Random Variables

Expected Value

$$ E[X] = \sum_x x \cdot P(X=x) \quad \text{(discrete)} $$

$$ E[X] = \int_{-\infty}^{\infty} x \cdot f(x) dx \quad \text{(continuous)} $$

Variance

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

Python

 1import numpy as np
 2
 3data = np.random.normal(0, 1, 10000)
 4
 5mean = np.mean(data)
 6variance = np.var(data)
 7std_dev = np.std(data)
 8
 9print(f"Mean: {mean:.3f}")
10print(f"Variance: {variance:.3f}")
11print(f"Std Dev: {std_dev:.3f}")

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