Proof of function of (multiple) random var.

Proof of function of (multiple) random var. Single: p 151-153 Sum of ind. Poisson: p 157,158 Sum of ind. Normal: p 160,161 Products of r.v: p165,166

if X is N(m, s)  Y is N(am, as) if X is LN(l, z)  Y is LN(lna + l, z) if X is N(m, s)  Y is N(am + b, as) if X1 and X2 are Poisson with mean rates n1 and n2 respectively  Z is Poisson with nz = n1 + n2 (see E4.5 on p. 175) Summary of Common Results

if X1 and X2 are N(m1, s1) and N(m2, s2) respectively  Z is N(mz, sz) where if X1 and X2 are s.i. r12 = 0 if Xi = N(mi, si) ; i = 1 to n  Z is N(mz, sz) where Summary of Common Results (Cont’d)

Y X1 X2 X3 X4 Normal Y or lnS is normal S is lognormal LN(lS,zS)

Formal solution of Ex 3.41 (c) • Cylinder concrete strength competition • Boy: Y~ N (80.8, 30) ; Girl: X~ N (80, 20) • Who is more likely to win?

Solution • Construct Z = Y – X • If Z > 0, boy wins; If Z < 0, girl wins • Z = N (80.8 – 80, (302 + 202)0.5 ) = N (0.8 , 36.05)

Moments of function of R.V. • E(X): • E(cX)=cE(X) • E(X+Y)=E(X)+E(Y) • E(XY)=E(X)E(Y)(X,Y are s.i) • var(X): • var(X)=E(X2)-E2(X) • var(cX)=c2var(X) • Var(X+Y)=var(X)+var(Y) (X,Y are s.i)

Proof of function of (multiple) random var.