A PDF is a probability distribution that describes a continuous random variable taking on values in different ranges.
The density value represents “probability per unit length/area/volume”, not probability itself, so it can be greater than 1 (e.g., for )

But the total area/volume under the curve must equal 1:
You can imagine poking the curve like a waterballloon… if it goes down somewhere, it must go up somewhere else to compensate.

It’s a model of a random process:
→ You can do hypothesis testing with data
→ You can estimate parameters (mean, std, …) with a small sample


for any measurable set
“densitiy w.r.t. base measure = counting measure → pmf, = lebesgue measure → pdf”

Density

is not a probability but the rate of accumulation of probability near : the derivative of the CDF at that point.
We call it density because it describes how tightly probability is packed around that point.
The relative density between two points and describes how much more likely it is to find a random sample near than near → We get

The probability of a random variable falling within a range is the integral of the PDF over that range:

→ Probabilities are only meaningful over intervals/ranges, not individual points.
→ The value of can be greater than , as it represents density not probability.
for any single point , as it represents an interval of width zero.

Example: For a standard normal distribution, , but , however, , representing the probability of falling within that interval.

So while taking sums of probabilities over individual points works in a discrete context (probability mass function):

It doesn’t work in a continuous context: