer
265 views

Difference between Gaussian distribution and Laplace distribution?

there. I always appreciate the members belong this site because very active support! Now I have some data set which were measured by same experimental setup, but their distributions were slightly ...
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2answers
63 views

Approximation of the binomial distribution

Let $S_n=\dfrac{B_n - np}{\sqrt{n\cdot p\cdot (1-p)}}$ be a random variable which has the standardized binomial distribution. From Chebyshev's inequality I know that $$P(|S_n| \ge x) \le ...
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1answer
33 views

Log-normal distribution

If $X \sim LogN(\mu,\sigma ^2) $, would the distribution for $aX \sim LogN(\mu+a,\sigma ^2) $ for $ a>0$? My solution: $log(X) \sim N(\mu,\sigma ^2) \\ log(aX) = log(a) + log(X) \\ log(aX) \sim ...
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37 views

A interesting question about moments.

Let $\{X_n\}$ be a random variable sequence and $X\sim N(0,\sigma)$. In general, the convergence $E(X_n^k) \stackrel{n}{\longrightarrow}E(X^k)$ doesn't implie that $E(X_n^{k+1}) ...
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1answer
68 views

$E|X-m|$ is minimised at $m$=median

For a continuous random variable $X$, I want to show that $E|X-m|$ is minimum implies $m$ is the median of the distribution. Assume that the distribution function is $F$ and the density function is ...
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1answer
31 views

Numerical approximation to the Wasserstein metric?

Are there numerical methods for approximating/calculating the Wasserstein metric in particular cases? Suppose that $f$ and $g$ are two density functions with the same support. How can I calculate the ...
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68 views

Weak convergence of order statistics

I've encountered the following problem: Let $U_1,...,U_n$ be iid uniformly distributed on $[0,1]$ and let $U_{n(k_n)}$ denote the $k_n$-th order statistic where $k_n$ is chosen, s.t. ...
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2answers
91 views

Limits of integration of manipulated probability distribution function

I have to compute $\mathbb E\left[e^X \right]$, where $X$ follows a uniform distribution on $[0,1]$. I have started by computing the probability density function of $Y = e^X$, by doing the following: ...
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162 views

Convergence in distribution to derive the expectation convergence

If $X_n\longrightarrow X$ in distribution, $\mathbb{E}(X)\lt\infty$, Do we have the following conclusion: $\mathbb{E}(X_n)\longrightarrow\mathbb{E}(X)$?
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1answer
83 views

Probability of five card stud flush

I'm not really good dealing with poker problems... I am doing a practice problem for a midterm coming up, and help would be appreciated. Consider the probability that, in a game of five-card stud, ...
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1answer
37 views

pdf of conditional order statistic

Suppose that I'm drawing random variables from a standard uniform distribution. If there are n draws, then the maximum order statistic is distributed according to a $\beta(n,1)$ distribution. ...
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2answers
63 views

Poisson Distribution: P(exceeds certain number)

A professor plans to schedule an open lab in order to provide answers and additional help to students in the hour before homework is due. The number of students who will come to open lab will vary ...
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1answer
221 views

What are some Poisson-like distributions over a finite range of integers?

I'm writing a program in which, in any given time step, a random number of messages is sent. The number of messages is always between $0$ and $n$. I want to be able to control the probability, so ...
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1answer
41 views

Cumulative binomial distribution sum manipulation

I have a binomial distribution, with Random Variable Y and n trials. r is an integer. How can I show that P(Y ≥ r) = P(X ≤ n − r), such that Y is a random variable with probability of success p, and X ...
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115 views

Studying the probability of an event with a continuous distribution

Let $W:=(W_1,W_2,W_3,...,W_k)$ be a random vector of dimension $k \times 1$ where each $W_j$ has a continuous uniform distribution in $[a,b]$, $0<a<b$. Let $1\{.\}$ be an indicator function ...
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1answer
46 views

Probability distribution help

I'm trying to understand what kind of probability distribution I need to use in order to calculate a very simple example using a deck of cards. Assume that there is a standard deck of cards (52 ...
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2answers
78 views

Question on negative binomial distribution

I was unable to see through this question on negative binomial distribution please help: A shipment of 2500 car headlights contains 200 which are defective. You choose from this shipment without ...
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1answer
30 views

In General what is the conditional density of Y given X=i when Y is continuous and X is discrete.

