16.322 Stochastic Estimation and Control Professor Vander Velde P(system failure)=l-P(system works)=1-P(WW.W)=1-I P(=1-l (1-P(FD) where P()=1-P(E) Example: 11-bit message pe- bit error probability A system may be able to tolerate up to k independent errors and still decode the message. Compute the probability of k errors. 口囗..口 P(k errors) 42(1-P)2k Conditional Independence Sometimes events depend on more than one random quantity, some of which may be common to two or more events, the others different and independent. In such cases the concept of conditional independence is useful Example: Two measurements of the sane quantity, x, each corrupted by additive independent noise m,=x+n, m, =x+n2 (m1),E2(m2) Since mi and m2 both depend on the same quantity, x, any events which depend on m1 and m2, E1(m1)and E2(m2), are clearly not independent P(EE2)+P(EP(E2)in general P(AB E)=P(AEP(BE) P(B)= P(B)if A, B are independent P(B|EA=(BE)=(EP81E)=P(4B)BE=P(B|E) P(EA) P(EP(AE) P(AJE)16.322 Stochastic Estimation and Control Professor Vander Velde Lecture 2 1 2 (system failure) 1 (system works) 1 ( ... ) 1 ( ) 1 (1 ( )) P P P WW W P W P F = − = − = −Π = −Π − n ic i i where ( )1 () PW P F i i = − . Example: n-bit message e p - bit error probability A system may be able to tolerate up to k independent errors and still decode the message. Compute the probability of k errors. e e ￾￾ ￾￾ ... ( errors) (1 ) k nk e e n Pk p p k − ⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠ Conditional Independence Sometimes events depend on more than one random quantity, some of which may be common to two or more events, the others different and independent. In such cases the concept of conditional independence is useful. Example: Two measurements of the same quantity, x, each corrupted by additive independent noise. 1 1 2 2 11 2 2 ( ), ( ) m xn m xn Em Em = + = + Since m1 and m2 both depend on the same quantity, x, any events which depend on m1 and m2, E1(m1) and E2(m2), are clearly not independent. 12 1 2 P EE P E P E ( ) ( )( ) ≠ in general i n are independent ( | ) ( | )( | ) ( | ) ( ) if A,B are independent ( ) ()( | ) ( | )( | ) (| ) (|) ( ) ()( | ) ( | ) P AB E P A E P B E PB A PB P ABE P E P AB E P A E P B E P B EA P B E P EA P E P A E P A E = ⇒ == = =