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5.708 distributions questions.

I have this statistics oriented question and I need some input on the approach. There is a new product introduced, a contact lence that has seleral ranges of attributes (powers) from which the users ...

Let X and Y be uniformly distributed on a unit disk such that $x^2 + y^2 + 2 \leq 1$ Let $R = \sqrt{X^2 + Y^2}$. Find the CDF and PDF of $R$. I start by writing $P(X^2 + Y^2 \leq -1)$, but this ...

A basketball player succeeds in making a basket three tries out of four. How many times must he try for a basket in order to have greater than 0.99 probability of making at least one basket? In this ...

Given $Z_1, Z_2$ are i.i.d standard normal random variables. Let $$V:=\frac{Z_1}{\sqrt{(Z_1^2+Z_2^2)/2}}$$ Derive the pdf of $V$. The numerator and denominator of $V$ are dependent, so the square ...

I have this homework question where there are 3 random variables $(U_0,U_1,U_2)$ which are independent and uniform in the interval $[-1,1]$. I have two other random variables $(X,Y)$ defined as ...

I have this vector data and I am trying to find distribution that this data fits. ...

What is the difference between several/Multivariate Random Variables (RVs) & a Sequence of RVs? Example: Picking a student from class and noting his/her height and weight is several RVs (...

I've been trying to prove the following: $$\phi'(x)(1-\Phi(x))+2\phi(x)^2>0$$ where $\phi$ is the standard Normal pdf and $\Phi$ its cdf. I've tried many simulations and I believe it's true in ...

What is the probability of a 1,1 distribution in Bridge when there are 2 cards outstanding? I understand there are 4 possible outcomes, for example Kx and 0, 0 and Kx, K and x, x and K. That is, 2 ...

I was thinking about the meaning of location-scale family. My understanding is that for every $X$ member of a location scale family with parameters $a$ location and $b$ scale, then the distribution of ...

In my notes from university I have written down that the residual standard error (from normal linear regression) has the following distribution $\frac{\hat{\sigma}^2}{\sigma^2}\sim \frac{\chi^{2}_{n-...

I am curious what the derivation for the expectation of the maximum of two jointly normal random variables $X$ and $Y$ with correlation coefficient $\rho$. I could start with the following but the ...

As part of a complicated project relating to widgets being sent out to incorrect factories ("missorts"), I am examining a process which involves operators standing on a conveyor belt, removing widgets ...

I am seeking an approach to sampling conditional joint distribution (new to probability). I will put my case in a simple way: Similar question for discrete variables is asked Here but not yet ...

I'm analysing some datasets that produce heavy tailed data when plotted as a histogram. My initial goal was to attempt to fit a known distribution to my dataset. Thereafter I use to the properties of ...

Why does the Scipy Generalised Gamma distribution have 2 shape parameters when other implementations only require one. See wikipedia and R(GAMLSS) for examples.

Consider a random variable with a known distribution. I want to know if we quantized the variable the how the probability density function(PDF) would change? Specifically, if the original PDF is ...

Suppose $X_1,\ldots,X_n,Y_1,\ldots,Y_n$ are i.i.d $\mathcal N(0,1)$ random variables. I am interested in the distribution of $$U=\frac{\sum_{i=1}^n X_iY_i}{\sum_{i=1}^n X_i^2}$$ I define $$Z=\...

In Poisson point processes, we first let $N$ be a random Poisson random variable, with parameter $\lambda$: $$ P(N=n)=\frac{e^{-\lambda}\cdot\lambda^n}{n!},\qquad n=0,1,\ldots $$ Without loss of ...

Let $D$ be a one-parameter distribution family whose support is the positive reals. Let $D_x$ be a distribution from this family parametrized by its mean $x \ge 0$, let $X \sim D_x$ and $Y \sim D_y$ ...

I know that the sampling distribution of the mean can be assumed to be normal if N>30, but does this have an implication on the "30" itself (the sample data)? I have three different time series with ...

I know that if $X_1$ and $X_2$ follow a Gamma distribution with parameters $(p_1,a)$ and $(p_2,a)$ respectively, then $\frac{X_1}{X_2}$ will follow a Beta prime distribution of parameters $(p_1,p_2)$. ...

Is it possible to derive the joint probability mass function of two discrete random variables (categorical in my specific case) IF we know that they are perfectly correlated? Let's assume for ...

