Let X be a random variable with probability density function...

Let $X$ be a random variable with probability density function $f(x)=\left\{\begin{array}{ll}c\left(1-x^{2}\right) & -1<x<1 \\ 0 & \text { otherwise }\end{array}\right.$
(a) What is the value of $c ?$
(b) What is the cumulative distribution function of $X ?$

Key Concepts

Probability Density Function (PDF)

A probability density function describes the likelihood of a continuous random variable taking on a given value. The function must be non-negative for all values and, when integrated over the entire range of the variable, must equal 1. This characteristic ensures that the total probability is properly normalized.

Normalization Condition

The normalization condition for a probability density function requires that the integral of the function over its entire support is equal to 1. This condition is used to determine any constant factors in the density function, ensuring that the total probability sums to 1, which is a fundamental property of all probability distributions.

Integration in Continuous Distributions

Integration is crucial in the context of continuous probability distributions, as it is used both in normalizing the probability density function and in computing the cumulative distribution function. Evaluating definite integrals allows one to accumulate the probabilities across an interval, thereby translating a density function into a cumulative probability measure.

Cumulative Distribution Function (CDF)

The cumulative distribution function of a random variable gives the probability that the variable will be less than or equal to a certain value. It is obtained by integrating the probability density function from the lower bound of the variable's support up to a variable point. The CDF is a non-decreasing function that ranges from 0 to 1.

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