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

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What is the probability of this question?

On a single draw from a deck of playing cards the probability of selecting heart is 1/4 the probability of selecting a black card is 1/2. what is the probability of selecting both a heart and a black ...
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0answers
6 views

Integration of a Random Process

Let {$X_t$} be a strictly stationary random process defined for all time $t$; in particular, the distribution (PDF) at any time is the same. Let $Y$ be a random variable defined by $Y = \int_0^5X_t ...
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0answers
20 views

The distribution of a function of uniform and a complex Gaussian random variable

Hope this question is clear and straight to the point, if not than I can edit it accordingly. Given the following independent random variables with distribution $$X_i\sim \mathcal{CN}(0,1) $$ ...
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0answers
52 views

Marginalizing product of multivariate normal distributions [migrated]

How should I marginalize $F_{i}$ from the following probability distribution $$p(y_{i}|F_{i},\alpha, \Lambda, \Phi, \Sigma) = N(\alpha + \Lambda F_{i}, \Sigma)$$ in order to obtain ...
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2answers
39 views

Formula the conditional probability of mables

I have a interesting question that need your help. I have two sets A and B. Set A have 10 marbles that numbered from 1 to 10. Set B have 6 marbles that numbered from 1 to 6. Randomly choose $g$ ...
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2answers
16 views

Find the marginal PMF of X given the following joint PMF

Oh boy, so this is a tough one. Let X and Y be 2 random variables with the joint pmf: $$p(x,y) = \frac{e^{-\lambda}\lambda^{x}p^{y}(1-p)^{x-y}}{y!(x-y)!}$$ $$y=0,1,\ldots,x $$ $$x = ...
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1answer
24 views

Satisfying the criteria for a joint pdf? [on hold]

Determine the values of $c$ for which the following functions satisfy the criteria for a joint pdf $f(x,y)=c(2x+y)$ for $x=0,1$ and $y=1,2,3$, $f(x,y)=cy$ for $0<y<x^2<1$.
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2answers
34 views

Relating a Gamma Distribution to an Exponential one?

Question related to Gamma and Exponential random variables. Suppose I have a Gamma random variables with shape and scale parameter $m$ and $\theta$ i.e $$X\sim\Gamma(m,\theta)$$ respectively. Can I ...
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0answers
14 views

How to approximate the formual of combination function with large number element in set

I am implement the function to get the value of W with input are $k, \epsilon,\Omega$. The function W is defined as Please don't worry about the complex of equation. It is very easy with three known ...
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1answer
19 views

Probability Distribution - snowfall prediction [on hold]

I have a list of probabilities of it snowing in 50 cities in the world. probabilities = [.32,.26,.23,.12,.16,0,.66,.45,.23,.12, .32,.26,.23,.12,.16,0,.66,.45,.23,.12, ...
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1answer
20 views

Probability distribution functions pdf and cdf [on hold]

I have tried to solve this problem by finding the pdf then converting to cdf then finding the probability, but for some reason I keep getting the wrong answer. Is there another method of solving where ...
2
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1answer
24 views

Help with Linear Transformation of a multivariate normal

Given X ~ $N_2$ (μ, Σ)$ Find the Distribution of $$ \begin{pmatrix} X+Y \\ X-Y \end{pmatrix} $$ Show independence if $Var(X) = Var(Y)$ Attempt: Given proper of ...
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0answers
38 views

How to calculate P (|X − Y | ≤ 1/6)? [on hold]

f (x, y) = 1 for 0≤x≤1,0≤y≤1 and 0 otherwise. How to calculate P (|X − Y | ≤ 1/6)?
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0answers
44 views

joint probability distribution function | Find constant c and determine if independent

I have the function: $f(x,y) = c(x+y)$ if $0 \leq x \leq 1$, $0 \leq y \leq 2$ and $0$ otherwise. 1) Find the constant c I had to set the double integral to be equal to 1. So I'm not sure if I ...
4
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0answers
86 views

Find a symmetric random walk on $\mathbb{Z}$ that is transient.

I wanted to know if it is possible find a symmetric random walk on $Z$ that is not recurrent. Let $X$ have the following distribution, with a probability $1/2^{i+1}$, $X=\pm b_i$. Let ...
3
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1answer
43 views

Where do I go wrong?

Suppose $X,Y$ are independent Uniform$(0,1)$ random variables. Find the probability $P(Y\geq X\mid Y\geq\dfrac{1}{2})$. Please note that I know the correct answer and that I have arrived at the ...
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1answer
19 views

Probability, Minmize Gaussian distribution

There is a one problem that bugs me a while: Two random variables with distribution $X$ is $\mathrm{Gaussian}(\mu=\frac{-3}{\sqrt5}, \mathrm{var}=\frac 9 5)$, and $Y$ is ...
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0answers
57 views

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ has the distribution of $aX$, find all characteristic functions of $X$.

