Provide an Appropriate Response $\mathrm { f } ( \mathrm { x } ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - \mathrm { x } ^ { 2 } / 2 }$

question 373

Essay

Provide an appropriate response.
-The standard normal probability density function is defined by $\mathrm { f } ( \mathrm { x } ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - \mathrm { x } ^ { 2 } / 2 }$ .
(a) Show that $\int _ { 0 } ^ { \infty } \frac { 1 } { \sqrt { 2 \pi } } x \mathrm { e } ^ { - x ^ { 2 } / 2 } \mathrm { dx } = \frac { 1 } { \sqrt { 2 \pi } }$ .
(b) Use the result in part (a) to show that the standard normal probability density function has mean 0 .

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Definitions:

Object Relations

A psychoanalytic theory focusing on how individuals relate to others and objects in their environment, shaped by early family interactions.

Ideology

A coherent set of beliefs, values, and norms that defines the way a group thinks about the world and their place within it.

Expectations

Beliefs or anticipations about the way events, situations, or behaviors will unfold in the future.

Psychological Dysfunction

Refers to a deviation from typical patterns of emotional, cognitive, or behavioral functioning, which can significantly interfere with an individual's daily life activities and well-being.