Questions tagged [probability-distributions]
Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions. Use this tag along with the tags (probability), (probability-theory) or (statistics).
446 questions from the last 365 days
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1
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38
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Predicting $Y$ from a correlated variable $X$
I was reading something. The context was we could measure the variables $X$ and $Y$ on individuals. And it appeared that $X$ and $Y$ were correlated with correlation: $\rho=0.3$.
The writer then ...
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-6
votes
1
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60
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Is this a probability density function? [closed]
Given $fx(x) = \{ \frac{1}{\pi} \; \text{for} \; x_1 + x_2 \le 1$
I am required to state if the function represents a density function and prove why. I know that to prove it I must check that $f(x) \...
3
votes
1
answer
70
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Conditional probability for linear combinations of independent exponentials
I am working on the following exercise.
Let
$$X_1 \sim \mathrm{Exp}\left(\tfrac12\right), \qquad
X_2 \sim \mathrm{Exp}\left(\tfrac12\right),$$
independent. Define
$$Y_1 = X_1 + 2X_2, \qquad Y_2 = 2X_1 ...
- 377
6
votes
1
answer
136
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Known properties of these generalized Cauchy distributions
Consider the following family of normalized probability densities parametrized by the strictly positive integer $k$:
$$
\begin{align}
\begin{aligned}
&f_k(x) = \frac{k}{\pi}\sin\left(\frac{\pi}{2k}...
- 619
0
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2
answers
67
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Finding shape and scale parameters of Gamma distribution given a confidence interval
This would have been a comment on Munki's question about the same thing but I just created my account so I don't have enough rep.
Suppose I have a confidence interval $(u, l)$ with respect to some ...
4
votes
1
answer
188
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Sum of two independent chi-squared random variables
Suppose $Z_1,\dots,Z_m$ are $m$ independent standard normal random variables. Also, $W_1,\dots,W_n$ are $n$ independent standard normal random variables. Define $X = \sum_{i=1}^{m}Z_i^2$ and $Y = \...
- 117
0
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1
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51
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Can mutual information be defined between a random variable and a conditional distribution
Quantities like mutual Information $I$, entropy $H$,etc. are typically defined as taking random variables as input. However, they are actually just functions on probability distributions - e.g. the ...
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6
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1
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146
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What is the probability distribution for the distance between two points placed randomly inside circles of different radii?
It is known that the probability density function for the distance, $s$, between two points located uniformly randomly inside a circle of radius $R$ is given by:
$$
f(s)=\frac{4s}{\pi R^{2}}cos^{-1}\...
- 571
0
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0
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57
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How to measure the divergence of a random geometric series (relative to its expected asymptotic growth)?
Suppose that $(a_i)_{i=0}^\infty$ is an iid sequence of positive real numbers. Assume for regularity that there is a constant $a > 1$ such that $P(1/a < a_i < a) = 1$ for all $i$, and ...
- 7,626
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0
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63
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Definition of conditional random variables [duplicate]
Let $X$ and $Y$ be two random variables. Then, define $X\mid\{Y = y\}$ as the random variable that takes outcomes from a subset of the sample space defined by the event $\{Y=y\}$. Assume further that $...
- 21
2
votes
2
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128
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Intuitive Explanation for Convergence in Probability and Convergence in Distribution
Having a bit of trouble with the definitions for convergence in probability and convergence in distribution for random variables. The textbook (Degroot) defines each as follows:
Convergence in ...
- 21
0
votes
0
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85
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How to derive the distribution of a 2D random walk?
A few years prior, an acquaintance of mine tackled a problem inspired by something in our statistics class, which basically was the idea of "what is the expected distance from the starting point ...
- 3,692
0
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0
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28
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$Y_n$ is Cauchy distributed if $S_n/n \sim Y_1$ and $Y_i$ are symmetric [duplicate]
A problem from Le Gall's Measure Theory, Probability and Stochastic Processes (Chapter 9, Exercise 9.11(4)), which I'm not really sure what it is asking:
Let $(Y_n)$ be a sequence of i.i.d. real ...
- 1,598
3
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0
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57
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References for a probability law [closed]
Consider a symmetric simple random walk starting at $0$ and denote by $p_{n,k}$ the probability the walk occupes $k$ at time $n$. Denote also $q_n=\left(q_{n,k}\right)_{k\geq 0}$ defined by
$$
q_{n,k}=...
- 31
0
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0
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56
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Can we have a random variable with mixed joint distribution resulting in a singular and a non-singular marginal distribution?
This question may be a little trivial, but I was wondering if we can construct a bivariate (or multivariate) probability distribution function in a way that we have a mix of a singular and an ...
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