The Joint Probability Density Function for Two Continuous Random Variables

question 18

Multiple Choice

The joint probability density function for two continuous random variables X and Y is $f ( x , y ) = \left\{ \begin{array} { l l } \frac { 1 } { 6 } e ^ { - x / 6 } e ^ { - y } & \text { if } x \geq 0 \text { and } y \geq 0 \\0 & \text { otherwise }\end{array} \right.$ Find the expected values $E ( X ) \text { and } E ( Y )$

• A) \(E ( X ) = 6 ; E ( Y ) = 1\)
• B) \(E ( X ) = 1 ; E ( Y ) = 5\)
• C) \(E ( X ) = 1 ; E ( Y ) = 6\)
• D) \(E ( X ) = 7 ; E ( Y ) = 1\)

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Definitions:

Overeat

The consumption of food in quantities greater than the body's physical need, often leading to discomfort, weight gain, and health issues.

Anal Expulsive

A psychoanalytic concept suggesting a personality type that results from a child's experience during the anal stage of psychosexual development, often associated with messiness or disorganization.

Anal Retentive

Pertains to a personality trait characterized by orderliness, meticulousness, and a controlling attitude, believed by Freud to originate during the anal stage of psychosexual development.

Personality

The characteristic set of behaviors, cognitions, and emotional patterns that evolve from biological and environmental factors.