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

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0answers
37 views

How to find the appropriate limits for finding this particular probability? [duplicate]

Suppose there are three statistically i.i.d continuous random variables $X_1$, $X_2$, $X_3$ each are uniformly distributed in the range $[0,1]$. How to find the probability $P(X_1 + X_2 < X_3)$? I ...
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1answer
34 views

Is it valid for a probability density function take the value $\infty$ somewhere?

In particular, it seems that the gamma random variable, with pdf (from wikipedia): becomes $\infty$ when $x=0$ and $\alpha<1$ Does it make sense for a pdf to take the value $\infty$ at some ...
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1answer
906 views

Determine the Asymptotic Distribution of the Method of Moments Estimator of $\theta$, $\tilde{\theta}$

I am having difficulty understanding what it means to find the asymptotic distribution of a statistic. I have the correct answer (as far as I know), but I am unconvinced that I understand the process ...
4
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1answer
164 views

Random variable exponentially distributed?

I just want to be sure about this: If I read the phrase ' a random variable is exponentially distributed'( which is often said in probability theory and then it is never explictely stated what $X$ ...
0
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1answer
64 views

how many ways are there to distribute 10white and 10black balls into 20 distinct boxes so that at most one box is empty

Question: how many ways are there to distribute 10white and 10black balls into 20 distinct boxes so that at most one box is empty. I understand case-1. But I cannot understand a part of answer ...
3
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1answer
136 views

Exponentially distributed random variables

Given two exponentially distributed random variables $ X_1 $ and $X_2$ (assuming rates $\lambda_1$ and $\lambda_2$ respectively), determine the probability that one is smaller than the other. So ...
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2answers
30 views

problem finding marginal distribution for a PDF

I need to find the marginal distribution function $f_y$ for $$f_{xy}(u,v)= \begin{cases} 1\over u, & \text{$u\ge 1, 0\le v \le {1 \over u}$} \\ 0, & \text{else} \\ \end{cases}$$ my problem ...
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1answer
13 views

find a CDF for a liner transformation for indipendent random variable

given that $X_1 \sim U[-2,1]. X_2=0.5e^{-|t|}, -\infty<t<\infty$. find $F_Y$ if: $$ Y = \begin{cases} X_2, & \text{$X_1<-1$} \\ X_1, & \text{$-1 \le X_1<0$} \\ 3, & ...
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1answer
19 views

Which distribution would fit this binomial type series of events?

We have arbitrarily many lottery scratch-n-win tickets Each ticket has a $p=0.1\%$ chance to win 1 prize (a success event) Scenario A, Scratch n tickets and record the number of prizes won $X$: ...
-1
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2answers
543 views

Uniform Distribution in [0,1] where P[x1+x2<=x3]

Consider the following question : X1, X2, X3 are 3 independent random variables having uniform distribution between [0,1] then P[x1+x2<=x3] to the greatest value is ? Now this is not a homework. ...
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1answer
75 views

Finding the MLE of pareto dist., and trouble interpreting $\prod$ notation properly.

I am generally having trouble understanding how to use product notation when calculating Maximum Likelihood Estimators. The example bellow is from a random sample $X_1,...,X_n$. Find the MLE of ...
3
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2answers
89 views

Conjugate priors make calculations easier but at what cost to the model?

As I understand, when we have a parametric pdf and need to estimate the parameter based on some observed fact, we tend to choose a conjugate prior of the pdf for the parameter. Because conjugate prior ...
1
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1answer
119 views

Stein's Method and Coupling of random variables

Suppose a particle starts at position 5 on a number line and at each period the particle moves one position to the right with probability p and, if the particle is above position 0, moves one position ...
0
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1answer
47 views

How to find the distribution for the following random variable $X$?

If $(X-20)\sim\operatorname{Poisson}(a)$, where $a$ is the model parameter for Poisson distribution. Then how is the random variable $X$ distributed? How to write down the pmf for random variable $X$? ...
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2answers
18 views

Giving the distribution law of the number of experience

the probability that a certain event occurs is 0,6. The experience is conducted until the event occurs but no more than 4 times. I should give the distribution of the number of experience It is ...
1
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1answer
24 views

If F is long-tailed then there exists $h(x)$ st $\frac{\bar{F}(x + h(x))}{\bar{F}(x)} \rightarrow 1$ as $x \rightarrow \infty$.

