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

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1answer
19 views

Convergence in distribution - using moment generating function

Let $X_{n1},X_{n2}...X_{nn} $ be independent random variables with a common distribution given as follows: $P(X_{nk} = 0)= 1 - \dfrac{1}{n} - \dfrac{1}{n^2} \quad,P(X_{nk} = 1)= \dfrac{1}{n} \quad ...
0
votes
1answer
28 views

The expected revenue problem

Question : A travel agent company organizes a tour with ticket price $\$50$ and the ticket is non-refundable. The company has a bus with $20$ seats. The company knows that the participant might not ...
1
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2answers
34 views

Binomial-like probability problem between two independent groups

Question A study is conducted to monitor the health of two independent groups in a year where there are 10 participants for each group. Each participant will quit from the study with probability 0.2, ...
0
votes
1answer
52 views

Is the Gamma Function a jointly sufficient statistic?

A random sample $X_{1},...,X_{n}$ are pulled from a gamma distribution. Are there jointly sufficient statistics based on these observations for the two unknown parameters? The definition of a gamma ...
0
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1answer
24 views

Common distribution of order statistics

Let $X_1,\dots,X_n$ be $iid$ with distribution function F and $Y_1,\dots,Y_n$ $iid$ with distribution function G. I've proved for some function $g$ that $X_1$ is equal in distribution to $g(Y_1)$, ...
1
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0answers
10 views

Order Statistics interval sizes

Suppose an i.i.d. sample of size $n \geq 2$ drawn from a known distribution with density $g$. Let us note the associated order statistics as $(X_{(1)}, \ldots ,X_{(n)})$. I am interested in the number ...
0
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1answer
25 views

How do you show a sequence of random variable are not independent or identically distributed?

Consider the i.i.d (independent identically distributed) sequence $X_1,X_2,X_3,..$ of random variables such that $X_i \in {1,2,3,...}$ and for all $i$ $P(X_n=i) = p_i > 0$ Let $Y_n = 1$ with ...
0
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1answer
22 views

Can anyone help me find the variance of this expression?

I have a vector of the form \begin{align} {\bf a }= \frac{1}{\sqrt{N}}[1, e^{jA}, e^{j2A},\cdots, e^{j(N-1)A}]^T \end{align} where A and N are constants. I also have a vector N of i.i.d ...
0
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1answer
41 views

Is there a formula for $\mathbb{E}\left[X \mid a \leq X \leq b\right]$?

Suppose $X$ is a random variable which has nonnegative support and $0 \leq a \leq b$. Is there a formula for $$\mathbb{E}\left[X \mid a \leq X \leq b\right]\text{?}$$ Specifically, this came up in an ...
3
votes
2answers
40 views

Variation of Chebsyhev: How to prove that?

I have the "job" to prove that for any random variable with standard deviation $\sigma$ and expectation $\mu$ and for any $t>0$ we have $$Pr[X-\mu \geq t \sigma] \leq \frac{1}{1+t^2}.$$ I thought ...
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votes
3answers
60 views

Why is a probability density function nonnegative?

Let $X$ be a random variable and its density $f$ be defined to be the derivative of its distribution function $F$, i.e. $$\Pr(a< X\le b)=F(b)-F(a)=\int_a^bf(x)\operatorname{dx}$$ Now let ...
1
vote
1answer
39 views

Expectation of truncated Poisson Distribution

I have found that if I have a $Y \sim \mathrm{Poi}(\lambda)$ and $Z=Y \mid Y>0$ then I say $$f_Z(k)=g(k)=Pr(Y=k\mid k>0)=\frac{\lambda^k}{k! (e^\lambda-1)}$$ Now I am trying to compute ...
0
votes
1answer
20 views

Finding the probability using probability distributions.

A contractor is required by a county planning department to submit 1-5 forms in applying for a building permit. Let $Y$ be the number of forms required of the next applicant. The probability that $y$ ...
0
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1answer
12 views

Are functions of multiple independent random variables independent?

Suppose X, Y, and Z are independent geometric random variables with parameter $ \theta $. Now suppose V=G(X,Y) and U=F(Z). It seems intuitive that V and U would also be independent. The variation in ...
4
votes
2answers
47 views

CDF of probablity distribution with replacement

I want to get every color of gumball in a gumball machine (where there are 16 types of gumballs, each with a 1/16 chance of obtaining a particular color [assume there are an infinite amount of ...
0
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1answer
24 views

conditional probability Bayes Rule

A digital communications system consists of transmitter and a receiver. During each short transmission interval the transmitter sends a signal which is to be interpreted as A zero, or it sends a ...
0
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1answer
10 views

How to find the marginal density given these restrictions?

I know how to find the marginal density I'm just a little confused how to do it with these restrictions. This is the problem: $ f(x,y) = \begin{cases} \ 24y(1-x-y), & \text{if x>0, y>0, ...
2
votes
1answer
14 views

A detail in the proof of Central limit theorem

For the sake of completeness i will give the whole theorem: Let $X_1,X_2...$ be i.i.d random variables with finite expectation and finite $\sigma^2$, and set $S_n = X_1 + X_2+....+X_n \enspace, n ...
0
votes
2answers
15 views

Marginal P.M.F and Conditional Expectation?

