1. South Dakota School of Mines & Technology Introduction to Probability & Statistics Industrial Engineering

2. Introduction to Probability & Statistics Continuous Review

3. Expectations Continuous Review Mean and Variance
Cumulative and Inverse Functions
Mean and Variance
Expected Value properties for one variable
Expected Value properties for two variables
Central Limit Theorem

4. Continuous Distribution
f(x) A x
a b c d
1. f(x) > , all x 2.
3. P(A) = Pr{a < x < b} =
4. Pr{X=a} = f x dx a d ( )   1 b c

5. Normal Distribution f x e ( )  1 2          65% 95%      1 2     

   
65%
95%
99.7%

6. Std. Normal Transformation
Standard Normal
f(z)    Z X     f z e ( )   1 2 
N(0,1)

7. Example
Suppose the distribution of student grades for university are approximately normally distributed with a mean of 3.0 and a standard deviation of What percentage of students will graduate magna or summa cum laude? x
3.0
3.5

8. Example Cont. ÷ ø ö ç è æ - ³ = 3 . 5 Pr s m X
3.0
3.5
Pr{magna or summa} = Pr{X > 3.5}} ÷ ø ö ç è æ - = 3 . 5 Pr s m X
= Pr(z > 1.67) = =

9. Example
Suppose we wish to relax the criteria so that 10% of the student body graduates magna or summa cum laude. x
3.0 ?
0.1

10. Example
Suppose we wish to relax the criteria so that 10% of the student body graduates magna or summa cum laude. x
3.0 ?
0.1
0.1 = Pr{Z > z}
z = 1.282

11. Example x = m + sz = 3.0 + 0.3 x 1.282 = 3.3846 But s m - = X Z 3.0 ?
0.1
But s m - = X Z
x = m + sz
= x 1.282 =

12. Example
Let X = lifetime of a machine where the life is governed by the exponential distribution. determine the probability that the machine fails within a given time period a.
, x > 0,  > 0 f x e ( )   

13. Example  f x e ( )   F a X ( ) Pr{ }     e dx   e   1 e a
Exponential Life
2.0 f x e ( )   
1.8
1.6
1.4
1.2 F a X ( )
Pr{ }  
f(x)
Density
1.0
0.8
0.6     e dx x a
0.4
0.2
0.0
0.5 1
1.5 2
2.5 3   e x a  a
Time to Fail   1 e a 

14. Complementary  F a X ( ) Pr{ }     e dx  e
Exponential Life
Suppose we wish to know the probability that the machine will last at least a hrs?
2.0
1.8
1.6
1.4
1.2
f(x)
Density
1.0
0.8
0.6 F a X ( )
Pr{ }  
0.4
0.2
0.0     e dx x  a
0.5 1
1.5 2
2.5 3 a
Time to Fail   e a 

15. Example
Suppose for the same exponential distribution, we know the probability that the machine will last at least a more hrs given that it has already lasted c hrs. a c
c+a
Pr{X > a + c | X > c} = Pr{X > a + c  X > c} / Pr{X > c}
= Pr{X > a + c} / Pr{X > c}    e c a  ( )