If F is a CDF for some rv X, and is long tailed ie $\frac{\bar{F}(x+1)}{\bar{F}(x)} \rightarrow 1, x \rightarrow \infty$ then there exists a function $h(x) \geq 0, h(x) \rightarrow \infty$ as $x ...
1
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1answer
64 views

Prove long-tailed distribution is heavy-tailed

Consider a sequence of iids $X_is$. I know a distribution is heavy-tailed if the $E(e^{tX_i}) = \infty$ for all $t>0.$. Additionally a distribution is long-tailed if ...
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2answers
46 views

PDF of the addition of several outcomes from Poisson distribution

We draw $n$ values from a Poisson distribution and add them. - What is the expected of this addition - What is the PDF of this addition It seems quite intuitive to me that if we add $n$ Poisson ...
0
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1answer
32 views

Probabilty Mass Function. Function which depends on past outcomes $X$

I randomly draw numbers according to the probability mass function (PMF) $X$ in which all negative values have probability zero. Each value that is drawn from $X$ can be thought of the lifespan of one ...
1
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1answer
156 views

Hypergeometric Distribution : probability that more than half is good

To simplify the context, let's say that 34 % of people are ugly. haha... lets take a sample of 15 people. (n = 15) a) What is the probability that 3 or less out of the 15 are ugly ? I went on and ...
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1answer
37 views

F distribution function continious in $x\Leftrightarrow \mu(\partial (-\infty,x])=\mu(\left\{x\right\})=0$

Consider $(\mathbb{R},\mathcal{B},\mu)$ with $\mu$ probability measure. Let $F$ be the distribution function of $\mu$. Show that $F$ is continious in $x$ exactly then when $$ \mu(\partial ...
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2answers
79 views

Maximum of two skewed normal distributions

Does there exist a means to approximate the maximum of two skewed normal distributions in terms of another skewed normal distribution? To make it clearer, given 2 skewed normal distributions ...
3
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1answer
148 views

Probability of arriving at office before $9$ am

I'm having trouble answering this question: A person leaves for work between $8:00$ A.M. and $8:30$ A.M. and takes between $40$ and $50$ minutes to get to his office. Let $X$ denote the time of ...
3
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1answer
52 views

Generating a uniform distribution in the volume of a box

Suppose I have a three dimensional box, of volume $V$, and with lengths $x, y$, and $z$. I then change the box volume by $\Delta V$, such that $(V + \Delta V) = (x + \Delta x)(y + \Delta y)(z + ...
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1answer
50 views

what is the expected return of this game when played with best strategy?

In a two people's game, you start with one dollar and you are betting 1 dollar coin at the very start of the game, then if you win you have double amount of money and you can choose of choose not to ...
2
votes
1answer
182 views

Coupling Pairs of Random Variable.

Let $\{X_i\}_{i=1}^{n}$ and $\{Z_i\}_{i=1}^{n}$ be sets of independent random variables with coupling $\{X^{\hat{}}_i\}_{i=1}^{n}$, $\{Z^{\hat{}}_i\}_{i=1}^{n}$ respectively. It then states ...
1
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1answer
51 views

Finding X with cdf $F(x)=1-\exp\left(-x^3\right)$

Suppose the following cdf $F(x)=1-\exp\left(-x^3\right), x \geq 0$ . How can I generate a stochastic variable $X$ with this cdf using the function runif() in R? Is ...
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3answers
125 views

Does $\rho>0$ imply $P(X>a,Y>a)$$>$$P(X>a) P(Y>a)$?

Consider two stochastic variables X and Y that are distributed $N(0, \sigma_1)$ and $N(0, \sigma_2)$ and are correlated $\rho>0$. Is it true that $P(X>a,Y>a)>P(X>a)$$P(Y>a)$? Does ...
0
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1answer
23 views

Establish that U1 and U2 are independent?

Suppose that $Y_1$ has a gamma distribution with parameters $\alpha_1$ and β, that $Y_2$ is gamma distributed with parameters $\alpha_2$ and β, and that $Y_1$ and $Y_2$ are independent. Let $U_1 = ...
2
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1answer
565 views

Multiplying two Gamma distributions over the same variable

I am looking at a software library where there is a function that multiplies two Gamma distributions defined over the same random variable. So, it is basically multiplying two Gamma densities with ...
1
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1answer
48 views

Random variable transformation function

I am stuck with a random variable transformation problem ($Y=\phi(X)$). The random variable $X$ has a uniform distribution $U(-1,1)$, and I want to transform it into $Y$ which is also an uniform ...
0
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0answers
81 views

Error Propagation - functions of the mean.