I need help understanding why it would be the case that $$f_{Y\mid X}(y\mid i)=\frac{P(X=i\mid Y=y)f_Y(y)}{P(X=i)}.$$ Though I've just started studying conditional distributions, I am comfortable ...
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1answer
53 views

Determine the Hypergeometric probability function using sample space in which the selection is ordered

I'm unable to think through this question please help. Suppose our sample space distinguishes points with different orders of selection. For example suppose that $S =\{SSSSFFF\ldots\}$ consists of ...
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1answer
34 views

Different ways of calculating the conditional probability in the continuous case

For simplicity, assume a joint pdf of 2 variables $f(x,y)$. Say we have two events $A$ and $B$. How would one calculate: $$\Pr[A \mid B]$$ if we have continuous pdfs in question? I would have ...
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1answer
48 views

What is the appropriate statistical test to see if a quantity has been distributed differently into discrete bins?

Say I have $10^6$ balls, $3$ bins $A,B,C$, and $2$ machines $X$ and $Y$ that distribute the balls into the bins according to an internal set of rules (i.e. a probability distribution). If I run both ...
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0answers
39 views

Sampling into two sets

Let $N$ be the set of integers $1,\cdots,n$ and let $A$ be a set of numbers sampled independently from $N$ such that each element of $N$ has probability $p=0.5$ to be selected. I am trying to answer ...
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determine liklihood function in GPD distribution

if $W_i|W_{i - 1} = w_{i - 1}$ have GPD distribution with parameters $\xi$ and $\exp(a+bw_{i-1})$ whose $a$, $b$, and $\xi$ can be estimated by maximising $\prod\limits_{i = 2}^n f_{W_i\mid W_{i - ...
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1answer
36 views

what family of distributions is this?

I want to know what type of family this distribution belongs to: The variable is $x$. There is a parameter $\alpha$ with $\alpha>0$ There is a constant $H > 0$. The pdf is $$ f(x,\alpha) = ...
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1answer
35 views

Where do the forumlas for expectation and variance for geometric and Poisson distributions come from?

Okay so I have been given a list of 4 distributions and their respective mean(expected) and variance. I can see where the Bernoulli and Binomial ones come from using the definition of expectation and ...
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1answer
341 views

Getting Rich Off Roulette

If you bet 1 dollar on number 13 at roulette then you win $35 if that number comes up, an event of probability 1/38, and you lose your dollar otherwise. You play roulette 70 times. (a) What is the ...
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1answer
78 views

Calculating $E[x]$ from even probability density function $f_{XY}$

I'm a new user on this site, and I have a question about calculus and probability. I want to prove that $E[x] = \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} xf_{XY}(x; y)dxdy = 0$ when $E[x]$ is ...
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2answers
83 views

PDF of 6 six-sided fair dice sum

I am trying to create a pdf for the sum of 6 six-sided fair dice. The sums range from 6 to 36 (either have all 1s or all 6s). My attempt at the solution: $x$ = sum of the 6 dice $P(x=6) = C(6,0) ...
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1answer
46 views

Find a sequence of events $A_n$ for which all three inequalities…

Let $(\Omega,F,P)$ denote the probability triple for the discrete uniform distribution on the set $\{1,2,3,4\}$. Find a sequence of events $\{A_n\}$ for which these inequalities hold: (i) For each ...
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28 views

Finding measure given constant margins

Suppose $g:[0,1]^2\to R$ and $g$ can have finitely many discontinuities. $F$ is continuous and atomless c.d.f on $[0,1]$ $$\int_{[0,1]} g(x,y)dF(y)=1/2, \forall x$$ $$\int_{[0,1]} g(x,y)dF(x)=1/2, ...