Verify that the functions are probability density functions...

Verify that the functions are probability density functions for a continuous random variable $X$ over the given interval. Determine the specified probability.
$f(x)=\frac{3}{2} x(2-x)$ over $[0,1], P(0.5>X)$

Key Concepts

Cumulative Distribution Function

The cumulative distribution function (CDF) is derived by integrating the probability density function from the lower limit of the distribution up to a specific value. It gives the probability that the random variable will take a value less than or equal to that specific value.

Definite Integration for Probabilities

Determining the probability that a continuous random variable falls within a specified interval involves computing the definite integral of its probability density function over that interval. This integration process aggregates the contributions of the density function over the range of interest.

Normalization Condition

For any probability density function to be valid, it needs to satisfy the normalization condition, meaning the integral of the function over its entire domain must equal 1. This ensures that the total probability over all outcomes sums (or integrates) correctly to 1.

Continuous Random Variable

A continuous random variable is a variable that can take on an infinite number of possible values within a given interval. Unlike discrete random variables, probabilities for continuous variables are calculated by integrating the probability density function over an interval, as the probability at an exact point is typically zero.

Probability Density Function

A probability density function (pdf) is a function that describes the likelihood of a continuous random variable falling within a particular range. It must be nonnegative for all values in its domain and its integral over the entire space must equal 1, ensuring that it properly represents a probability distribution.

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