Continuous Random Variables - Probability & Statistics

PDF and CDF Relationship

For continuous random variables, the PDF is the derivative of the CDF, and the CDF is the integral of the PDF.

$$ f_X(x) = \frac{d}{dx} F_X(x) \quad \text{and} \quad F_X(x) = \int_{-\infty}^x f_X(t) \, dt $$

Select Distribution

The probability that $X$ falls in an interval $[a, b]$ is the area under the PDF curve: $$ P(a \leq X \leq b) = \int_a^b f_X(x) \, dx $$

Lower bound (a): 0.25

Upper bound (b): 0.75

P(0.25 ≤ X ≤ 0.75) = 0.500

If $Y = g(X)$ where $g$ is monotonic and differentiable, the PDF of $Y$ is:

$$ f_Y(y) = f_X(x) \left| \frac{dx}{dy} \right| \quad \text{where } x = g^{-1}(y) $$

Select Transformation

$$ f_Y(y) = \frac{1}{2} \text{ for } y \in [1, 3] $$

$$ E[X] = \int_{-\infty}^{\infty} x f_X(x) \, dx $$

The "center of mass" of the distribution.

$$ \text{Var}(X) = \int_{-\infty}^{\infty} (x - E[X])^2 f_X(x) \, dx $$

Or equivalently: $\text{Var}(X) = E[X^2] - (E[X])^2$