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Results tagged with pr.probability
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user 75985
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
2
votes
2
answers
310
views
How does pairwise independence restrict dependence to a third variable?
Let $X,Y,Z$ be random variables such that
$X,Y,Z$ have mean $0$ and variance $1$.
$X$ and $Y$ are pairwise independent.
$Z$ can be arbitrarily dependent on $X$ and $Y$.
My question is: What can we s …
- 73
0
votes
0
answers
64
views
"Marginal" Three-Series Theorem?
Let $\{X_t\}_{t=1}^\infty$ be a sequence of random variables and define the partial sums by $S_t=\sum_{s=1}^t X_s$.
Kolmogorov's three-series theorem states that
Theorem (Davidson, 2021, Theorem 21.8 …
- 73
0
votes
0
answers
75
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Parameter derivative of the confluent hypergeometric function
Let $X$ follow a normal distribution with mean $\mu$ and variance $\sigma^2$. Mathematica gives
$$
E[\log|1+X|]=\frac{1}{2}\biggl(-\gamma-\log 2+2\log\sigma-\frac{\partial}{\partial a}{}_1 F_1\biggl(0 …
- 73