Probability, Statistics, and Reliability for Engineers and Scientists MULTIPLE RANDOM VARIABLES

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1 CHATER robability, Statistics, and Reliability or Engineers and Scientists MULTILE RANDOM VARIABLES Second Edition A. J. Clark School o Engineering Department o Civil and Environmental Engineering 6a robability and Statistics or Civil Engineers Department o Civil and Environmental Engineering University o Maryland, College ark CHAMAN HALL/CRC CHATER 6a. MULTILE RANDOM VARIABLES Slide No. Introduction In engineering, it is common to deal with two or more random variables simultaneously in solving problems. I the load applied to a structure is considered to be a random variable, then the structural response will also be a random variable.

2 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. Introduction The load and the response can be modeled separately as random variables; however, it is more prudent to model the uncertainty jointly. More inormation can be etracted rom the joint distributions. Thus, it is necessary to etend the discussion to multiple random variables. CHATER 6a. MULTILE RANDOM VARIABLES Slide No. Introduction In general, multiple random variables are encountered in the ollowing two orms:. Joint occurrences o multiple random variables that can be correlated or uncorrelated. Random variables that are known in terms o their unctional relationship with other basic random variables

3 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 4 Joint and Their robability Distributions The outcomes, E, E,, E n, that constitute a sample space S are mapped to an n-dimensional (n-d) space o real numbers. The unctions that establish such a transormation to the n-d space are called multiple random variables (or random vectors). CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 5 Joint and Their robability Distributions Multiple random variables are classiied into two types: Discrete random variables Continuous random variables A distinction is made between these two types because the computations o probabilities depend on their type.

4 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 6 robability or Discrete Random Vectors Joint robability Mass Function (JMF) The joint probability mass unction or a discrete multiple random variable or random vector = (,,.., n ) is given by ( ) = ( =, =,..., ) = Note that ( =, =,..., n = ) n n n CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 7 robability or Discrete Random Vectors Joint Cumulative Mass Function (JCMF) The joint cumulative mass unction or a discrete random variable or random vector = (,,.., n ) is given by F ( ) = (,,..., n n ) = (,,..., ) all n n (,,..., ) 4

5 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 8 robability or Discrete Random Vectors roperties o JCMF. F (all ) =. F (,,, i -,, n ) =, or any i =,,, n. F (,,, i -,, k -,, n ) =, or any values o i,, k 4. F (,,, i +,, n ) = F j ( j : j =,,, n and j i), called the marginal distribution o all the random variables ecept i 5. F (,,, i +,, k +,, n ) = F j ( j : j =,,, n and j i to k), called the marginal distribution o all the random variables ecept i to k 6. F (all + ) = 7. F () is a nonnegative and nondecreasing unction o CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 9 robability or Discrete Random Vectors roperties o JCMF The irst, second, and third properties deine the limiting behavior o F (); as one or more o the random variables approach -, F () approaches zero. The ourth and ith properties deine the possible marginal distributions as one or more o the random variables approaches +. The sith property is based on the probability aiom. The seventh property is based on the cumulative nature o F (). 5

6 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. robability or Two Discrete For simplicity, the presentation o the materials in the remaining part o this section is limited to two random variables. The presented concepts can be generalized to n random variables CHATER 6a. MULTILE RANDOM VARIABLES Slide No. robability or Two Discrete Conditional robability Mass Function where The conditional probability mass unction or two random variables and is given by (, ) ( ) = ( ) ( ) given that =. results in the probability o ( ) = marginal mass unction or = 6

7 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. robability or Two Discrete Conditional robability Mass Function where The conditional probability mass unction or two random variables and is given by (, ) ( ) = ( ) ( ) given that =. results in the probability o ( ) = marginal mass unction or = CHATER 6a. MULTILE RANDOM VARIABLES Slide No. robability or Two Discrete Marginal Distributions The marginal mass unction or that is not equal to zero is = ( ) ( ), all The marginal mass unction or that is not equal to zero is ( ) = ( ), all 7

8 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 4 robability or Two Discrete roperties I and are statistically independent (uncorrelated) random variables, then and, ( ) = ( ) ( ) = ( ) (, ) = ( ) ( ) The important relationship can be obtained : CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 5 robability or Two Discrete Eample: Two Discrete RV s The time to produce a typical engineering drawing, represented by a random variable, and its quality, represented by a random variable, are under consideration. Suppose can be 7, 8, 9, or hours. The quality o a drawing can be considered to be moderate, good, and ecellent, and can be considered to be,, and, respectively. Suppose that such drawing are evaluated and the inormation provided the net table is obtained. 8

9 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 6 robability or Two Discrete Eample (cont d): Two Discrete RV s CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 7 robability or Two Discrete Eample (cont d): Two Discrete RV s. Find the joint MF o and.. lot the marginal MF o and.. I only ecellent quality drawings are acceptable (i.e., = ), plot the conditional MF o. 9

10 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 8 robability or Two Discrete Eample (cont d): Two Discrete RV s. The joint MF, (, ) o and CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 9 robability or Two Discrete Eample (cont d): Two Discrete RV s =,. The marginal MF o ( ) ( ) all ( 7) = =. ( 8) = =. ( 9) = =. ( ) = =. 6

