Let $X _ { 1 } , X _ { 2 } \text {, and } X _ { 3 }$

Essay

Let $X _ { 1 } , X _ { 2 } \text {, and } X _ { 3 }$ represent the times necessary to perform three successive repair tasks at a certain service facility. Suppose they are independent normal random variables with expected values $\mu _ { 1 } , \mu _ { 1 } \text {, and } \mu _ { 3 } \text { and variances } \sigma _ { 1 } ^ { 2 } , \sigma _ { 1 } ^ { 2 } \text {, and } \sigma _ { 3 } ^ { 2 } \text {, }$ respectively.
a. If $\mu = \mu _ { 2 } = \mu _ { 3 } = 65 \text { and } \sigma _ { 1 } ^ { 2 } = \sigma_ { 2 } ^ { 2 } = \sigma_ { 3 } ^ { 2 } = 20 \text {, }$

Calculate $P \left( X _ { 1 } + X _ { 2 } + X _ { 3 } \leq 210 \right)$
What is $P \left( 150 \leq X _ { 1 } + X _ { 2 } + X _ { 3 } \leq 210 \right) ?$
b. Using the $\mu _ { 2 } ^ { \prime } s \text { and } \sigma _ { 2 } ^ { \prime } s$
given in part (a), calculate $P ( \bar { X } \geq 59 ) \text { and } P ( 62 \leq \bar { X } \leq 68 )$
c. Using the $\mu _ { 2 } ^ { \prime } s \text { and } \sigma _ { 2 } ^ { \prime } s$
given in part (a), calculate $P \left( - 10 \leq X _ { 1 } - .5 \mathrm { X } _ { 2 } - .5 X _ { 3 } \leq 5 \right)$
d. If $\mu _ { 1 } = 40 , \mu _ { 2 } = 50 , \mu _ { 3 } = 60 , \sigma _ { 1 } ^ { 2 } = 10 , \sigma _ { 2 } ^ { 2 } = 12 \text {, and } { \sigma_ { 3 } ^ { 2 } }= 14 \text {, }$
calculate $P \left( X _ { 1 } + X _ { 2 } + X _ { 3 } \leq 160 \right) \text { and } P \left( X _ { 1 } + X _ { 2 } \geq 2 X _ { 3 } \right)$