Random Variables¶

Types of Random Numbers¶

Random Distribution Functions¶

Central Limit Theorem¶

PDF of sample mean with sample size \(n>30\) tends to normal distribution, regardless of what the underlying distribution is

Interpretation: Given a sufficiently large sample

1. Mean of sample means \(\approx\) normal-distribution
2. Mean of sample means \(\approx\) population mean
3. Variance of sample means \(\approx\) Population Variance/Sample Size

Moment-Generating Function¶

For a random variable \(x\) $$ \begin{aligned} M_x(t) &= E[ e^{tx} ] \ \implies \underbrace{\dfrac{d^{(k)} M_x}{dt^{(k)}} (0)}{\text{k th derivative } } &= \underbrace{E(x^k)} \ \implies M_x(t) &= \sum_{k=0}^\infty \dfrac{t^k}{k!} m_k, & m_k = E(x^k) \ t &\in R, k \in Z \end{aligned} $$ Note: Does not exist for all distributions (for eg: Log-Normal) $$ \begin{aligned} x, y \text{ have same dist} &\iff M_x(t) = M_y(t) \ x, y \text{ have same dist} &\implies {m_k}_x = {m_k}_y & \text{(Converse not necessarily true)} \end{aligned} $$ For a sequence of random variables $x_1, x_2, \dots, $}

Large of Large Numbers¶

Consider iid rv \(x_1, \dots, x_n\) with mean and variance \(\mu, \sigma^2\) $$ x = \dfrac{\sum x_i}{n} , n \to \infty \implies E(x) \to \mu_x $$

• This is how casinos’ make money for blackjack, as they have a higher expected value compared to the player
• But does not apply for Poker, as the casino makes money from round fees, since Poker is played against players, not the casino

Averaging Distributions¶

Given \(n\) identically-distributed RVs with variance \(\sigma^2\) and correlation \(\rho\), the variance of the mean is $$ {\sigma^2}' = \rho \sigma^2 + (1 - \rho)\dfrac{\sigma^2}{n} $$

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