Continuous Probabilities

Uniform: All intervals of the same length are equally likely.
Exponential: Models wait times (Memoryless).
Normal (Gaussian): The bell curve, central to statistics.

For continuous variables, individual points have zero probability. We use Density Functions ( $f(x)$ ).

The Probability Density Function (PDF)

$P(X \in A) = \int_A f_X(x) dx$

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For two continuous variables, we have a joint density $f(x, y)$ . Independence means the joint density factors: $f(x, y) = f_X(x)f_Y(y)$

We use convolution to find distributions of sums, introducing new families like Gamma, Cauchy, and Beta.