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

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1answer
18 views

Probability Density Function Example Help

I am reading through the text, "Introduction to Probability Theory with Contemporary Applications" by Lester Helms. I am stuck on the attached example. I understand how the author obtained the joint ...
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0answers
18 views

Stratified Sampling $E(E(X \mid \mu, \sigma))$

Let $X$, $\mu$ and $\sigma$ be random variables. I want to estimate $E(X)$ using Monte Carlo. I am able to sample from, and know in closed-form, both the conditional distribution of $X \mid (\mu, \...
4
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0answers
26 views

Transformation of random variables exercise

I want to know if my solution to the following exercise is correct: Let $X$ be a gamma distributed random variable with parameter 2, meaning with distribution $$P_X(\mathrm{d}x)=\mathbb{1}_{\{x>...
2
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0answers
40 views

On the distribution and the moments of $\max\{1/\sqrt{U_1},…,1/\sqrt{U_n}\}$, where $(U_k)$ is i.i.d. uniform on $(0,1)$

Let $U_1,U_2,...$ denote an i.i.d. sequence of random variables with the uniform distribution on $[0,1]$. For every integer $n\geq1$, we set $M_n = \max\{1/\sqrt{U_1},...,1/\sqrt{U_n}\}$. a) Compute ...
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1answer
33 views

Probability involving the maximum of i.i.d. uniform r.v.'s

The question is : $100$ numbers are independently and uniformly distributed on $(0,1)$.Then what is the probability that the maximum of these numbers will be at most $0.9$? How can I solve it? ...
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2answers
56 views

estimate a probability

Let $X_1....X_{48}$ be independent random variables, each follows a uniform probability distribution over [0,1]. What is the best way to estimate P($\Sigma_{i=1}^{48} X_i > 20)$?
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1answer
81 views

Distribution on a collision variant of Hypergeometric distribution?

I have the following scenario: there is a set $\Omega$ of $N$ elements, among which $K$ are marked — let $M\subseteq \Omega$ be this subset. Alice select uniformly at random (i.e., sampling without ...
3
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1answer
48 views

What does it mean for a pdf to have this property?

What does it mean for a probability density function $f(x)$ to have the following property? $$1+\int_{x=0}^{\infty}x^2 \left(\frac{f'(x)^2}{f(x)}-f''(x)\right)dx>0$$ I have tried a lot to ...
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1answer
14 views

Reference needed for properties of Convergence of Random Variables

Does anybody know a good reference for properties of convergence of random variables? For example, if $X_n$ converges almost surely (a.s) to $X$ and if $Y_n$ converges a.s to $Y$, then $X_n Y_n$ ...
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1answer
96 views

Number of urns containing a ball of each color: is there a probability distribution describing this?

There are $B$ urns. There are $n$ red balls and $n$ white balls with $n\leq B$. Each ball is independently put into each urn with equal probability. An urn can get at most one ball with the same color ...
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0answers
31 views

Issues regarding my take on proving $E(X) = \lambda$, where $X\sim Poisson(\lambda)$

My proof: Let $X\sim \mathrm{Poisson}(\lambda)$. Then $$f_{\Tiny{X}}(x) = \frac{\lambda^x}{x!} e^{-\lambda}.$$ Thus, $E(X) = \sum_{x=0}^{\infty} x f_{\Tiny{X}}(x) = \sum_{x=0}^{\infty} x \frac{\lambda^...
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0answers
11 views

what are the mean vector and covariance matrix of the multivariate truncated normal distribution?

Let $\mathbf{X}=(X_1,X_2, \ldots, X_p)^\prime$ has multivariate normal distribution with mean vector $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}$, i.e. $\mathbf{X}\sim N_p(\mathbf{\mu}, \...
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2answers
36 views

Transformation of the uniform distribution

I struggle to understand the transformation of a random variable with uniform distribution. For example: Let $X\sim \text{Uniform}(0,1)$ and $T=-2\ln(X)$ and I want to find the CDF of $T$, then I ...
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1answer
27 views

Finding probability sample proportion is less than 33% assuming null hypothesis is true

Candidates 1,2 and 3 are running for a position in a company. Candidate 1 claims 38% favourability among all the voters. Assuming this is true, what is the probability that in a random sample of 500 ...
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2answers
36 views

Need help understand independent identically distributed random variable.

Let $X_1, X_2, X_3$ be three independent, identically distributed random variables each with density function $$f(x)= \begin{cases} 3x^2 & 0 \le x \le 1 \\ 0 & \text{otherwise} \end{cases} $$ ...
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3answers
44 views

How could I find the covariance for $X$ and $Y$ in this case?

