Engineering Reliability - Random Variables | MEM 361, Study notes of Mechanical Engineering

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MEM 361: Engineering Reliability

(Random Variables)

Dr. Ajmal Yousuff Dept. MEM Drexel University

overview

• Random Variables

– Discrete

– Continuous

– Characterizations

– Distribution functions

• Probability distribution
• Cumulative probability

• examples

Discrete random variable

• Probability distribution:

• Cumulative probability function:

• Mean:
• Variance:
• Standard deviation: s

( (^) i ) ( (^) i ) { ( (^) i ) 1} i

f x  P X  x  f x 

k

F xk   i  f xi F xk  P X  xk

1 ^ 

n

   i  x fi xi

n

s   i  xi  f xi

Continuous random variables

• Cumulative probability function:

• Probability distribution (density):

– Mean:

– Variance:

– Standard deviation: s

F x ( )  P X (  x )

d

f x F x f x dx

dx



Graph of a pdf

Notes:

• Area under f(x) = 1.
• Centroid = .
• Moment of inertia, about  = s^2.
• Radius of gyration = s.
• @ mode , f(x) is max.
• @ median, area separates into two equal halves.
• For a given g(x) , its expected value is: - E(X) =  - E(X^2 ) = ( s^2 + ^2 )

E g ( )  (^) ^  g x f ( ) ( ) x dx



s (^) s

f(x)

x xmode

-  

left skew (^) right skew

x

f(x)

Example (2-16)

• Let t (in years) be the time-to-failure of a

washing machine. Suppose its pdf (for some

constant A ) is:

– Axiom of total probability: {F(∞)=1}

f t ( )  Ate 0.5 t ; 0  t  

0 0 ( ) ( ) 1 1 4

F   ^ f t dt  ^ Ate ^ tdt   A 

Example pdf, cpdf

• Skewness:

•  Left-skewed.

sk^13 ( t ) (^3 Ate 0.5 t ) dt > 0 s  ^ 

(^0 )

0.

0.

2 4 6 8 10  s

pdf (^) cpdf

0

1/ 2

1

pdf cpdf

discussions

• pdf, cpdf  visual presentation of MTTF.

– Failure probability rises during the 1st^ 2 yrs.

{F(2)~25%}

– MTTF = 4 yr.

• F(4) = 0.594 (  60% will fail by 4th^ year.)

– Within a warranty of 1 yr., 9% will fail.