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Unformatted text preview: IMPORTANT IDEAS FROM CHAPTER 2 Expected Values J
1. Let X be, a random variable, g a function from the real line to the real line. X discrete: E(g(X)) : Zacg(:t)p(w) where the sum is over the support of X
(provided the sum converges absolutely) X continuous: E( (g(X)) —f_oo 9(3) fx (2:) )da: (provided the integral converges
absolutely) 2. Properties of expectation: a) If P(X:c) = 1, then E(X) = c; b) If 9(3):) 5 h(:t:) for all x, then E(g(X)) s E(h(X)).
6) E(ag(X) + WXD = aE(9(X)) + bE(h(X)) 3. The km moment of a random variable X is E(Xk). The variance is
deﬁned as the second moment of X — 1a,, where ,u is the mean of X. FACT: If the It“ moment of X is ﬁnite, then the moments of order k; — 1,11: —
2,. . . ,1 are all ﬁnite. 4. Let X be a r.v. with c.d.f FX. The moment generating function (m.g.f.)
of X is deﬁned by M X (t) = E(etx), provided that this expectation exists for all t in a neigh
borhood of zero. a) If X has m.f.g. MX(t), then E(X“) = MENU), the nth derivative of the
m.g.f. evaluated at t=0. b) If X and Y are two r.v.’s with rn.g.f.’s Mx(t) and My(t), respectively,
and if M X(t) = My(t) for all t in a neighborhood of zero, then X and Y have
the same distribution. c) (For later use). Suppose X1, 73: 1,2, are r.v.’s with m.g.f.’s MX, (t),z‘ =
1,2, and c.d.f.’s in. Suppose lime,(t) = M(t) as 1' —> 00, for all t in a
neighborhood of 0. If M (t) is the m.g.f. for the r.v. X, then limFX,(m) = FX(:r)
at all x where FX is continuous. ST 561: Important topics from Chapter III Discrete mulitvariate random variables Let X1, X2, . . .,Xk be discrete. The joint proba
bility mass function of X1, . . . , X k is given by fx($1,...,$k):P(X1=31,...,Xk age). The marginal probability function of X1 is given by summing out all variables except an; in the
joint p.m.f.: not) = Zf(x1,...,xi). where the sum is over 321, . . . , $i_1, act+1, . . .,a‘n. The conditional p.m.f. of, say, X2, . . .,Xk given X1
is given as f($2,,93k[ﬂ=1)= f($1,$2,,$k)/f1($1)‘
If h: Rk —> R1, the expected value of h(X1, . . .,Xk) is given by E(h(X1,X2, . . .,Xk) = Zh($1,...,$k)fx(ﬂi1,.. .,:Ek) where the sum is over all variables $1, . . ”ask. The conditional expectation of, say, h(X2), given
X3 = 931, is given by E(h(X2)lX1 = $1) = Zhlxzilflhlﬁl), when X1 takes the value (1:1. The random variable E[h(X2)X1] is the random variable taking the
value E(h(X2)X1 : 3:1) Whenever X1 = x1. Useful facts here are that E(E(h(X2)lX1)) = E(h(X2))a Vadthﬂ) = ElVGT(h(X2)X1)) + VGT(E(h(X2)X1))v X1, . . .,Xk are mutually independent iif fx(:c1, . . . , m) = Hf=1fi(xi). When this is satisﬁed,
Elh1(X1)   hk(Xkll I HlLiElhilXill In particular, the moment generating function of the sum of independent random variables is the
product of the individual m.g.f.’s, One of the most important examples of a discrete multivariate random variable is the mnltinomial (n, pl, . distribution, in which P(X1=$1,...,Xk LE 2W
k) $81!...wklp1 where p; +. . .+pk = 1 and $1 +. . .+a:k : n. The marginal distribution of X5, here is binomial{n,pi).
Continuous random variables If X1, . . . ,Xk are continuous random variables, their joint distribution is described by the joint probability density function f(a:1,:c2, . ﬂak), which is non—
negative and integrates to 1 over R’“. For a set A in Rk, ”51%) P((X1,...,Xk) e A) =[4f(:c1,...,2:k)d$1...dosk, where the integral is k—dimensional. For continuous random variables, the formal deﬁnitions of the
marginal and conditional p.d.f.’s , expectation, conditional expectation, etc. are exactly the same
as in the discrete case, with integrals replacing sums. The deﬁnition of independence is exactly the
same. When k:2, an important example is the bivariate normal distribution, with joint p.d.f. 1 $1HM12W $1—I512ﬂ72yM22 m12".”22
2...... )2p( )( )+( H. 0'1 0'1 0'2 0'2 f(3311$2): exammm where 00 < #1,].L2 < 00, 0 < (71, 0 < oz, and —1 S p S 1. In this model, both the marginal and
conditional distributions are normal, and X1, X2 are independent iff p = 0. Covariance and correlation: The covariance of random variables X1 and X2 is given by
000(X11X2) : E{(X1 r“ #1)(X2 — #2)},
where “1, #2 are the means of X1, X2 respectively. Then
Var(X1 + X2) = Var(X1) + Var(X2) + 2000(X1, X2),
so the variances are additive if X1,X2 are independent. The correlation of X1, X2 is given by Cov(X1, X2) COT‘T(X1,X2) = 0102 where 01, 02 are the standard deviations of X1,X2 respectively. The correlation p must satisfy _1 g P S 1:
and if X1,X2 are independent, their correlation is zero. In the mnltinomial(n,p1, . . . ,pk) model,
the correlation of X, and X5 is given by —{(1gp€33(’1'_pj)}1/2. In the bivariate normal model, the parameter p is the correlation of X1,X2. Inequalities: Markov’s inequality: For X 2 0, P(X > t) 5 E(X)/t. Special cases of this
include Chebyshev’s inequality: Pox — m > t) s Var(X)/t2.
where ,u is the expected value of X. The CauchySchwarz inequality states that if X1, X2 have ﬁnite variances,
E(lX1llX2) S x/E(X12)E(X§) J ensen’s inequality states that if g is a convex function on R1, then Elng)) 2 9(E(X)) ...
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