Questions tagged [distributions]
A distribution is a mathematical description of probabilities or frequencies.
9,756 questions
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Conditioning on X to obtain distribution of parameter estimators in simple linear regression [duplicate]
Consider the simple linear regression model with the following assumptions:
I am trying to verify that $\dfrac{\hat{B}_1 - B_1}{\sigma / \sqrt{\sum_{i=1}^n (X_i - \bar{X})^2}}
\;\Big|\; X_1,\ldots,...
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Example of Distribution which is a Base Distribution for the Proportional Hazard family as well as Reversed Hazard rate family of Distribution
Let $X$ and $Y$ be two random variables one belongs to proportional hazard rate family and another is proportional reverse hazard rate family of distribution. Now my concern is whether there exists ...
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How to improve my understanding and intuition of Graphical Models
I am currently taking a course on graphical models. When understanding lecture material and attempting questions, I often find myself stuck as I cannot clearly see the link between a graph and the ...
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Chi-squared random variable
Given a random variable $X$ which is $\chi^2_{n}$, can I define $n$ independent standard normal random variables $Z_{1,...,n}$ on the probability space such that $X = Z_1^2 + Z_2^2 + ... + Z_n^2$ ...
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Why isn't this sum normal?
Define $A, B \overset{iid}{\sim} N(0, 1)$, and define $X=\vert A\vert$ and $Y = -\vert B\vert$. This answer on Math.SE shows why $X+Y$ is not Gaussian.
Huh? The $X$ and $Y$ cut a Gaussian in half, and ...
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How can negative log likelihood be properly compared between two sets with different sample sizes?
I have a dataset that I have divided into training and testing data, with approximately 160 samples in the training set and 40 in the testing set. I fitted a probability distribution to each dataset ...
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How to find confidence intervals for binary outcome probability?
I'm doing stats for medical chart review research. The binary outcomes vary in probability from less than 0.05 to greater than 0.5 depending on risk factors. For relatively more common outcomes like ...
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Density Estimation?
I'm not sure if the following question of mine sound silly. I thought I would just go ahead and ask. The question is the following. We often find in probability text books questions, for example, of ...
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The Expectation of the Difference of Sample Quantile and Population Quantile
Suppose a distribution function $F(\cdot)$ is continuous. For some $\tau \in (0, 1)$, the $\tau$th quantile is defined as
$$
Q_\tau = \inf \{ x : F(x) \ge \tau \}.
$$
For an i.i.d. sample $X_1, \dots, ...
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What is the statistics term for overlapping populations
Given this image from another question:
I have two populations, from which some software has given me the x̄ and s. I want to quantify the overlap, preferably with an equation or formula that can be ...
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Teaching Poisson approximation to Binomial still relevant?
Convergence of Binomial to Poisson:
If $X_n\sim \text{Bin}(m_n,p_n)$ and if $m_n\to\infty$ and $p_n\to 0$ such that $m_np_n\to\lambda$, then $X_n\stackrel{d}{\to}\text{Poi}(\lambda)$
The above result ...
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What is distribution of gaps between activated sites in a Bernoulli sequence?
Consider a sequence of $n$ independent Bernoulli random variables $X_1,\dots,X_n$, where each $X_i=1$ (site is on) with probability $p_i$, and $X_i=0$ (site is off) with probability $1-p_i$.
After ...
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Asymptotic Distribution of Weighted Empirical Distribution
Suppose that we observe an i.i.d. sample $(X_1, Y_i), ..., (X_n, Y_n)$ from $(X, Y)$. We assume that $X_i$ is bounded by $B$ and $E(X) = 0$. For some $\tau \in (0, 1)$, define the $\tau$th quantile of ...
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Intuition behind measure used in Steven Frank's "The common patterns of nature"
I have been working my way through The common patterns of nature by Steven Frank. I'm confused by the measure $m_y$ which he introduces in the section "The binomial distribution".
...[it] ...
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Why do we need the small-$\beta_t$ ( ≒ Gaussian $q(x_{t-1}\mid x_t)$ ) assumption in diffusion models
In diffusion models, the forward process is
$q(x_t \mid x_{t-1}) = \mathcal{N}\big(x_{t};\sqrt{1-\beta_t} x_{t-1},\beta_t I\big)$, and the reverse model is parameterized as
$p_\theta(x_{t-1}\mid x_t)=\...
