Basic probabilistic techniques

Warning: notes are incomplete and unchecked for correctness 1 Gaussian distribution 1.1 Definition The (standard) Gaussian distribution on the real line R is defined by its density function φ(z) := 1 √ 2π exp(−z 2 /2). The notation Z ∼ N (0, 1) means that the real-valued random variable Z follows the standard Gaussian distribution. The " N " probably comes from the fact that the distribution is also called the normal distribution, and the " (0, 1) " comes from the fact that Z has mean 0 and variance 1 (we'll see this shortly).