Given a number of measurements $\{x_i\}$ with values distributed according to a (known) probability distribution $\rho(x)$ with a theoretical mean $\langle x\rangle = \int dx x\rho(x) = f(y)$ and a ...
0
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0answers
28 views

Probability Mass function and Variance Question

Given the following probability mass function, can you obtain k such that Var(X)=1.5? X=0 , P(0)= 3k X=1 , P(1)= 2k X=2 , P(2)= k I know that Var(X)=E(X^2)-(E(X))^2 and that E(X)= ΣXP(X). I did ...
3
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2answers
112 views

Probability of getting 50 heads is equal to the probability of getting 51 heads

$100$ coins are tossed. Probability of getting $50$ heads is equal to the probability of getting $51$ heads.that probabiity is $.....?$ My TRY i gave a lot time to this problem.but i am very confused ...
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1answer
790 views

Relationship between Poisson and Exponential distribution, automobiles arrive per minute

I am struggling to understand the Poisson and Exponential distributions. The number of automobiles that arrive at a certain intersection per minute has a Poisson distribution with a mean of 5. ...
0
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1answer
31 views

transfer function over a uniform distribution to generate a two-fold variation, but result is also s uniform distribution

I think, I am stuck at a transformation problem where I believe there is a solution, but I don't know what it is. I have uniform distribution $u_1(x) = \frac{1}{b - a}$, where $b=-1$ and $a=1$, and ...
0
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2answers
66 views

PDF Questions - expected value? 2nd moment? variance?

I am trying to figure out how to solve the questions above for the given PDF but im not sure how to do the steps. I'm not looking for answers here as I want to ...
0
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1answer
32 views

Standard deviation / Bell Shaped Distribution

In a forest, it is known that the circumference of 10 year old oak trees when measured three feet above the ground is 20 inches with a standard deviation of 4.5 inches. It is also known that these ...
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1answer
26 views

Mean of a PMF with a variable

I am given the following PMF: and I am asked to find the mean. I'm a bit lost on what to do with the ...
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1answer
480 views

Central Limit Theorem exercise

I'm trying to solve this exercise: Drums labeled 30 L are filled with a solution from a large vat. The amount of solution put into each drum is random with mean 30.01 L and standard deviation 0.1 L. ...
2
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1answer
32 views

finding a cummulative distribution function from a uniform density function

how can I find a cummulative distribution function from a piecewise uniform density function?
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2answers
90 views

What is the probability that a student knows the answer given that he has answered it correctly? [closed]

A large class in stochastic processes at a school is taking a multiple choice test. For one particular question with m proposed multiple choice answers, the fraction of students who know the answer is ...
2
votes
1answer
211 views

Correlation Coefficient Distribution Function: An Apparent Discrepancy?

I'd like to explain an apparent discrepancy between: (1) The sample correlation distribution function between sample vectors for a bivariate, correlated random variable (correlation coefficient = ...
2
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1answer
147 views

Structure of the functional space $\int_ {- \infty} ^ \infty f (x) dx = 1 $

Please, help me with studying of useful practical features of the following functional space: $$\int_{-\infty}^\infty f(x) \, dx = 1$$ For example: 1) What basis types are most convenient for ...
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1answer
21 views

What is the expectation of the first time dice-rolls Z and Z+1 are similar?

We roll a dice. Let $Z=$ the first time the rolls $Z$ and $Z+1$ are similar. What is $E[Z]$ ? What is $E[Z^2]$ ? The problem sounds unnatural to me because i'm asked about a roll $(Z+1)$ that ...
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2answers
401 views

What is the probability that a student knows the answer given that he has answered it correctly,…?

A large class in stochastic processes at at a school is taking a multiple choice test. For one particular question with m proposed multiple choice answers, the fraction of students who know the answer ...
1
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1answer
40 views

Variance with correlated variables

A simple question that I don't manage to solve: I can use different methods to measure a magnitude $x$. The results of these methods are correlated and have some uncertainties. Combining the results ...
0
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0answers
39 views

Convergence of sequence of stationary distributions of Markov chains

I have a sequence of finite, discrete-time ergodic Markov chains indexed by a parameter $N$, and I want to prove that their stationary distributions are converging to a well-defined limit as $N\to ...
0
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1answer
220 views

Generating Random Serialnumber with least similarity

I want to generate 16-digits hexadecimal serial-number like: F204-8BE2-17A2-CFF3. (This pattern give me 16^16 distinct serial-number But I don't need all of them as I describe below) I need an idea ...
0
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1answer
41 views

Min of exponential distr

Let $X_1, X_2, \ldots X_n$ be independent random variables, where each of them follows an exponential distribution. What distribution has $Y=\min\{X_1,X_2, \ldots,X_n\}$ ? Attempt: We have a ...