Contents
- 1 How to calculate the product of probability density functions?
- 2 When is the joint probability density function more complicated?
- 3 Why is the Gaussian density function so common?
- 4 What is the Church-Rosser property of the lambda calculus?
- 5 How to define the set of free variables in lambda calculus?
How to calculate the product of probability density functions?
That gives us two probability densities r f ( r x) and r g ( r x) whose product is a probability density. One simple construction is to let f ( x) = g ( x) = k > 1 on the shared interval [ 0, 1 k 2], and then let f ( x), g ( x) have disjoint support otherwise.
When is the joint probability density function more complicated?
The joint probability density function for two independent Gaussian variables is just the product of two univariate probability density functions. When the data are correlated (say, with mean 〈 d 〉 and covariance [cov d ]), the joint probability density function is more complicated, since it must express the degree of correlation.
Is the product of random variables a probability distribution?
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions.
Why is the Gaussian density function so common?
This probability density function has mean 〈 d 〉 and variance σ2 ( Fig. 2.12 ). The Gaussian probability density function is so common because it is the limiting probability density function for the sum of random variables.
What is the Church-Rosser property of the lambda calculus?
Parallelism and concurrency. The Church–Rosser property of the lambda calculus means that evaluation (β-reduction) can be carried out in any order, even in parallel. This means that various nondeterministic evaluation strategies are relevant. However, the lambda calculus does not offer any explicit constructs for parallelism.
When was the untyped lambda calculus first published?
Subsequently, in 1936 Church isolated and published just the portion relevant to computation, what is now called the untyped lambda calculus. In 1940, he also introduced a computationally weaker, but logically consistent system, known as the simply typed lambda calculus.
How to define the set of free variables in lambda calculus?
The set of free variables of a lambda expression, M, is denoted as FV ( M) and is defined by recursion on the structure of the terms, as follows: 1 FV ( x) = { x }, where x is a variable. 2 FV (λ x. M) = FV ( M) \\ { x }. 3 FV ( M N) = FV ( M) ∪ FV ( N ).