I am a stats rookie and have a relatively basic question regarding the normal distribution of data. If I measure a continuous variable under several different conditions/treatments, I look at the ...

Suppose $X_1,X_2,\dots,X_n$ are p-dimensional random variables with distribution $N_p(\mu,\Sigma)$. Let $X_{n\times p}$ is the data matrix. $\mathbb{Z}_{n\times p}= (n-1)^{-1/2}\left(I_n - n^{-1}\...

Let's say we have a random variable $Y$ defined as the sum of $N$ Bernoulli variables $X_i$, each with a different, success probability $p_i$ and a different (fixed) weight $w_i$. The weights are ...

I am doing bayesian analysis. Exploratory analysis shows the parameter might has a left skewed shape. So what kind of distribution should I used as prior distribution for this parameter? Any kind of ...

I have a rather funny situation here. There is this random complex number, $z$, with an unknown distribution $\rho_\lambda(z)$ which depends on a parameter, $\lambda$. I can compute the moments and ...

Let, I have the following Frequency Distribution: Which one would be the correct Grouped frequency Table if class-width = $1.0$? $Table\#1$ $Table\#2$ Explain "Why". What would be the answer if $\...

If we observe two censored distributions, where all observations above a cutoff are set at the value of the cutoff, how can we test whether the observable distributions suggest that the two censored ...

Suppose we have a known discrete probability distribution $X$ over $\{0,1\}^k$. Given a sequence of binary values $e = (e_1, ..., e_n)\text{, where } e_i\in \{0,1\}^k$, what is the probability (or the ...

I have an empirical distribution that looks like the the image below and I hoping to model it with some parametric distribution. The X axis measures "number of ...

I need to compare a few distributions. I believe they all are non-normal with fat tails. How can I be sure of it? I plotted the quantile plot for all of them, and I can see that they all have a fat ...

Suppose that I visualized a distirbution and noticed that it was multimodal. For example, I collected the heights of a bunch of students. I notice that the distirbution is multimodal. How would I ...

The title says it. If one wishes to analyze ternary outcomes—that is categorical outcomes with specifically three values (-1,0,1)—are $\chi^{2}$ / contingency table tests and multinomial-logistic ...

I'm experimenting with replacement for normal distribution for my models. So far, I tried Gamma and Inverse Gamma distributions. Gamma significantly underestimates the probability of the extreme ...

I am currently working with a dataset in which I have provided a numerical value (0 to 3) to different regions on a subject. The values correspond to the severity of the region affected (0 - not ...

Problem Description I have simulated the movement of cars in a lane. The simulation model was first calibrated based on an observed car traffic. Now, I have vectors of speeds (collected every 0.1 ...

I have $ B $ bins and $ G $ balls, and each of the balls has a weight. I toss the balls uniformly into the bins. I select the $ K $ bins that have the highest weights as determined by the weights of ...

ESPN has partnered with IBM to use Watson to predict NFL player performance. I took at the underlying data and it includes a score distribution, which Watson indicates is a Fisk distribution. The ...

I'm just currently getting into the world of Bayesian machine learning, with a lot of frequentest stats background, and I frequently find myself limited in my modeling by my lack of knowledge about ...

This is an obfuscated version of a real problem: Each day I speak with some number of (distinct) girls. I compute the Jaccard similarity index between two consecutive days: $$ J(t)=J(t,t-1)=\...

Analyzing an an enterprise environment where large number of Windows end user devices have low disk space. To identify causes of the low disk spaces I ran scripts which provides the total size for ...

Let $f(x;\theta)$ be probability density function. Suppose that this pdf contains a proportionality constant $c$, so that $f(x;\theta) = c\cdot g(x;\theta)$, where $g$ is an integrable function. Is ...

Generally, I normalize variables using standard normal variates or (x-xmin)/(xmax-xmin)But this only works well for variables that are not truncated, for example ...

I apologize if I do not know the proper terminology, I will do my best to describe the problem. I have a computer model that gives me a set of 1000 possible outcomes for a single event. Currently, I ...

Quoting: As can be seen in Table 1, the experimental and control group distribution were sufficiently normal for the purpose of conducting a $t$-test (skewness $<|2.0|$ , kurtosis $<|9.0|...

I am very new to this subject, but i've tried searching other questions and haven't found what i need. I'm working on an application that will be promoted in physical meetings. Lets say, for example, ...

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