Probability density function vs continuous probability distribution

Question

I am trying to understand the difference between the two concepts mathematically and also graphically.

Regarding the probability density function (pdf), I know that it shows the probability of a continuous random variable having a value that belongs to an interval. What I don't know is how we make the transition from probability mass function (pmf) to pdf. I would like to know this, as additional knowledge, but it's not the point of my thread, but I'd appreciate links/pdf etc regarding this.

My problem is to understand the difference between pdf and the continuous probability distribution. In fact my problem is understanding the continuous probability distribution.

Example : we have the normal pdf. It's a function that whose value depends on a continuous variable and we can plot it, and we would get the bell shaped graph. So I understand the meaning of pdf, how we express it mathematically and how it looks graphically.

Now I want to know the same about the normal continuous probability distribution.

What it means?

How we express it mathematically?

How it looks graphically?

By knowing the answer to the above 3 questions then I can understand how it's different from the pdf.

Answer 1

I suspect what you are calling a "continuous probability distribution" is more commonly called a cumulative distribution function. (If not, I'd be interested in knowing which source seemed to indicate that they are different.)

Indeed, the probability density function $f$ and the cumulative distribution function $F$ are the most important tools for working with continuous random variables. To give the meaning of $F$ (as you've done for $f$), it is simply \begin{equation} F(x) = \mathrm{Pr}(X < x). \end{equation} Mathematically, you can go from one to the other with \begin{align} f(x) &= \frac{d}{dx} F(x) \\ F(x) &= \int_{-\infty}^x f(y) dy \end{align} which are consistent with eachother by the fundamental theorem of calculus. Graphically, if it is still hard to get a sense of the shape from the above formula, you can go to the normal distribution Wikipedia article where both the pdf $f$ and cdf $F$ are plotted. The latter looks like a smoothed out step function. Its minimum and maximum values are 0 and 1 respectively as required for a probability.