The area under the graph of the PDF. $$ \boxed{P(a \leq X \leq b) = \int_a^b f_X \, (x) \, dx} $$
To qualify as an PDF, $f_X(x) \geq 0$ for every $x$, and $f_X$ must satisfy the normalize equation: $$ \int_{- \infty}^{\infty} f_X(x) \, dx = P(- \infty < X < \infty) = 1 $$
For any single value $a$, $P(X = a) = \int_a^a \, f_X(x) \, dx = 0$.
Thus $$ P(a \leq X \leq b) = \underbrace{P(X = a)}_0 + \underbrace{P(X = b)}_0 + P(a < X < b) = P(a < X < b). $$
Jumps in the CDF correspond to points $x$ for which $P(X = x) > 0$. Thus the fact that the CDF does not have jumps is consistent with the fact that $P(X = x) = 0$ for all $x$.
$\delta > 0$, small $$ \boxed{\mathbf{P}(x < X < x + \delta) = \int_x^{x + \delta} f_X(t) \, dt \approx f_X(x) \cdot \delta} $$
$$ \Rightarrow f_X(x) = \frac{P(x \leq X \leq x + \delta)}{\delta} = \frac{\text{Probability mass}}{\text{Unit length}} $$
The CDF $F_X(x)$ “accumulates” probability “up to” the value of $x$.
$\{X \leq x\}$ is always an event $\Rightarrow$ Any random variable associated with a given probability model has a CDF, regardless of whether it is discrete, continuous, or other.
continuously varying form
$F_X$ is monotonically nondecreasing: $$ \text{if } x \leq y \text{, then } F_X(x) \leq F_X(y). $$
$F_X$ tends to 0 as $x \to -\infty$, and to 1 as $x \to \infty$.
the PDF and the CDF can be obtained from each other by integration or differentiation $$ \boxed{F_X(x) = \int_{-\infty}^{x}f_X(t) \, dt} $$
$$ \boxed{f_X(k) = \frac{dF_X}{dx} (x) = F^{\prime}_X(x)} $$
For all $a \leq b$, $$ \boxed{P(a < X \leq b) = P(X \leq b) - P(X \leq a) = F_X(b) - F_X(a)} $$
Important in the theory of probability: Centrail limit theorem.
If you have a phenomenon in which you measure a certain quantity, but that quantity is made up of lots and lots of random contributions.
Then your random variable is actually the sum of lots and lots of independent little random variable. And no matter what kind of distribution the little random variables have, their sum will turn out to have approximately a normal distribution.
$\rightarrow$ This makes the normal distribution to arise very nartually in lots and lots of context. Whenever you have noise that’s comprise of lots of different independent pieces of noise, then the end result will be a random normal variable.
