Yeah I get the confusion, in short: the probability distribution is the big picture, while the PDF is the "formula" that helps describe it for continuous cases

Longer version: the probability distribution is the overall description of how probabilities are assigned to different possible outcomes of some random variable, it gives you a complete picture of how likely different values are. The PDF is a specific function that represents the probability distribution of a continuous random variable, it tells you how "dense" the probability is at different values. And since continuous variables can take infinitely many values, the PDF itself doesn’t give probabilities directly, but must be integrated over an interval to find the probability of the variable falling within that range

Sometimes, they are used interchangeably in text, which is fine. As the other response pointed out, the probability distribution is the concept (e.g., Gaussian distribution), whereas the PDF is the mathematical function. A probability distribution has a PDF, a CDF, and other properties.

The concept of probability distribution is a wider notion that defines how probabilities are apportioned.Applies to both discrete (PMF) and continuous (PDF) cases. The PDF is a specific function used for continuous random variables within a probability distribution.

A Probability Density Function shows how likely it is for a continuous random variable to fall within a certain range of values. A probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. Probably they are the same thing :)

A distribution function F(x) is the probability that a random variable X will take a value less than x. So, F(x)=P(X

The probability distributions refers the chances of occurrence of the event, it may contains discrete and continuous random variables, but probability density function refers probability of occurrence of the event under continuous random variables only.

Depending on context, and on whatever convention is being used by whoever is writing what you are reading, a probability density function can be a so-called generalised function, and thus might be used for any distribution. An example of a generalised function is a Dirac delta-function. This approach allows certain simple formulae applicable to purely continuous distributions to be applied to the more general case. An example here is the definition of the characteristic function.

However, in non-sophisticated contexts, it is usual to assume that a probability density function will be an ordinary function.

Let X be a random variable, that means a function from an outcome space S with associated sigma-algebra SS and probability measure P to R, the set of real numbers . Elements of SS are called events . Because X is a random variable we have that by definition { s in S such that X is smaller than or equal to x } where x is a real number, is also an event .

Therefore the probability of this event is well defined. We call this number F(x) , for every x . Sometimes you are lucky and then the function F is differentiable . We call this derivative equal to f. Now f is the density of the random variable X. Just consult a decent book about probability theory that takes the framework of Kolmogorov seriously! It is wise to have such a book on your shelve!

There are a number of ways of mathematically specifying a probability distribution over numerical values, of which the probability density function and/or cumulative distribution function is only one approach.

Others are:

1) a mechanistic approach via algebraic combinations of other basic random quantities with known properties;

2) via a characteristic function.

Of course a probability distribution can refer to a random distribution over non-numerical outcomes.