If $X \sim U(-1, 1)$ (so $X$ is uniformly distributed between $-1$ and $1$) and $Y = X^2$, what is the covariance between $X$ and $Y$? Are they independent? So the formula for covariance is: $\...
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0answers
20 views

Need help with cumulative distribution function problem.

A commercial water distributor supplies an office with gallons of water once a week. Suppose that the weekly supplies in tens of gallons is a random variable with pdf $$f(x)= \begin{cases} 5(1-...
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1answer
17 views

Joint distribution of $n$ Bernoulli variables equal to binomial distribution, how? [closed]

Is the joint distribution of $n$ Bernoulli variables equal to binomial distribution? I am confused by this questions and I would like to understand this. What about if Bernoulli variables dependent?
3
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1answer
31 views

Asymptotic Moments of the Binomial Distribution, $E(X/(np))^k = 1 + O(k^2/n)$?

Let $X \sim \text{Binomial}(n, p)$ be the sum of $n$ Bernoulli($p$) random variables. What is the value of $E(X/(np))^k$, where $k$ is a large integer, as $n$ grows large? From calculations the ...
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0answers
14 views

(Binomial) Distribution problem.

I have two questions concerning the following: A pension (B&B) has three guest-rooms. The pension is opened from April - November (244 days, 35 weeks). Last year the distribution of the number of ...
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1answer
16 views

Find relating equation of M and N

P$X$(X)= Me^(-2|x|) + Ne^(-3|x|) is the probability density function for the real random variable X over the entire axis , M andN Both are positive real number . What will be the equation relating M ...
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1answer
26 views

Value of cumulative distributed function at the origin

Consider a Gaussian distributed random variable with zero mean and standard deviation sigma. The value of its commulative distributed function at theorigin will be .... In this question, 4 options ...
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1answer
20 views

sufficient statistics to estimate the unknown parameters

I am a beginner in statistical inference and am learning sufficient statistics. As far as I know the distributions conditional on the sufficient statistics doesn't depend on the unknown parameters. I ...
0
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1answer
56 views

If $X \sim Exp(1)$ and $Y \sim Exp(1)$, prove $(\frac{X}{Y}, Y)$ is continuos without using the change of variables theorem. [closed]

I've been thinking about this problem for a while and I'm not sure which way to go. Let $X$ and $Y$ be two independent random variables with exponential distribution of parameter 1. Let $U = \frac{X}...
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0answers
22 views

Polynomial Chaos with Beta Distribution - Standard Beta Random Variable, Transformation of Beta Random Variable

Background: I am dealing with a non-intrusive polynomial chaos expansion (e.g. here [Hosder,Walters;2010]). This means I want to represent an uncertain output $U(\xi)$, dependent on a vector of ...
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0answers
27 views

compound probability and conditional expectation

I'm stuck on a formula which looks obvious but that I fail to prove: If $Z$ is a real-valued random variable with distribution $\mu$, $T_z$ is a random time for each $z\in \mathbb{R}$ and $B$ is a ...
0
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1answer
16 views

Central moments of a Bernoulli distribution

Consider a discrete random variable distributed as a Bernoulli: $$ Y=\begin{cases} 1 & \text{with probability } p\\ 0 & \text{with probability } 1-p \end{cases} $$ The $n$-th central moment ...
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0answers
10 views

Wishart plus scalar multiple of identity

Is the sum of a Wishart distributed matrix and a scalar multiple of identity matrix, another Wishart distributed matrix? I guess it is not. If not, what is the distribution called and can its density ...
0
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1answer
25 views

Why not represent discrete multivariate probability distribution as univariate?

For example, a bivariate distribution over binary variables can be represented with a 2x2 matrix of probabilities: \begin{bmatrix} p_{0,0} &p_{0,1} \\p_{1,0} &p_{1,1} \end{bmatrix} The same ...
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1answer
24 views

Finding the value of a for a density probability function

Below is the graph of the density of a random variable $X$. I know that i have to set up two integrals equals to 1, because the whole area is equals to 1. So this is my attempt. $\int_2^4 2a\,dx =...
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1answer
10 views

Need help with hyper geometric distribution problem?

Consider a suitcase with 7 shirts and 3 pants. Suppose we draw 4 items without replacement from the suitcase. Let X be the total number of shirts we get. Compute $P(X ≤ 1)$ . This is a problem Marcel ...
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0answers
48 views

Birth-death process and transience.

