**TI82** TxtView file generated by CalcText - Kouri6 'ZPROB'%ÿPreuve probaPreuve: E(X+b)=E(X)+b et E(aX)=aE(X) Soit b E R.On a E(X)=p1x1+p2x2+...+pnxn E(X+b)=p1(x1+b)+p2(x2+b)+...+pn(x+b) =p1p2+p1b+p2x2+p2b+...+pnxn+pnb =p1x1+p2x2+...+pnxn+p1b+p2b+...+pnb =E(X)+b*1=E(X)+b Soit a E R,a=!0. E(aX)=p1(ax1)+p2(ax2)+...+pn(axn) =a(p1x1+p2x2+...+pnxn) =aE(X) Preuve V(aX)=a^2*V(X) et V(X)=(i=nSi=1) Soit a E R,V(aX)=(ax1-E(aX))2*p1+(ax2-E(aX))2*p2+...+(axn-E(aX))2*pn =(ax1-aE(X))2*p1+(ax2-aE(X))2*p2+...+(axn-aE(X))2*pn =(a(x1-E(X)))2*p1+(a(x2-E(X)))2*p2+...+(a(xn-E(X)))2*pn =a2(x1-E(X))2*p1+a2(x2-E(X))2*p2+...+a2(xn-E(X))2*pn =a2((x1-E(X))2*p1+(x2-E(X))2*p2+...+(xn-E(X))2*pn) =a2*V(X) V(X)=(x1-E(X))2*p1+(x2-E(X))2*p2+...+(xn-E(X))2*pn =(x1^2-2*x1*E(X)+E^2(X))*p1+(x2^2-2*x2*E(X)+E^2(X))*p2+...+(xn^2-2*xn*E(X)+E^2(X))*pn =x1^2*p1-2x1E(X)p1+E^2(X)p1+x2^2*p2-2x2E(X)p2+E^2(X)p2+...+xn^2*pn-2xnE(X)pn+E^2(X)pn =p1x1^2+p2x2^2+...+pnxn^2-2E(X)(p1x1+p2x2+...+pnxn)+E^2(X)(p1+p2+...+pn) =p1x1^2+p2x2^2+...+pnxn^2-2E(X)*E(X)+E^2(X)*1 =p1x1^2+p2x2^2+...+pnxn^2-E^2(X) =(i=nSi=1)pixi^2-E^2(X)ÿË