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

question 1

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 $P \left( X _ { 1 } + X _ { 2 } + X _ { 3 } \leq 160 \right) \text { and } P \left( X _ { 1 } + X _ { 2 } \geq 2 X _ { 3 } \right)$

Recognize how personal coping strategies, including problem-focused and emotion-focused coping, gender, and culture, influence stress management and outcomes.

Understand the biological components involved in stress response, including the role of catecholamines and the nervous system structure.

Understand the factors influencing patient adherence.

Recognize the role of social support in patient adherence.

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Definitions:

Nuts

Hard-shelled seeds of certain plants that are generally edible and nutritious, often eaten raw, roasted, or used as ingredients in cooking.

Berries

Small, pulpy, and often edible fruits, typically juicy, rounded, brightly colored, and seed-containing, consumed worldwide in various forms.

Utility Function

A mathematical representation of how individual preferences or consumption bundles yield levels of satisfaction or utility.

Budget Constraint

A representation of all the combinations of goods and services a consumer can afford to buy given their income and the prices of the goods.