I have a joint density function that is formulated as follows: $$ f_{X,Y}(k,y) = \begin{cases} \frac{\partial{P(X=k, Y\le y)}}{\partial y} = \lambda \frac{(\lambda y)^k}{k!}e^{-2\lambda y} & ...
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0answers
15 views

joint probability distributions

The problem is: I have two partially overlapping histograms of different shapes, each corresponding ultimately to a certain fraction, let's say one histogram represents the value 50% and the other ...
0
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0answers
23 views

Why does marginalizing out normal error just change the variance?

Assume that data follow the following model: $$Y_i \sim Normal(\alpha_0 + \alpha_1\mathbb{I}(F) + \zeta,\hspace{1mm} \sigma^2),$$ where $\zeta \sim Normal(0,\hspace{1mm}\tau^2)$. (In the problem, $F$ ...
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0answers
14 views

Simple Bernoulli distribution

Suppose an urn contains 6 yellow balls and 9 green balls. Draw one ball at random from the urn. Let x=1 if a yellow ball is drawn and x=0 if a green ball is drawn. Is this just a Bernoulli ...
0
votes
2answers
52 views

Sum of the product of two combinations

Could anyone explain how this statement is true? You may notice that this statement is part of the process of adding two independent binomial r.v.'s. $$ \ \sum_{x=0}^\infty{n \choose x}{m \choose ...
0
votes
1answer
32 views

Density of division of random variables

Given two independent random variables $X$ and $Y$, with densities $f_X (x)$ and $f_Y (x)$. What is the density of $X / Y$? Assume $P(Y = 0 ) = 0$.
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0answers
17 views

Gamma distribution problem

$X$ is a gamma distributed random variable with (n, 1) parameters. I have to show that $P(X<2n)\geq\frac{(n-1)}{n}$. Where should I begin? Any help is appreciated!
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0answers
7 views

One dimensional marginal distribution of spherically symmetric Gaussian?

Let $f$ be a spherically symmetric distribution on $\mathbb{R}^n$. What is meant by the one-dimensional marginal distribution of $f$? Is it just the distribution restricted to $\mathbb{R}$ (on any ...
4
votes
1answer
46 views

Slowly varying function without limit at infinity

A function $f:\mathbb R \to \mathbb R$ is slowly varying at infinity if for any $t>0$ $$ \lim_{x\to +\infty}\frac{f(xt)}{f(x)}=1. $$ Is there a bounded function slowly varying at infinity whose ...
-1
votes
0answers
19 views

Conditional independence in Markov family. [closed]

Suppose that X , ($\Omega, \mathbb{F}$), ${\lbrace}{P^x}{\rbrace}_{x \in \mathbb{R}^d} $ is a markov family with shift operators ${\lbrace} \theta_s{\rbrace}_{s\geq 0}$. Using the fact that for every ...
3
votes
1answer
80 views

Prove that the maximum of $n$ independent standard normal random variables, is assyptotically equivalent to $\sqrt(2\log n)$ almost surely.

Lets $(X_n)_{n\in\mathbb{N}}$ be an iid sequence of standard normal random variables. Define $$M_n=\max_{1\leq i\leq n} X_i.$$ Prove that $$\lim_{n\rightarrow\infty} \frac{M_n}{\sqrt{2\log ...
1
vote
1answer
32 views

Why are linear combinations of independent standard normal random variables also normally distributed?

My professor has given a list of questions that will not be appearing on my test, with this being one of them. I still feel this is extremely important to understand. How can I prove the following ...
2
votes
0answers
26 views

Expectation of $\frac{1-\Phi(\frac{Y-\mu_z}{\sigma_z}-\sigma_z)}{1-\Phi(\frac{Y-\mu_z}{\sigma_z})}$ of normal random variable.

The question is: Suppose that X and Y are independent Normally distributed random variables, $$X\sim N(a,\sigma_1^2)$$ $$Y\sim N(b,\sigma_2^2)$$ and $Z = \rho X + \sqrt{(1-\rho^2)}Y$, find ...
0
votes
1answer
31 views

How to prove Gaussian integral in normal distribution can be scaled to a standard curve?

If I want to solve the gaussian integral for normal distribution problems I only need to scale it to a standard normal distribution curve and consult a table. I want to know why this is valid (the ...
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0answers
17 views

Distribution of Multivariate normal divided by its squared norm

Let $X \sim MVN(\mu, \tau^2 I)$, what is the distribution of $X/\|X\|^2$? I know that $\|X\|^2$ would be a (non-central) chi-squared, but $X$ and $\|X\|^2$ are dependent. If I'm not wrong, the ...
-1
votes
1answer
51 views

Cumulative Distribution Function of a Die

Im trying to understand this question: Find the cumulative distribution function of the outcome of a single die roll that has the number 2, 4, 6, 8, 10, and 12. Also draw a graph. All I have came up ...
2
votes
3answers
79 views

Distribution of number of unique elements

I've been stuck on the following problem for a few days and would really appreciate some help. This isn't homework. The context of this problem is that $m$ and $n$ may be extremely large but there's ...
0
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2answers
41 views

I need to find the smallest lambda for which $P(X\ge 2)=0.99$ when $X\sim \text{Poisson}(λ)$

What I have tried to do is this: Since $P(X>=2)=0.99$, then $P(X<2)=0.01$, Hence $P(X=0)+P(X=1)=0.01$, so replacing in the pmf, I got $\exp(-λ)+λ\exp(-λ)=0.01$, $g(λ)=\exp(-λ)+λ\exp(-λ)-0.01$. ...
1
vote
4answers
31 views

What does X-2 mean given continous probability distribution X?

