6.041 / 6.431 6. Discrete Random Variable Examples & Joint PMFs

LECTURE 6 • Readings: Sections 2.4-2.6 Lecture outline • Review: PMF, expectation, variance • Conditional PMF • Geometric PMF • Total expectation theorem • Joint PMF of two random variables Review • Random variable X: function from sample space to the real numbers • PMF (for discrete random variables): pX(x)= P(X = x) • Expectation: E[X]= ! x xpX(x) E[g(X)] = ! x g(x)pX(x) E[!X + "]= !E[X]+ " • E " X − E[X]# = var(X)= E $(X − E[X])2% = ! x (x − E[X])2 pX(x) = E[X2] − (E[X])2 Standard deviation: #X = &var(X) Random speed • Traverse a 200 mile distance at constant but random speed V 1 200 1/2 v pV (v ) 1/2 • d = 200, T = t(V ) = 200/V • E[V ]= • var(V )= • #V = Average speed vs. average time • Traverse a 200 mile distance at constant but random speed V 1 200 1/2 v pV (v ) 1/2 • time in hours = T = t(V )= • E[T ]= E[t(V )] = 'v t(v)pV (v)= • E[TV ] = 200 "= E[T ] · E[V ] • E[200/V ]= E[T ] "= 200/E[V ]. Conditional PMF and expectation • pX|A(x)= P(X = x | A) • E[X | A]= ! x xpX|A(x) 1 x pX (x ) 4 1/4 2 3 • Let A = {X # 2} pX|A(x)= E[X | A]= Total Expectation theorem • Partition of sample space into disjoint events A1,A2, . . . , An A1 B A2 A3 P(B)= P(A1)P(B | A1)+···+P(An)P(B | An) pX(x)= P(A1)pX|A1(x)+···+P(An)pX|An(x) E[X]= P(A1)E[X | A1]+···+P(An)E[X | An] • Geometric example: A1 : {X =1}, A2 : {X> 1} E[X]= P(X = 1)E[X | X = 1] +P(X> 1)E[X | X> 1] • Solve to get E[X]=1/p Geometric PMF • X: number of independent coin tosses until first head pX(k) = (1 − p)k−1 p, k =1, 2,... E[X]= $! k=1 kpX(k)= $! k=1 k(1 − p)k−1 p • Memoryless property: Given that X> 2, the r.v. X − 2 has same geometric PMF 1 k pX (k) ... p(1-p)2 k pX |X>2(k) ... 3 p p k pX-2|X>2(k) ... 1 p Joint PMFs • pX,Y (x, y)= P(X = x and Y = y) 4 3 2 1 y • ! x ! y pX,Y (x, y)= • pX(x)= ! y pX,Y (x, y) • pX|Y (x | y)= P(X = x | Y = y)= pX,Y (x, y) pY (y) • ! x pX|Y (x | y)= 1 2 3 4 x 1/20 4/20 3/20 2/20 2/20 2/201/20 1/20 1/20 1/20 2/20 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.