1. South Dakota School of Mines & Technology Expectations for Exponential

2. Expected Life   E [ x]  x  e dx   x e dx    ( ) 2  1 
For a producted governed by an exponential life distribution,
the expected life of the product is given by 
2.0 E [ x]   x  e   x dx
1.8
1.6
1.4
1.2 f (x
t )   e   x
1.0      x e dx 2 1
Density
0.8
0.6
0.4
0.2
0.0 X
0.5 1
1.5 2
2.5 3    ( ) 2
1/  1 

3. Variance      ( ) x dF     E x [( ) ] =     ( ) x p   2    ( ) x dF   2   E x [( )
] =   2    ( ) x p   2      ( ) x f dx

4. Property        ( ) x f dx    ( ) x f dx     x f dx xf (2      ( ) x f dx     ( ) x f dx 2      x f dx xf 2 ( ) 

5. Exponential Example       E X [ ]   1   x e dx ( )   x e
For a producted governed by an exponential life distribution,
the expected life of the product is given by   2   E X [ ]
2.0
1.8   2 1    x e dx ( )
1.6
1.4
1.2 f (x
t )   e   x
1.0
Density
0.8      x e dx 3 1
0.6  1 2 
0.4
0.2
0.0 X
0.5 1
1.5 2
2.5 3    ( ) 3
1/ 1 2   1 2  =