Lecture 2: Random variables in Banach spaces
... Given ε > 0 we can find a compact set K ⊆ E with µ(K) = P{S ∈ K} > 1 − ε. The set L := 12 (K − K) is compact as well, and arguing as in the proof of Lemma 2.5 we have P{Sn 6∈ L} 6 P{S 6∈ K} + P{S − 2Sn 6∈ K} = 2P{S 6∈ K} < 2ε. It follows that P{Sn ∈ L} > 1 − 2ε for all n > 1, and therefore the seque ...
... Given ε > 0 we can find a compact set K ⊆ E with µ(K) = P{S ∈ K} > 1 − ε. The set L := 12 (K − K) is compact as well, and arguing as in the proof of Lemma 2.5 we have P{Sn 6∈ L} 6 P{S 6∈ K} + P{S − 2Sn 6∈ K} = 2P{S 6∈ K} < 2ε. It follows that P{Sn ∈ L} > 1 − 2ε for all n > 1, and therefore the seque ...
Useful Probability Distributions
... X is the number of successes p is the probability of success in any individual trial, and q=1-p P(X=x), x=0,1..,n is denoted pi ⎛n⎞ Pr( X = x) = ⎜⎜ ⎟⎟ p x (1 − p ) n − x ⎝ x⎠ ⎛n⎞ = ⎜⎜ ⎟⎟ p x q n − x ⎝ x⎠ ...
... X is the number of successes p is the probability of success in any individual trial, and q=1-p P(X=x), x=0,1..,n is denoted pi ⎛n⎞ Pr( X = x) = ⎜⎜ ⎟⎟ p x (1 − p ) n − x ⎝ x⎠ ⎛n⎞ = ⎜⎜ ⎟⎟ p x q n − x ⎝ x⎠ ...
Chapter 10 – Statistical Estimation
... Not making this distinction is the primary reason for mistakes in this material.) Statistics are themselves random variables. Their exact value depends on the sample we have randomly chosen from the population. There is not “a sample mean” in the same sense as there is “a population mean” -- we will ...
... Not making this distinction is the primary reason for mistakes in this material.) Statistics are themselves random variables. Their exact value depends on the sample we have randomly chosen from the population. There is not “a sample mean” in the same sense as there is “a population mean” -- we will ...
Central limit theorem
In probability theory, the central limit theorem (CLT) states that, given certain conditions, the arithmetic mean of a sufficiently large number of iterates of independent random variables, each with a well-defined expected value and well-defined variance, will be approximately normally distributed, regardless of the underlying distribution. That is, suppose that a sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic average of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the computed values of the average will be distributed according to the normal distribution (commonly known as a ""bell curve"").The central limit theorem has a number of variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations, given that they comply with certain conditions.In more general probability theory, a central limit theorem is any of a set of weak-convergence theorems. They all express the fact that a sum of many independent and identically distributed (i.i.d.) random variables, or alternatively, random variables with specific types of dependence, will tend to be distributed according to one of a small set of attractor distributions. When the variance of the i.i.d. variables is finite, the attractor distribution is the normal distribution. In contrast, the sum of a number of i.i.d. random variables with power law tail distributions decreasing as |x|−α−1 where 0 < α < 2 (and therefore having infinite variance) will tend to an alpha-stable distribution with stability parameter (or index of stability) of α as the number of variables grows.