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

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25 views

definition of distribution function of random variable

please help me to understand fully following definition : i am using this book http://www.math.harvard.edu/~knill/books/KnillProbability.pdf page 79,i can't understand some part,in spite of this ...
1
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1answer
36 views

Use a probability function of X to prove the Moment Generating Function

So my question reads: Given the probability function of $X$ as follows: $f(x) = \frac{1}{2} \left(\frac{2}{3}\right)^x$ , $x=1, 2, 3, \dots$ (a) Use the definition $M_x(t)= E(e^{tx})$ to show ...
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0answers
16 views

MLE for mean of symmetric but otherwise unknown distribution

Given i.i.d. draws $x_1,...,x_n$ from $X$, where: $X$ has a finite mean $E[X]=\mu$, $X$ is symmetric about its mean, meaning $f_X(\mu+c)=f_X(\mu-c)$ for all $c$., The probability density function ...
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0answers
108 views

Conditional Probability with Gamma Distribution

I have a stochastic process $X$ that is distributed according to the Gamma distribution: $X$ ~ $\Gamma(\alpha, \beta)$ I would like to compute the following : Probability = $P(X_t > 0.75 | X_{t-1} ...
4
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1answer
34 views

Properties of cumulative binomial distribution

Let $F\left(k, n, p\right) = \sum_{i=1}^k\binom{n}{i}p^i\left(1-p\right)^{n-i}$ denote the cumulative binomial distribution function. If $F\left(k, n, p\right)-F\left(k, n, p'\right) \geq ...
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1answer
90 views

How to mathematically prove that we are sampling from same distributions?

The content of this question is about rigorously proving something which is otherwise considered easily correct intuitively. Let's assume we have a multivariate distribution $g(x_1,x_2,...,x_n)$ over ...
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0answers
17 views

Cumulative distribution function of a model similar to the multinominal distribution

I would like to solve a problem similar to the multinominal distribution (http://en.wikipedia.org/wiki/Multinomial_distribution): For k independent trials each of which leads to a success for ...
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1answer
53 views

Probability roots of quadratic lie in unit disc

$A,B\sim\mathscr{U}(0,1)$ and independent. We consider: $$x^2+2Ax+B=0$$ Given that both of the roots of this equation are real, what is the probability that they lie in the unit disc? ...
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0answers
242 views

Formula when drawing from multiple urns (Probability)

I'm trying to work out a general solution for a probability problem, but I can't seem to figure out how to go about with it. I manage to calculate the individual probabilities on a per problem basis, ...
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1answer
11 views

Probabilty of random number distribution

Given a random number generator generating integer numbers in the range 1 to N. What is the probability that a given number appears Q times (not necessarily sequentially, but in any order) in a ...
1
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1answer
32 views

Independence proof

$X,Y\sim\mathscr{E}(1)$ (exp. with parameter $1$) and independent. I'd like to show that $\min\{X,Y\}$ and $|X-Y|$ are independent. Let $Z=\min\{X,Y\}$ and $W=|X-Y|$. The transformation gives a ...
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3answers
270 views

Specific statistics problem: Ranking universities

What do you think is the best way to rank universities participating in a competition if we know for each university the number of recognitions that it's students won. Each university can participate ...
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1answer
195 views

Probability Density Function and Proof

Given the Probability Density Function: $f(x)=kx(2-x), 0\leq x\leq 1$ Prove that $k=\frac 3 2$ Looks like it should be a Beta Distribution, but all examples of a beta distribution use the format: ...
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0answers
28 views

Conditional Poisson PMF ~ the joint PMF not independent?

Let X denote the number of customers who arrive during a service time and Y the first service time. Customers arrive in a Poisson process with rate $\lambda$. Then: $P(X=x|Y=y)= e^{-(\lambda y)} ...
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0answers
15 views

Identically distributed subset of samples selected dependently

Assume we have a set of (vectorial) data point drawn from an unknown distribution $p(x)$. we select a subset of these samples one by one. Selecting a new point is such that make it dependent on the ...
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0answers
17 views

Raffle between different groups composed by different numbers

I've got this issue, I need to prepare a raffle between teams for a cars race. Cars are grouped by teams. Rounds are 1:1, composed by different manches until the cars are done. Total number of cars is ...
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1answer
55 views

I dont understand probability functions.

