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

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32 views

How to calculate Fourier Transform of logarithmic function?

Given a random variable (RV) $S$ equal to the sum of two mutually independent (RVs) $X_1,X_2$,i.e.$S=X_1+X_2$ and piece-wise probability density functions (PDFs) of $f_{X_1},f_{X_2}$ are as follow: ...
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1answer
114 views

Expected Number of points in Point Poisson Process

Let $\lambda$ be the intensity of points, distributed as point poisson process, in a square grid of area $A$. Then, the Cumulative disributive function is given by: $$ P(r \leq R) = 1 - e^{-\lambda ...
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1answer
148 views

Beta/Dirichlet question

A generalization of the beta distribution is the Dirichlet distribution. In its bi-variate version, (X,Y) have pdf $f(x,y) = Cx^{a-1}y^{b-1}(1-x-y)^{c-1}, 0<x<1, 0<y<1, ...
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0answers
36 views

maximum likelihood of a dirichlet prior

Suppose $\theta \sim D(\alpha)$ where $D$ denotes the Dirichlet distribution and $\alpha = (\alpha_1,\ldots,\alpha_K)$ its hyperparameter, in which case: $$p(\theta) = \frac{\Gamma(\sum_k ...
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2answers
137 views

Probability of Snow in New York

In New York, snow is reported 25% of days in February. If this trend continues, what is the probability that it will snow exactly 9 days this coming February and is not a leap year? Solve this ...
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1answer
253 views

Show independence of number of heads and tails

I am independently studying Larry Wasserman's "All of Statistics" Chapter 2 exercise 11 is this: Let $N \sim \mathrm{Poisson}(\lambda)$ and suppose we toss a coin $N$ times. Let $X$ and $Y$ be the ...
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1answer
48 views

Letter Arrangements of M,A,R,Y

List all possible arrangements of the four letters m,a,r,and y. Let $\; C_1 \;$be the collection of the arrangements in which y is in the last position. Let $\; C_2\;$ be the collection of the ...
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1answer
70 views

Uniformly at random polynomial

We have a polynomial of degree $d$, and multiply it by a polynomial whose coefficients are chosen uniformly at random and its degree is equal to or less than $d$. My question is whether the result is ...
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1answer
114 views

Birthday Problem: Big Numbers and Distribution of the Number of Samples involved in Collisions

A lot of questions about the birthday problem can be found here, but none seems to address my problem: Background I am thinking of a hash-type data structure design which accepts a certain number of ...
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1answer
89 views

How to calculate conditional probability with inequality

I know that: \begin{equation}\displaystyle P(A=x|A+B=y) = \frac{P(A=x \cap A+B=y)}{P(A+B=y)}\end{equation} Assuming $A$ and $B$ are independent, the intersection of the two events can be resolved as ...
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1answer
111 views

Z~U[0,1] and X=f(Z) and f is:

I have found the f(z): Now, I need to find pdf of X. And I can see that 0< f(Z)=X<1, I don't know how I am going to get f(X), I just can see that f(X)=0 when X<0 and x>1, but I can see a ...
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2answers
30 views

Simple finding the PDF given function

I am a little confused on how to go about finding the PDF given a condition for a function. So I have the function $$ Y(x)=ae^{-bx} \,\,\,\,\,\,\, a,b,x \geq0 $$ and I need to find the value for X ...
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1answer
54 views

Central limit theorem kind of statement for records

I am trying to prove the following statement, but I do not know how to go on: Let $F(x)$ be an arbitrary continuous distribution function. Then there are constants $A_n, B_n > 0$ such that, as ...
2
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0answers
95 views

Relationship between chi-squared and standard normal distributions.

It is well known that if $Z \sim N(0,1)$ then $Z^{2} \sim \chi^{2}(1)$. However, if we know that $X^{2} \sim \chi^{2}(1)$, under what conditions is it true that $X \sim N(0,1)$? As far as I know, this ...
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1answer
24 views

Using a joint distribution table to find probability?

I have the following joint distribution table. I am trying to answer the following questions. A,B,C,D For (a) I put $P(X=1, Y=2)=1/20$ (B) $p(x=0,1\le y<3)= 1/4+1/8$ But I am not sure how to ...
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2answers
44 views

Does the sampling distribution coincide with the population distribution if every possible sample is taken?

