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1[Aug 27]Tutorial [Ariel Kleiner].pdf
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详细说明：在不久的将来，深度学习将会取得更多的成功，因为它需要很少的手工工程，它可以很容易受益于可用计算能力和数据量的增加。目前正在为深度神经网络开发的新的学习算法和架构只会加速这一进程。Probability: Foundations
A probability space(Q2, F, P)consists of
a set S2 of"possible outcomes
a set of events. which are subsets of n2
a probability measure P: F[0, 1] which assigns
probabilities to events in F
EXample: Rolling a Die
Consider rolling a fair Six-sided die. In this case
2={1,2,3,4,5,6}
F={0,{1},{2},…{1,2},{1,3},,}
P(0)=0,P({1})=,P({3,6})=,
Actually, F is a o-field. See Durrett's Probability: Theory and examples
for thorough coverage of the measure-theoretic basis for probability theory
Probability: Random Variables
A random variable is an assignment of (often numeric)
values to outcomes inΩ.
o For a set a in the range of a random variable x, the
induced probability that X falls in A is written as P(XE A
Example Continued: Rolling a Die
Suppose that we bet $5 that our die roll will yield a 2. Let
X: 1, 2, 3, 4, 5,6>5,5 be a random variable denoting
our winnings: X=5 if the die shows 2, and X=5 if not
Furthermore,
P(X∈{5})= - and P(X∈{-5})
6
Probability: Common Discrete Distributions
Common discrete distributions for a random variable x:
Bernau(p):p∈0,1];X∈{0,1}
P(X=1)=p,P(X=0)=1-p
Binomial(p,m):p∈[0,1],n∈N;X∈{0,,,n}
P(X=x)=()p(1-p)
o The multinomial distribution generalizes the Bernoulli and
the Binomial beyond binary outcomes for individual
experiments
Poisson():入∈(0,∞);X∈N
P(X=X)
e-X入
Probability: More on Random Variables
o Notation: X P means"X has the distribution given by P
o The cumulative distribution function(cdf) of a random
variable x∈ IRm is defined for x∈ IRm as f(X)=P(X≤X)
We say that X has a density function p if we can write
P(X≤x)=p()吵y
In practice, the continuous random variables with which we
will work will have densities
o For convenience in the remainder of this lecture we will
assume that all random variables take values in some
countable numeric set, R, or a real vector space
Probability: Common Continuous Distributions
Common continuous distributions for a random variable x:
Uniform(a,b):a,b∈R,a
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