Defining the pdf, calculating probabilities and quartiles

Understanding Probability Density Function (pdf)

A Probability Density Function (pdf) represents a probability distribution for a continuous random variable. It plots the likelihood of different outcomes.
The total area under a pdf curve is always equal to 1, as it represents the total probability of all possible outcomes.
The probability of a single, exact point from the continuous random variable is always zero.
The value of a pdf at any two or more distinct points can be unequal.

To calculate the probability that a random variable X falls in a certain interval, find the area under the pdf curve above that interval.
Use integral calculus to determine this area.
The probability that X is between a and b is given by the integral of the pdf from a to b.
Remember, single-point probabilities in a continuous distribution are zero.

Quartiles divide a rank-ordered data set into four equal parts.
The first quartile (Q1), also known as the lower quartile, is the value below which lies 25% of the data.
The second quartile (Q2), or the median, is the value below which lies 50% of the data.
The third quartile (Q3), or upper quartile, is the value below which lies 75% of the data.
To calculate the quartiles of a continuous probability distribution, solve for the value of the variable X in the equations P(X <= x) = 0.25 (for Q1), P(X <= x) = 0.5 (for Q2), and P(X <= x) = 0.75 (for Q3).
Here, P() denotes probability, and you calculate it via the integral of a pdf.

Understanding the pdf is essential in fields like physics, engineering, computer science and economics to model real-world scenarios.
Quartiles are useful in descriptive statistics for providing a rough sketch of the distribution of the data. They also allow comparisons between different data sets.
The concept of quartiles is used in the construction of a box plot which pictorially represents data.
Always remember, the spread of data and understanding it, is critical for the concepts of pdf and quartiles.