Say you have a population. You take random samples repeatedly, and the distribution of all the means of those random samples is the sampling distribution. Right? So does that mean, that if you take ...
4
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1answer
115 views

Distribution of $\frac{X}{|Y|}$, where X and Y are standard normal r.v.'s

Let X and Y be independent standard normal random variables. What is the distribution of $\large \frac{X}{|Y|}$? Attempt: Let $\large U = \frac{X}{|Y|}$ and $ V = |Y|$. This transformation is not ...
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1answer
23 views

Finding the cumulative distribution [duplicate]

How can I find the cumulative distribution function for the following prob density. $$ f(x) = \begin{cases} x & \text{if } 0<x<1 \\ 2-x & \text{if } 1 \leq x <2 \\ 0 & ...
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1answer
36 views

Probability question using Poisson

Here is my Question: A country bus driver picks up passengers randomly and independently at a mean rate of 12 per hour. (i)Find, correct to 3 decimal places, the probability that he picks up ...
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1answer
41 views

Joint density calculation-Spot the error

Suppose $X_1 $ and $X_2$ are i.i.d standard normal r.v.s and $Y=X_1^2+X_2^2$, then we know $Y \sim \chi_2^2 $ and $f_Y(y)= \frac{1}{2}e^{\frac{-y}{2}}$. Using the identity $f_{X,Y}=f_{X\mid Y} \cdot ...
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0answers
63 views

Asymptotic distribution of MLE of the parameters for an ARMA(1,1) model

I was just have trouble interpreting and understanding the asymptotic distribution of the MLE of the parameters for and ARMA(1,1) model (and an ARMA(p,q) model in general). It has been given that ...
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1answer
33 views

Binomial Distribution Proof

What is that $I(\cdot)$ in the 3rd step means? $p_{x_n}(y_n-y_{n-1}) = p(X_n=y_n-y_{n-1}) = p(X_n)$ belongs to the interval $\{0,1\}$, since it is a random variable. Then, how are getting to that ...
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0answers
227 views

Upper and lower bound for variance given mean and median

I have a random variable $X$ taking values in the interval $[0,1]$ with mean 0.2 and median 0.3. What are the lower and upper bounds of the set of possible variances of $X$? I am able to solve this ...
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1answer
43 views

Bernoulli distribution solving for n

So we have this missile protection system that has $n$ radar sets that are all independent. Each have a probability of $0.9$ of detecting a missile. How large must $n$ be if we want the probability ...
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3answers
363 views

Probability to pass multiple-choice test, with two type of questions

First i want to say there are a lot of questions related to this, but i couldn't find a similar case. Suppose we have the typical problem where we need to compute the probability of pass a ...
3
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0answers
636 views

Law of a geometric brownian motion first hitting time (proof checking)

I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all subsequent simulation. Could someone ...
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2answers
58 views

Uniform Random Number

Two uniform random numbers are chosen one after the other. what is the probability of second number second random number greater than first number? I tried this way Please correct me if I am wrong. ...
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2answers
40 views

Integrating a function wrt different measures [duplicate]

Suppose that $(\Omega, \mathcal E, P)$ is a probability space and $X\colon \Omega \to \mathbb R$ is a RV defined on $\Omega$. Denote as $\mu\colon \mathcal B \to [0,1]$ the probability measure on ...
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1answer
56 views

Splitting intervals with cut-off

I would like determine the cumulative distribution function (cdf) of the following random-variable X: Suppose we have the following process: The unit interval is split into two pieces at a point $u$, ...
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1answer
43 views

How can I find the expected value of a random variable with terms that increase until infinity?

Here is the question A company buys a policy to insure its revenue in the event of major snow storms that shut down business. The policy pays nothing for the first such snowstorm of the year and ...
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1answer
36 views

Let $T$ be exponential with parameter $\lambda$. Let $X$ be discrete defined by $X= k$ if $k \leq T < k+1$, $k=0,1,2,\dots$. Find the pdf of $X$.

To be honest, I am lost on this question. Here is what I have so far: $$ \ F_T(t)=- e^{-\lambda t}=P[T\le t] \ $$ $$ \ P[X=k]=P[k\le T \lt k+1] \ $$ I am not sure how to go about finding the pdf for ...
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1answer
78 views

Confidence interval for the conversion on site

I am the developer of web service and I'm trying to to build some plots for the inner dashboard. I raised two questions that I can not solve on their own. Suppose ...
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0answers
23 views

Checking some work on an expectation value problem

I am working on a pretty simple problem (or so it seems it should be) from Griffith's QM text. The problem states: for the probability density function $\rho (x) = Ae^{-\lambda(x-a)^2}$ a) find A ...
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3answers
48 views

Find the $P(X>1)$ for the given pdf?