My question reads: Given that the probability function of X is as follows: $$ {\rm f}\left(x\right) \equiv {\sqrt{\,x + 9\,} \over 10}\,,\qquad x = -8, -5, 0, 7. $$ (a) Find the mean of $X$. (b) ...
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2answers
4k views

Is it possible to calculate the mean and standard deviation from a median and quartiles?

Any advice would helpful. I understand that the reporting of median and quartiles for small samples is an indication of skewed data. If such is correct, then is it useless to try to work out the mean ...
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1answer
53 views

systematic way of finding the bounds for change of variables (multivariable case), Jacobian

Let's say that $X,Y$ are independent standard normal random variables. I am interested in the distribution $P(X+Y\le 2t)$. Clearly, the domain of integration in this case is $-\infty<x<\infty$ ...
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1answer
49 views

Find the marginal pdf of $X$

The joint pdf of $X$ and $Y$ is given by $$ f_{X,Y}(x,y) = \frac{1}{8} (y^2 - x^2) e^{-y} I_{(0, \infty)}(y) I_{(-y,y)}(x)$$ Find the marginal pdf of $X$ My attempt: $f_X(x) = \int_0^{\infty} ...
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0answers
173 views

Unconditional mean from conditional distribution

Consider the conditional distribution $$f(y|x)=\frac1{(a+b*x)}e^\frac{-y}{a+b*x}, (y\geq0,0\leq x\leq1)$$ Show that the conditional mean E[Y|X] is equal to $a+b*x$. [Done] Let X in $f(y|x)$ be ...
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1answer
37 views

Cauchy random values in a interval [a, b]

How do I generate random numbers following a Cauchy distribution in a given interval [a, b]. I tried using explained here Trucated distribution, but did not succeed
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1answer
23 views

What is this distribution formulated with w, m and sum sign?

I have a binary classification problem, part of which is defined as follows : p(x|y=1) $\sim w (m_1 , \sum_1$) and p(x|y=0) $\sim w (m_0 , \sum_0$) Where $\sum_1$ is a covariance matrix : $$ ...
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0answers
25 views

$\operatorname{Cov}(X, g(Y))$ of bivariate normal variables.

I was reading some lecture notes and stumbled upon a formula that I can't prove $$ \operatorname{Cov}(X,g(Y)) = \mathbb{E} [g'(Y)] \operatorname{Cov}(X,Y).$$ Where does it come from? It is only given ...
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1answer
2k views

binomial distribution(overbooking plane tickets)

I am having trouble with binomial distribution and this problem: an airplane has 200 seats, but 202 tickets are sold. Assume passengers do not show up with a probability of .03 independently. What is ...
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0answers
80 views

Bayesian sequential updates of normally distributed variables

Suppose that you can observe data that are independently and identically distributed as $N(\mu, 1)$. Your prior distribution for $\mu$ is $N(m, v)$. After observing $n_1$ data with sample mean ...
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1answer
66 views

Generate random numbers following the exponential distribution in a given interval $[a, b]$

I know that to genarete ramdom variables following exponential distribution just do: $$X=-\frac{1}{\lambda}ln(U)$$ where $U\sim U(0,1)$ Now, to find a distribution restricted to the interval $(a, ...
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2answers
239 views

How can I calculate the CDF of this random variable?

$X_1$, $X_2$, $X_3$ are random variables distributed following non-identically independent exponential distribution. The PDF $X_i$, $f_{X_i}(x)$=$\frac{1}{\Omega_i}\exp(\frac{x}{\Omega_i}), ...
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2answers
427 views

What is the appropriate probability distribution to model this situation?

I want to model a random variable which represents the number of failures before success in a repeated Bernoulli trial. I will conduct only utmost N trails and I am ...
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2answers
696 views

Binomial/Geometric Distribution explanation

I've found the following exercise in my Stats coursework. I only have solutions to it, but no explanation. And I would really like to know how to get to the answer. An urn holds 5 white and 3 black ...
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1answer
102 views

Generate exponential random values in a given range [duplicate]

Need to generate random values ​​that follow an exponential distribution on an interval [a, b​​]. I tried using explained here Trucated distribution, but did not succeed
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2answers
1k views

Binomial theorem in probability

We know according to binomial probability theorem , $$P= \binom{n}{r} p^r (1-p)^{n-r} \tag{1}$$ Now If I flip a coin 10 times and want to get the probability for 4 heads then we get according to the ...
4
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1answer
108 views

Tricky probability problem

I am having trouble with proving the following assertion: $X,Y$ are i.i.d. with mean $0$ and variance $1$. If $X+Y$ and $X-Y$ are independent then $X,Y$ are normally distributed. Should I be ...
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2answers
1k views

Sufficient Statistic for a Geometric R.V.

