By William Geraint V. Rosser
Advent to Statistical Physics (Mathematics and Its functions) [Paperback
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Extra resources for An introduction to statistical physics (1982)
Then HP ||M (Q) ≤ HP ||M (R) and HP (Q) ≤ HP (R) Comments: The lemma can also be stated in terms of random variables and mappings in an intuitive way: Suppose that U is a random variable with finite alphabet A and f : A → B is a mapping from A into another finite alphabet B. Then the composite random variable f (U ) defined by f (U )(ω) = f (U (ω)) is also a finite random variable. If U induces a partition R and f (U ) a partition Q, then Q < R (since knowing the value of U implies the value of f (U )).
1, which proves the first inequality. 2). ✷ The next result provides useful inequalities for entropy considered as a function of the underlying distribution. In particular, it shows that entropy is a concave (or convex ) function of the underlying distribution. Define the binary entropy function (the entropy of a binary random variable with probability mass function (λ, 1 − λ)) by h2 (λ) = −λ ln λ − (1 − λ) ln(1 − λ). 4: Let m and p denote two distributions for a discrete alphabet random variable X and let λ ∈ (0, 1).
Since the qn (f ) are nondecreasing, so are the Em (qn (f )) and this sequence must either converge to a finite limit or grow without bound, in which case we say it converges to ∞. In both cases the expectation Em f is well defined, although it may be infinite. 22) provided this does not have the form +∞ − ∞, in which case the expectation does not exist. It can be shown that the expectation can also be evaluated for nonnegative measurements by the formula Em f = sup Em g. 6. EXPECTATION 13 The expectation is also called an integral and is denoted by any of the following: Em f = f dm = f (x)dm(x) = f (x)m(dx).
An introduction to statistical physics (1982) by William Geraint V. Rosser