(A) Show That Is the Probability Density Function of a Random Variable

question 104

Essay

(a) Show that $$f ( x ) = \left\{ \begin{array} { l l } 4 x + 12 x ^ { 2 } & \text { if } 0 \leq x \leq \frac { 1 } { 2 } \\0 & \text { otherwise }\end{array} \right.$$ is the probability density function of a random variable.(b) What is the mean for this distribution?
(c) Calculate the median of $f$

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Definitions:

Responsible

Having an obligation to do something, or having control over or care for someone, as part of one's job or role.

Determined Actions

Refers to actions whose outcomes are influenced or dictated by prior conditions or laws, suggesting a lack of free will.

Compatibilism

The thesis that both determinism (in some interpretations) and free action can be true. Determinism does not rule out free action, and the possibility of free action does not require that determinism be false. They are compatible positions.

Kant

Immanuel Kant was an 18th-century German philosopher whose work in epistemology, ethics, and aesthetics had a profound influence on modern Western philosophy, especially through his doctrine of transcendental idealism.