I have a problem that I know I am very close to the solution for, but I think I just need some more formatting to make it a really clean proof. The problem goes like this: Suppose X is a discrete ...
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1answer
903 views

Simple geometric distribution (solution verification)

The question is: In a hockey competition, a player scores $80\%$ of his shots. What is the probability that the player will not miss until his $10^{th}$ try? So I did the following ...
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1answer
30 views

Craps game odd with a pair of dice

In a dice game craps, Alex rolls a pair of fair dice. If he gets $7$ on the first roll, he wins immediately If the result is any number $\neq 7$, he keeps rolling the dice until he gets that number ...
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1answer
424 views

Probability generating function of geometric distribution

For a geometric distribution with $p_{x}(x)=p(1-p)^x, x=0,1,2,3,...$ I have been asked to find the probability generating function. I know that the way to find this is by finding $E(s^x)$ (the ...
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1answer
343 views

Show that the least squares estimator of the slope is an unbiased estimator of the `true' slope in the model.

Under the assumptions of the classical simple linear regression model, show that the least squares estimator of the slope is an unbiased estimator of the `true' slope in the model. Anyone have any ...
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1answer
206 views

How do I find $P(X+Y = k)$ for a geometric distribution?

If $X$ and $Y$ are independent identically distributed random variables where $P(X=k) = P(Y=k) = pq^{k-1}$ where $q = 1-p$. How do you find $P(X+Y=k)$? Is it acceptable to say that $$P(X+Y=k) = ...
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2answers
60 views

Operations on distributions

Say we have two r.v X and Y which are independent and differently distributed ( for e.g X follows a bell curve and Y follows an exponential distribution with parameter $\lambda > 0$ What are the ...
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1answer
33 views

Probability of score greater than 40 by Normal Distribution analysis

Part A is straight forward. Part B am not able to do stuck with the thought that a score (value)X has to be give to John for z= X-mean/std. Pointers would be helpful There are 20 true-false ...
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0answers
26 views

Determining a distribution funtion

This related question: Conditional expectation of conditional sum(Not fully complete) The number $B(\geq 0)$ of bats that leave a cave at the time of a nuclear explosion has a geometric ...
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1answer
54 views

Conditional expectation of conditional sum(Not fully complete)

I have a question: Determine the conditional expectation $\mathrm{E}(A|B)$ for: The number $B(\geq 0)$ of bats that leave a cave at the time of a nuclear explosion has a geometric distribution ...
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0answers
9 views

Exponential Distribution and Capacity

How do you find maximum channel capacity when message transmission time is exponentially distributed with mean 1/p?
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1answer
231 views

Basic core probability question

Two balls are chosen at random from a box containing 12 balls, numbered 1;2; : : : ;12. Let X be the larger of the two numbers obtained. Compute the PMF of X, if the sampling is done (a) ...
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0answers
28 views

how to compute the variance of the random variable Y in binomial distribution?

sorry to bother but I just saw some slides provided by the harvard university. One of those show the binomial distribution with the VAR(Y)=$\frac{\pi(1-\pi)}{N}$ Im bit confused because usually I see ...
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0answers
29 views

Probability distribution of location of maximum of random process

I have the following problem: Given a complex function $H(x)$ at positions $x_1, x_2, x_3,..., x_n$ The function values at each position are independent random circularly Gaussian variables, this ...
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2answers
95 views

Probability of defective cogs in a carton

A company that manufactures cogs sells them in cartons of 100. It is historically known that about 1% of the cogs manufactured by the company are defective. How do I find an expression for the ...
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1answer
48 views

If speeds of two cars are Normal RV s, what is the distribution of the distance between them?

The speeds of two cars are random variables that follow $N(\mu_1,\sigma_1)$ and $N(\mu_2,\sigma_2)$ distributions.They start simultaneously. Let X be the distance between them after m hours. (Note ...
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0answers
24 views

Policy Adjustment in Markov Decision Process

I was using MDP on my work to make optimal decision. I used discrete time, finite state MDP. I assumed that I will have an initial parameters, like the Reward/Cost, state transition probabilities and ...