11 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. robability or Two Discrete ( ) Eample (cont d): Two Discrete RV s ( ) = ( ), Marginal MF o all CHATER 6a. MULTILE RANDOM VARIABLES Slide No. robability or Two Discrete Eample (cont d): Two Discrete RV s The marginal MF o ( ) ( ) =, all ( ) = = ( ) = = () = =. 47

12 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. robability or Two Discrete ( ) Eample (cont d): Two Discrete RV s Marginal MF o ( ) = ( ), all CHATER 6a. MULTILE RANDOM VARIABLES Slide No. robability or Two Discrete Eample (cont d): Two Discrete RV s. Conditional robability o ( ) = i ( 7 ) ( 8 ) ( 9 ) ( ) = (,) i ().5 = = = = = = ( ) = =. 47 (, ) ( )

13 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 4 robability or Two Discrete ( ) Eample (cont d): Two Discrete RV s ( ) (,) Conditional MF o Y = i = i () CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 5 robability or Continuous Random Vectors Joint robability Density Function (JDF) The joint probability density unction or a continuous multiple random variable or random vector = (,,.., n ) is used to deine Note that n l u ( ) =... ( ) l l l n + + ( < < + ) =... ( ) u + u u d d... d d d... d = n n

14 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 6 robability or Continuous Random Vectors Joint Cumulative Distribution Function (JCDF) The joint cumulative distribution unction o a continuous random variable is deined by F n ( ) = ( ) =... ( ) d d... d n CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 7 robability or Continuous Random Vectors roperties o JCDF. F (all ) =. F (,,, i -,, n ) =, or any i =,,, n. F (,,, i -,, k -,, n ) =, or any values o i,, k 4. F (,,, i +,, n ) = F j ( j : j =,,, n and j i), called the marginal distribution o all the random variables ecept i 5. F (,,, i +,, k +,, n ) = F j ( j : j =,,, n and j i to k), called the marginal distribution o all the random variables ecept i to k 6. F (all + ) = 7. F () is a nonnegative and nondecreasing unction o 4

15 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 8 robability or Continuous Random Vectors The joint density unction can be obtained rom the a given joint cumulative distribution unction as ollows: ( ) That is =... n n F ( ) (,,..., ) n F... n (,,..., )... n n CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 9 robability or Two Continuous For simplicity, the presentation o the materials in the remaining part o this section is limited to two random variables. The presented concepts can be generalized to n random variables 5

16 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. robability or Two Continuous Conditional robability Density Function The conditional probability density unction or two random variables and is given by (, ), ( ) = ( ), (, ) joint density unction o and. = ( ) = marginal density unction or that is not equal to zero. where CHATER 6a. MULTILE RANDOM VARIABLES Slide No. robability or Two Continuous Conditional robability Density Function The conditional probability density unction or two random variables and is given by (, ), ( ) = ( ), (, ) joint density unction o and. = ( ) = marginal density unction or that is not equal to zero. where 6

17 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. robability or Two Continuous Marginal Distributions The marginal density unction or that is not equal to zero is + =, ( ) ( ) d The marginal mass unction or that is not equal to zero is + =, ( ) ( ) d CHATER 6a. MULTILE RANDOM VARIABLES Slide No. robability or Two Continuous roperties I and are statistically independent (uncorrelated) random variables, then and ( ) = ( ) ( ) = ( ) (, ) = ( ) ( ) The important relationship can be obtained : 7

18 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 4 robability or Two Continuous Eample: Two Continuous RV s The joint density unctions o two random variables and Y can be epressed as c 4 y, Y (, y) = ( )( 9) or and y elsewhere (a) Determine the constant c. (b) Determine the marginal density unction or. (c) Determine the marginal density unction or Y. (d) Are and Y statistically independent? (e) Determine the probability o the ollowing event: F, Y (, ) CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 5 robability or Two Continuous Eample (cont d): Two Continuous RV s (a) c( 4)( y 9) ddy = or or or 6 y c 9y c = 96 6 c( y 9) 4 dy = c( y 9) =. dy =. 8

19 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 6 robability or Two Continuous Eample (cont d): Two Continuous RV s (a) c( 4)( y 9) ddy = or or or 6 y c 9y c = 96 6 c( y 9) 4 dy = c( y 9) =. dy =. CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 7 robability or Two Continuous Eample (cont d): Two Continuous RV s (b) ( ) = ( 4)( y 9) dy = ( 4) 96 6 (c) (d) Y ( ) y = ( 4)( y 9) d = ( y 9) 96 8 ( ) ( y) = ( 4) ( y 9) Y 6 8 = ( 4)( y 9) = ( ) Y ( y) 96 and Y are statistically independent random variables. 9

20 CHATER 6a. MULTILE RANDOM VARIABLES Slide No. 8 robability or Two Continuous Eample (cont d): Two Continuous RV s (e) F, Y = 96 (,) = ( 4) d ( y 9) dy. 6875