Continuous Random Variables

When outcomes are not just countable integers, but can take any value in a range.

1. Probability Density Function (PDF)

For continuous variables, the probability of any single exact value is 0. Instead, we define probability over intervals using a Probability Density Function (PDF), $ f_X(x) $.

Key Property

$$ P(a \le X \le b) = \int_{a}^{b} f_X(x) \, dx $$

Normalization

$$ \int_{-\infty}^{\infty} f_X(x) \, dx = 1 $$

2. The Normal Distribution

The most important distribution in statistics. Also known as the Gaussian distribution or "Bell Curve". It is defined by two parameters: Mean ($ \mu $) and Variance ($ \sigma^2 $).

$$ f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} $$

Interactive Normal Explorer

Adjust Mean ($ \mu $) to shift the curve, and Standard Deviation ($ \sigma $) to change its spread.

Mean (μ): 0

Std Dev (σ): 1

3. Calculating Probabilities

Probability is the area under the PDF curve. Use the tool below to visualize $ P(a < X < b) $ for a Standard Normal Distribution ($ \mu=0, \sigma=1 $).

Area Under the Curve

Lower Bound (a): -1

Upper Bound (b): 1

P(a < X < b)= 0.6827

Exponential Distribution

Models time between events in a Poisson process. Memoryless property!

f(x) = λe^(-λx) for x ≥ 0

Time until next bus
Lifetime of a lightbulb

Uniform Distribution

All intervals of the same length are equally likely.

f(x) = 1/(b-a) for a ≤ x ≤ b

Random number generator
Waiting time if bus comes every 10 mins