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ have the same distribution as $aX$ for some real $a$, what are the possible characteristic functions of $X$? Let $\varphi_X(t)$ be ...
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0answers
27 views

Show that the derivative of a fraction is always negative.

I'm looking for a neat way to show that the derivative with respect to n of: $$\frac{0.5^n}{\sum_{x\in G}{x^n+(1-x)^n}}$$ is always negative, when G is a finite set such that $x \in G \Rightarrow ...
2
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0answers
59 views

Is this limit finite?

What is the limit of $$\lim_{u+v\rightarrow 1}\frac{\ln \int f_0(y)^{1-v} f_1(y)^{v}\mathrm{d}y- \ln \int f_0(y)^u f_1(y)^{1-u}\mathrm{d}y}{1-(u+v)}$$ where $f_0$ and $f_1$ are some density ...
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0answers
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Poisson Distribution finding $E(x^4)$ [closed]

Given $E(x^n) = E((x+1)^{n+1})$ with mean $=1$. Thanks
4
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2answers
64 views

What is the expected value of $\min\{|X|,|Y|\}/\max\{|X|,|Y|\}$ assuming $X$ and $Y$ are independent?

So I need to compute $$E\left[\frac{\min\{|X|,|Y|\}}{\max\{|X|,|Y|\}}\right]$$ given $X,Y \sim$ Normal$(0,1)$ and independent. What I am having trouble seeing is whether $\min\{|X|,|Y|\}$ and ...
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0answers
26 views

Is $r_1 \cdot f_1 + r_2 \cdot f_2 $ uniformly distributed?

Consider $f_1$ and $f_2$ are fixed polynomials, $r_1$ is a random linear polynomial, $r_2$ is a random polynomials, degree($r_2$)=degree($f_i$)=$d$. We define $f_i$ and $r_i$ over $R[x]$ where $R$ can ...
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1answer
23 views

Find $\mathbb{P}(X > t+h \mid X>h)$. [on hold]

Random variable has a probability distribution with density: $g(x)=xe^{-x}\mathbf{1}_{x \geq 0}$. Find $\mathbb{P}(X > t+h \mid X>h)$.
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17 views

Limiting Gumbel Form

Given a set of values $x_1 > x_2 > ...$, which represent existing observed order statistics from a large sample drawn from a distribution $F$ with the upper tail in the Gumbel domain of ...
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0answers
11 views

How to generate a random number according to a user defined distribution.

Assume we have a user defined distribution as following: $Q(d)=\intop_{d-\frac{1}{2}}^{d+\frac{1}{2}}G(t,\mu,\sigma)dt$ Where G is Gaussian distribution with mean $\mu$ and variance $\sigma^2$. How ...
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0answers
14 views

some well known distribution?

I calculated the discrete probabilities for my project. I have two parameters k and l and I varied the third one which is the y axis (x values are log of the probabilities) in this enclosed picture. ...
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0answers
29 views

Is there a better way to mathematically use this data than the way I am doing it?

I am trying to use math to predict NFL fantasy football scores. My current process for projecting a players score is as follows: For every team (32 teams), I list the average points it gives up to ...
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0answers
21 views

Poisson Distribution: Trouble understanding this example [closed]

first time posting on this stack exchange, could you please help me with this question, I haven't covered Poisson distribution in a while and I am struggling to answer this: The number of code ...
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0answers
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Find $P(0.5\le X\le 2, 0\le Y\le 1)$ given $X$, $Y$ continuous random variables and pdf

$X$ and $Y$ are continuous random variables with joint pdf; $$ f(x,y) = \begin{cases} \dfrac{6}{11}x(x^2 +y^2) & 0\le x\le 1; 0\le y\le 2\\[2ex] 0 & \text{otherwise} ...
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2answers
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Variance of $2X_1 +X_2+3X_3$ with $X_i \sim \operatorname{Poisson}(i x \lambda)$

$X_1, X_2, X_3$ are independent random variables such that $X_i \sim \operatorname{Poisson}(i x \lambda)$, $i=1,2,3$. What is the variance of $2X_1 + X_2 +3X_3$? I know how to find ...
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0answers
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To prove given $ r \cdot f_1+f_2\cdot (s+1)$ one who knows $f_2$ cannot find out what $f_1$ is

We define the polynomials $r,f_1,f_2,s\in R[x]$. Where $r$ is a random degree 1 polynomial and $s$ is a random polynomial such that: $\deg(s)=\deg(f_1)=\deg(f_2)$. Let $R$ be $\mathbb {Z}_q$ where $q$ ...
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1answer
22 views

How to prove Markov's inequality $P_X(X\geq t) \leq \frac{\mathbb{E}[X]}{t}$?

