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Unformatted text preview: IMPORTANT IDEAS FROM CHAPTER 1 Probability Functions 1. For our purposes, a probability function P(.) is deﬁned on the class of all
subsets of a sample space 8. A probability function must satisfy: a) P(A) 2 O for all events A (an event is a subset of S); b) P(S) = 1; c) If A1,A2, . . ., are pairwise disjoint, then P(UE?_,1A,) = P(Ai).
2. A probability function has certain properties: a) P(S) = 1; b) 0 S P(A) S 1; onmﬂ—Ho{ d) P(A U B) = P(A) + P(B) — P(A n B); e) IfA C B then P(A) 5 P(B); f) If {Agh);1 is a partition of 8, then P(B) — °° P(B 0 Ad). i=1 Conditional probability and independence 1. P(AB) is the “conditional probability of A, given B” and equals
P(An B)/P(B), if P(B) > 0. _ 2. Bayes Rule: If {Ai} is a (ﬁnite or inﬁnite) partition of S and P(B) > 0,
then ‘
P(BlAi)P(Ai) me=enwawnr 3. Events A and B are independent iﬂ’ P(A n B) = P(A)P(B). Events
{A1, . . . , An} are independent ifgiven any subset {2451... .,A,,} of {A1, . . . , An}, magnetJ.) = Hg=1P(Azj). Equally liker spaces and counting rules 1. For a ﬁnite sample space S in which each element has probability 1/8f,
the probability of any event A is just IAf/ IS 2. Counting principle: If a job consists of p separate tasks and the it“ task
can be done in a; ways, the number of ways of doing the entire job is 11’? Int. 1% 3. Two fundamental distinctions for counting problems:
a) Is sampling with or without replacement? b) Is order relevant? 4. The number of ordered samples (permutations) of r objects selected with
out replacement from 11 objects is (n), = n(n — 1) . . . (n — r + 1). 5. The number of unordered samples (combinations) of r objects selected
without replacement from 11 objects is = Random variables 1. A random variable is a function from the sample space S to the real line. 2. Every random variable X'has a cumulative distribution function Fx (t) =
P{s68 : X (s) S t}. This is a rightcontinuous, nondecreasing function deﬁned
on the entire real line, with the properties that lim Fx(t) :0 est—i —oo, lim FX(t) = 1 ast—) co. 3. If S is ﬁnite or counts.ny inﬁnite, all random variables deﬁned on 8 are
discrete. The probability distribution of a discrete random variable is deﬁned
by its probability mass function p: p(m) = P(X = 2:) for any x in the range of X (the range of X is the support of
the distribution.) Important examples include the binomial(n,p) distribution: p(m) = pm(1—
p)"""", w = 0,1, . . ., n, the Poisson distribution: p(a:) = EMF/ml, a = 0, 1,2, . . .,
and the geometric distribution, p(:r) = p(1  p)‘”‘1, :r = 1,2, . . .. For a discrete random variable, the c.d.f and the p.d.f. are related by FX (t) =
29:.“ p(a:), where the sum is taken over all x in the support of X. 4. If S is uncountable, random variables deﬁned on 8 may be continuous.
For these variables, the c.d.f. is continuous and often diiferentiable everywhere.
The distribution of such a random variable X is described by its probability den
sity function, or p.d.f., fx(a:), with the properties: a) fx(m) 2 0 for all x (by convention, the p.d.f. is assumed deﬁned for all
x, and {x : fx(m) > 0} is called the support of X); foob) Ii (3 < b, P(a S X 5 b) is deﬁned to be I: fx(a:)d.1:. In particular,
)5»: a: do; = 1.
~00 The c.d.f. and p.d.f. for these variables are related by 00 ﬁg (3:)ds = F); (t),
or equivalently by (d/dt)FX(t) = fx(t). Note that for a continuous random variable X with a p.d.f. fx (.), we have
P(X = t) = 0 for any t, even though fx (t) may be > 0. Important examples include the Unifom(n, b) distribution, with p.d.f. f (ac) =
ﬁ—al(a,b)(n:); the Namelm, 0)) distribution, with p.d.f. f(3) = ﬁe—(“wwzaz' F
(3—1 e—mfﬁ and the Gommo(a,[3) distribution, with p.d.f. f(:t) = Whomﬁm). ...
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