(Requires Calculus $\frac { p ( 1 - p ) } { n }$

question 8

Essay

(Requires calculus.)Let Y be a Bernoulli random variable with success probability Pr(Y = 1)= p. It can be shown that the variance of the success probability p is $\frac { p ( 1 - p ) } { n }$ Use calculus to show that this variance is maximized for p = 0.5.

---

Definitions:

Environmental Influences

The impact of surrounding conditions, including social, physical, and cultural settings, on an individual's behavior, attitudes, and development.

Natural Selection

A process in biology by which organisms with traits well-suited to their environment have higher survival rates and reproduction, shaping evolution.

Inherited Traits

Characteristics or qualities passed down genetically from parents to offspring.

Reproduction

The biological process by which new individual organisms are produced, allowing for the continuation of a species.