Explain what is meant by a probability density function.

A probability density function (PDF) is a function that describes the likelihood of a continuous random variable taking on a particular value or occurring within a specific range of values. It does so by assigning probabilities to different intervals on the real line. The PDF is used to describe continuous probability distributions, where the outcomes can take infinitely many values within a given range.

There are two main properties of a probability density function that must be satisfied: 1. The PDF must be non-negative for all values of the random variable: For a continuous random variable X and its PDF f(x), we must have f(x) ≥ 0 for all x. This is because the probability of any event cannot be negative. 2. The integral of the PDF over the entire range of the random variable must equal 1: Since the PDF assigns probabilities to all possible values of the random variable, it must be the case that the sum of probabilities of all possible outcomes is 1. In mathematical terms, for a PDF f(x) and a random variable X with the range [a, b], the integral of f(x) from a to b must equal 1: ∫_{a}^{b} f(x) dx = 1

A probability density function does not give the probability of a specific value directly, but rather the probability for the value to be in a given interval. To find the probability of a continuous random variable being within a certain range, we calculate the integral of the PDF over that range. For example, if X is a continuous random variable with PDF f(x) and we want to find the probability that X lies within the range [c, d], we would compute: P(c ≤ X ≤ d) = ∫_{c}^{d} f(x) dx It is essential to note that the probability of a continuous random variable taking a specific value is zero. This is because there are infinitely many values that the random variable can take on, so the probability of it being equal to one specific value is infinitesimally small. In summary, a probability density function is a function that characterizes the distribution of a continuous random variable. It adheres to specific properties, and its integral is used to compute probabilities of the random variable within specified ranges.