6.041 / 6.431 7. Multiple Discrete Random Variables

LECTURE 7 Review • Readings: Finish Chapter 2 pX(x)= P(X = x) Lecture outline pX,Y (x, y)= P(X = x, Y = y) • Multiple random variables pX|Y (x | y)= P(X = x | Y = y) – Joint PMF – Conditioning – Independence pX(x)= ! pX,Y (x, y) • More on expectations y • Binomial distribution revisited pX,Y (x, y)= pX(x)pY |X(y | x) • A hat problem Independent random variables Expectations pX,Y,Z(x, y, z)= pX(x)pY X(y | x)pZX,Y (z | x, y) E[X]= ! xpX(x)||x E[g(X, Y )] = !! g(x, y)pX,Y (x, y) • Random variables X, Y , Z are x yindependent if: pX,Y,Z(x, y, z)= pX(x) · pY (y) · pZ(z) • In general: E[g(X, Y )] =#g"E[X], E[Y ]# for all x, y, z y 1 2 3 4 x 1/20 4/202/20 2/202/20 3/20 2/201/20 1/20 1/20 1/20 • E[!X + "]= !E[X]+ " 4 • E[X + Y + Z]= E[X]+ E[Y ]+ E[Z] 3 2 • If X, Y are independent: 1 – E[XY ]= E[X]E[Y ] – E[g(X)h(Y )] = E[g(X)] E[h(Y )]· • Independent? • What if we condition on X ! 2 and Y " 3? Variances • Var(aX)= a2Var(X) • Var(X + a) = Var(X) • Let Z = X + Y . If X, Y are independent: Var(X + Y ) = Var(X) + Var(Y ) • Examples: – If X = Y , Var(X + Y )= – If X = −Y , Var(X + Y )= – If X, Y indep., and Z = X − 3Y , Var(Z)= Binomial mean and variance • X = # of successes in n independent trials – probability of success p E[X]= n! k=0 k"n k # p k(1 − p)n−k • Xi = $% & 1, if success in trial i, 0, otherwise • E[Xi]= • E[X]= • Var(Xi)= • Var(X)= The hat problem • n people throw their hats in a box and then pick one at random. – X: number of people who get their own hat – Find E[X] Xi = $% & 1, if i selects own hat 0, otherwise. • X = X1 + X2 + ···+ Xn • P(Xi = 1) = • E[Xi]= • Are the Xi independent? • E[X]= Variance in the hat problem • Var(X)= E[X2] − (E[X])2 = E[X2] − 1 X2 =! i X2 i +! i,j:i#=j XiXj • E[X2 i ]= P(X1X2 = 1) = P(X1 = 1)·P(X2 =1 | X1 = 1) = • E[X2]= • Var(X)= MIT OpenCourseWare http://ocw.mit.edu 6.041 /6.431 Probabilistic Systems Analysis and Applied Probability Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.