I have the continous probability distribution X: $f(x) = 2x e^{-x^2} \, x \geq 0$ and zero everywhere else. One of my homework problems is to find the probability distribution of X-2, -2X, and X^2 but ...
0
votes
1answer
33 views

The quantile function $F^{-1}(p) = c$ for all $p$ in the interval $(p_0, p_1)$ has the condition that $Pr(X=c) = p_1 - p_0$

Let $X$ be a random variable with c.d.f. $F$ and quantile function $F^{-1}$. Assume the following three conditions: (i) $F^{-1}(p) = c$ for all $p$ in the interval $(p_0, p_1)$, (ii) ...
0
votes
3answers
50 views

Finding $P(N>E(N))$

I know how to do the (i) and I will put out the results, just in case the second part is related to the first part answers. The C.D.F : $$F(t) = 1 – (t – 1)^{-2}$$ $P(T>5)$ : $$= 1 – F(5) = 1 ...
1
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0answers
18 views

Definition of inverse binomial distribution

I am trying to succinctly define the inverse binomial distribution. Not the normal approximation, but the real thing, which will be discrete. So far I have this: $F^{-1}(\alpha;N,p) = k,\ \ s.t.\ \ ...
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2answers
33 views

Condition on two distributions. All N(0,1).

This problem The RV $X_1$,$X_2$,$X_3$ are independent $N(0,1)$. Consider $Y_1=X_2+X_3, Y_2=X_1+X_3,Y_3=X_1+X_2$. Find the conditional density of $Y_1$ given $Y_2=Y_3=0$ (From Gut An intermediate ...
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vote
1answer
37 views

Integration and Laplace-Stieltjes of a multiplied Weibull and Exponential distribution Function

I have a trouble for integrating a multiplied weibull and exponential distribution. The expression is as follows: $$ Y(t) = \int_0^t e^{-\lambda T}e^{-(T/\mu)^z}dT\,. $$ Then, I need to take ...
2
votes
0answers
30 views

Determine the limit distribution

I have this question here that I could use some help with. Let $X_1$, $X_2$, . . . be a sequence of random variables such that $P(X_n=\frac{k}{n})=\frac{1}{n}$, for $k=1,2,...,n$ Determine the limit ...
0
votes
1answer
65 views

PDF and CDF of probability theory [closed]

The continuous random variable X has pdf $$f(x) =\begin{cases} x/2, \ 0<=x<=2 \\ 0, \ \text{elsewhere} \end{cases} $$ Two independent determinations of X are made. What is the probability ...
0
votes
0answers
22 views

Expected length of a line segment

Suppose we select points $a$ and $b$ from $[0,1]$ at random with uniform probability. What is $E(|a-b|)$? Running a bunch of trials shows that the expected value should be close to $1/3$, but I'm not ...
0
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1answer
19 views

Discrete distributions; find E(X)

For each of the following distributions, find: $\mu=\mathsf E(X)$ $\mathsf E[X(X-1)]$ and $σ^2=\mathsf E[X(X-1)]+\mathsf E(X)-\mu^2$: a) $f(x)= \frac{3! }{ x!(3-x)! }(\frac 1 4)^x (\frac 3 ...
3
votes
1answer
46 views

What is meant by“…on se ramène par régularisation…”?

I am currently attempting to translate the paper 'Sur l'équation de convolution $\mu = \mu \ast \sigma$' by Choquet and Deny. In the paper, a locally compact abelian group $G$ and a positive measure ...
1
vote
1answer
33 views

Conditional expectation of the maximum of two independent uniform random variables given one of them

Given $X_1$ and $X_2$ two independent random variables both uniformly distributed on $[0,1]$. What is the conditional expectation of $\max\{X_1,X_2\}$ given $X_2$? And the conditional expectation of ...
1
vote
1answer
42 views

Tips for evaluating $P(X\gt Y\gt Z)$

Does anyone know of any references for how to evaluate stochastic inequalities? Surprisingly, I can't find any good references for general problems. For example, suppose we have three random ...
1
vote
1answer
17 views

Is the joint distribution of two independent, normally distributed random variables also normal?

Say I have $X \sim N(\mu_1, \sigma_1^2)$ and $Y \sim N(\mu_2, \sigma_2^2)$, also $X$ and $Y$ are independent, then is the joint distribution of $X$ and $Y$ multivariate normal? I.e., $$\begin{bmatrix} ...