As the subject states, how can Markov's inequality $P_X(X\geq t) \leq \frac{\mathbb{E}[X]}{t}$ be proven? Is the proof distribution-dependent or there is a general way to prove it?
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0answers
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An expected value of a function on binomial random variable $E(\frac{1}{1+x})$ [closed]

I would like to know what is the expected value $E(\frac{1}{1+x})$ of a function on binomial random variable with parameter n. Thanks your help
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0answers
17 views

Normal pdf/cdf inequality

Let $\Phi$ be the cdf and $\phi$ the pdf of the standard normal distribution. I want to show that: $$ \Phi(z)[1-\Phi(z)]\geq \phi(z)^2, \quad z\in\mathbb R. $$ How can I do this? I tried by looking at ...
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2answers
7 views

normal distribution question with percentages

how a can i solve a normal distribution without the mean ? suppose a truck of river sand delivered by a company has normal distribution with a standard deviationof 100kg.if 20% of loads are at least ...
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13 views

implications of unimodality of distribution

Suppose $X$ is a continuous r.v. such that $X\ge0$ and its distribution is unimodal. What sort of consequences does that entail for $X$ in terms of its other properties (e.g., moments, etc.)?
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22 views

Convergence in law of random variables

Let $X_1,\ldots,X_n \overset{i.i.d}{\sim} \ Uniform(0,1)$ and write $M_n = \max(X_1,\ldots,X_n)$. If we have proved that $M_n \overset{a.s.}{\rightarrow} 1$, then we know that $(M_n)$ converges in ...
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1answer
32 views

values of p, so that$ f(0)\ge f(1)\ge…\ge f(10)$

Let $f(x)= {10 \choose x} p^x (1-p)^{10-x} $,$ x=0,1...,10$, zero elsewhere. Find the values of p, so that$ f(0)\ge f(1)\ge...\ge f(10)$. Here is my solution: $x=0 , {10 \choose 0} p^0 (1-p)^{10-0}; ...
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1answer
14 views

Given the joint pdf $f_{X,Y}(x,y) = 2e^{-(x+y)}$

HW problem here. I want to check my first three answers, as well as get help on the last part. Given the joint pdf $$f_{X,Y}(x,y) = 2e^{-(x+y)}, \ \ \ \ 0\leq x \leq y, \ \ \ \ \ \ \ \ y \geq 0$$ ...
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1answer
11 views

Finding density functions from conditional distribution

I'm currently taking a statistics course, but I'm having trouble with a specific concept, and hope this is a good place to ask. Essentially, for random variables $y_{1},y_{2}$, how do you get from ...
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1answer
77 views

Roll a fair die until a 6 appears for the third time. What is the chance that all six values have occurred?

The question in the title is a homework question that I have been stumped on for some time. My approach thus far was to treat it as an occupancy problem. From class we derived the following formula ...
3
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1answer
23 views

elementary question about probability distributions

If $X \sim N(0,1)$, then what is the joint probability distribution of $(Y=X+1,X)$? An attempt: $f(y,x)=f(x+1,x)=f(x+1|x)f(x)=f(x)$... which doesnt make sense to me, since the distribution is now ...
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0answers
35 views

Stock price, probability, binomial model [closed]

Find the probability (in terms of n and p) that the price of the stock in the binomial model goes down at least twice in the first n time periods.
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15 views

Is there a symmetric alternative to Kullback-Leibler divergence?

I have two samples of probability distributions that I would like to compare. I have previously heard about the Kullback-Leibler divergence, but reading up on this it seems like its non-symmetricity ...
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0answers
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Probabilty of even number of games won/lost uses auxiliary variables for a quadratic equation. Why?

In a problem of finding the probability that an even number of games (even S) not being lost in $l$ games, I read the following explanation : "We form the equation, $x^2 - 4rx + 2r^2 = 0$, and ...
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3answers
65 views

Find the Distribution of the following;

Give that $Z_1,Z_2,...,Z_n$ are independent identically distributed standard Gaussian random variables with mean 0 and variance 1. Find the Distribution of $$X=\dfrac{(Z_1+Z_2)^2}{(Z_1-Z_2)^2}$$ ...
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2answers
62 views

Version of iterated expectations conditioned on subsets: Simple proof?

Thanks for any help with this. It is from the Stokey and Lucas (1988) Recursive Methods text (pg. 208) and uses notation from a Dynamic Modeling course taught at Carnegie Mellon and at Florida ...
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0answers
10 views

This question involves sampling distribution, standard errors, and probablilities of a random sample

Two candidates are running for the office of mayor for Statscity. The race is very close, and currently, Mr. E has 54% of the vote. A random sample of 1,000 voters is asked whether or not they will ...
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0answers
25 views

Airplane Overbooking Problem

Sometimes customers will make a reservation and then not turn up. To off-set this problem some companies may decide to “overbook” so they are not left with empty places. For example, an airline ...