I am unable to tackle part c) and d) can anyone help/ sugesstions? A Markov chain with state space ${0,1,2,...}$ is called a “birth-and-death chain” if the only non-zero transitions from state $i$ ...
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0answers
18 views

Positive moments of independent variables are also independent

Suppose we have $X$ and $Y$ which are random variables and they are also independent and we also have $i, j \in \mathbb{N}_{+}$. Is it true that $X^{i}$ and $Y^{j}$ are independent? Actually I need ...
2
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1answer
43 views

Uniformly distributed differences

Is there a collection of random variables $X_1,X_2,\ldots,X_n$ such that $Y_1=X_1-X_2,~Y_2=X_2-X_3,\ldots,~Y_n=X_n-X_1$, are independently uniformly distributed on $[-1,1]$. How $X$'s should be ...
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1answer
32 views

Closed Form E[exp(x'Ax)]

Is there a (general) closed form available for the following expression? $$\mathbb{E}\left[e^{x^{T}Ax}\right]$$ Where: $$x=\left\{ x_{1},x_{2},...,x_{N}\right\} \sim\mathcal{N}\left(0,\varSigma_{N}\...
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1answer
67 views

Drawing numbers on a plane *uniformly*

I am not being very precise here, because I do not know what would be the more precise terminology. I would definitely appreciate any comments on my method and my terminology. By choosing uniformly ...
4
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0answers
178 views

Is this function increasing?

I'm stuck in showing that the following function is increasing over the domain $\left[0,x_0\right]$: $$\Pi\left(z\right) = \int_{0}^{\phi\left(z\right)}\int_{x}^{\bar{x}}\left(2y-b\left(x\right)-x\...
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0answers
41 views

Is this integral analytically solvable?

I am trying to find the average of exponential of a random quantity which obeys log normal distribution. For that I need to evaluate $$\int_{-1}^{\infty } \frac{\mathrm{e}^{-\left(\log (\delta +1)+\...
2
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1answer
141 views

Evaluating the distribution involving a Brownian motion.

I'm trying to solve one exercise closely related to this question. Since I don't have an answer yet, I thought to post a new question with my thoughts about the problem. I hope this does not break any ...
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2answers
112 views

Stationary distribution of “probabilistic geometric series” with two alternative ratios

I have an iterative process starting at $X_0=2$. In each iteration $i=1,2,\ldots$, the value of $X_i$ is determined based on the value of $X_{i-1}$ as follows: With probability 0.5, $X_i=qX_{i-1}+1$, ...
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1answer
58 views

Probability of getting n heads when n is very large [closed]

How to show that the probability of getting n heads when 2n times coin is toss is very small? Moreover, how to show that the probability of more than n heads is close to 0.5?
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2answers
25 views

Notation inconsistency? Standard uniform distribution U(0,1)

Very simple and quick question. Usually distribution notation is such that you give the name of the distribution, then its mean, and finally the variance, for example for normal distribution: $$N(0,1)...
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38 views

Drawing tickets from a hat

I am trying to solve this question. Suppose we draw two tickets from a hat that contains tickets numbered 1, 2, 3, 4. Let $X$ be the first number drawn and $Y$ be the second. Find the joint ...
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3answers
44 views

How to calculate $P(X^2 < X)$

I need help with this: Let $X \sim \mathcal{exp(1)}$. we need to calculate $P(X^2 < X)$. All I know so far $P(X^2<X) = P(-$$\sqrt{X}<$X$<$$\sqrt{X}$) = P($X$<$\sqrt{X}$) whats now?
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1answer
24 views

Finding the Integral´s interval for a Probability Function

The probability density function of a given random variable is given by the graph below. How can I set up the integral in order to find P(X>0, P(X>3/4). I tried to set up the integral and this is my ...
0
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1answer
35 views

Use of language on wikipedia - what kind of distribution?

I have an interesting problem and was wondering whether anyone would be able to point me in the right direction. I am wondering whether the use of a word in the english language on Wikipedia is ...
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0answers
30 views

Algebraic manipulation of probability distributions

Let $X$ and $Y$ be random vectors that have the same continuous distribution. If $A$ and $B$ are constant matrices and $AX$ and $BY$ have the same distribution does this imply that $A=B$? Are there ...
2
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1answer
37 views

Joint distribution of dependent Bernoulli Random variables

I have $N$ Bernoulli random variables $X_1, ..., X_{N}$ with known parameters $p_1, ..., p_{N}$. I want generate a joint distribution in which these random variables are not independent as I know that ...
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24 views

How to generate multivariate random variables given probability distribution?

Suppose you can generate uniformly distributed random numbers $x_i\in[a,b]$. To shape probability distribution of these numbers as you like using inverse transform sampling. But what if you need to ...