A part of this question asks me to find the $\Pr(X>1)$ given that $$f_X(x) = \begin{cases}\frac{1}{\sqrt{4x}} & 1 <x<4 \cr 0 & \text{otherwise} \end{cases}$$ I solved this by taking ...
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1answer
134 views

Binomial distribution cdf as the number of trials tends to infinity

I am trying to establish the behavior of the cdf as the number of trials tend to infinity. With a certain probability of success and K number of successes, if we increase the number of trials to ...
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1answer
24 views

Probability Need help understanding how to work problem out

What number would complete this probability distribution? And could you explain how, I am new to this and my textbook isn't helping. x 3 7 11 P(X)0.38 0.29 ?
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1answer
65 views

Convergence in distribution and probability

Suppose ${X_{n}}$ is a sequence of non-negative random variables with cumulative distribution function given by $F_{X_{n}}(x) = 1 - 1/(1+nx)$ for $x\geq 0$. Examine if $\{X_{n}\}$ converges in ...
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1answer
18 views

Finding the probability of a probability density function

Suppose that $f(x) = e^{−x}$ for $0 < x$. find $P(1 < X)$ I know typically we integrate $f(x)$ from $1$ to $\infty$ but in this case $x = 1$ is not included, how do I go about doing this? All ...
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1answer
95 views

Central limits without replacement in a finite population.

"Everybody knows" that there are lots of variations on the theme of the central limit theorem. The most frequently seen form seems to be this: Suppose $X_1,X_2,X_3,\ldots$ are i.i.d. random variables ...
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1answer
76 views

Marginal distribution functions of two discrete random variables

I am currently taking a probability course based on the book A first course in probability by Sheldon Ross. I have been trying to solve the following problem: $f_{X,Y,Z}(x,y,z) = c$ where $x = 1, ...
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1answer
591 views

Joint distribution of range $(R=X_n-X_1)$ and mid-range $(V=\frac{1}{2}(X_1+X_n)$order statistics

Let $X_1,X_2, · · · , X_n$ be independent and identically distributed Uniform random variables on the interval (0, a) for a > 0, each having a density function $f(x) = \frac{1}{a}$, $0<x<a$. Let ...
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1answer
116 views

Probability: Random Variables and Probability Distributions

1) The function: $F(x)=k(1-(1/2)^{[x]})$, $x > 0$ Is the distribution function for a discrete random variable X. Here, [x] denotes the integer part of x (i.e., the greatest integer less than or ...
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0answers
73 views

Likelihood Ratio Test distribution

I am reading the third edition of Testing Statistical Hypotheses by Lehman and Romano. I am reading the last part of chapter 12, where there is the theorem that gives the asymptotic distribution of ...
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1answer
124 views

Exchangeable/Independent Bernoulli Distribution

Let P be a uniform random variable on the interval $(0,1)$ with density function f(p) = 1, $0<p<1$. Let $X_i|P$, i = 1,2,...,n be independent and identically distributed random variables having ...
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1answer
204 views

Travelling from one destination to another

This is the problem : Manish has to travel from A to D changing buses at stops B and C enroute. The maximum waiting time at either stop can be 8 minutes each, but any time of waiting up to 8 minutes ...
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6answers
1k views

Looking for the logic of a sequence from convolution of probability distributions

I am trying to detect a pattern in the followin sequence from convolution of a probability distribution (removing the scaling constant $\frac{6 \sqrt{3}}{\pi }$: ...
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1answer
67 views

Apparently same probability questions with different answers.

I was reading A first course in probability by Sheldon Ross when and then I came up with this question. This is how he introduces the famous problem of points Independent trails, resulting in a ...
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2answers
48 views

Flipping several biased coins

Assuming I'm flipping $M$ biased coins with different probability for heads $p_i, i=\{1,...,M\}$. What is the probability of having $k$ times head? Is there a distribution function known for this?
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2answers
98 views

Finding $f_Y$ such that $Z=Y\cos(X)\sim\mathcal{N}_{0,\sigma}$ for $X\sim\mathcal{U}[0,2\pi]$

I need to choose the probability distribution $f_Y(y)$ of a random variable $Y$ such that the variable $Z=Y\cos(X)$ is normally distributed with zero mean, i.e. ...
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2answers
45 views

How can I do a constructive proof of this:

Say Z is a non-negative R.V, and P(Z>0)>0. Then exists a a>0 and an b>0 such P(Z>a)>b. I am not sure how to start with the proof, I've been assigning numbers than can qualify for some CDFs but I don´t ...