1 random variables

Random variable rv is a function mapping the sample space to some other set. X: \Omega \to \mathbf{R}. \mathcal{X} denotes the range of X: \mathcal{X} = X(\Omega)

1.1 cumulative distribution function

F(x) = P(X \le x) = P(\{\omega \in \Omega : X(\omega) \le x\})

1.1.1 properties

1. \lim_{x\to -\infty} F(x) = 0
2. \lim_{x\to \infty} F(x) = 1
3. cdf is non-decreasing
4. cdf is right continuous

1.1.2 propositions

• P(a < X \le b) = F(b) - F(a)
• P(a \le X \le b) = F(b) - F(a^-)
• P(a < X < b) = F(b^-) - F(a)
• P(a \le X < b) = F(b^-) - F(a^-)

1.2 discrete rv

A rv that can take on at most a countable number of possible values

1.2.1 probability mass function pmf

p(a) = P(X = a)

• p(x) \ge 0
• \sum_{x\in X}p(x) = 1

1.3 continuous rv

Uncountable set of possible values

1.3.1 probability density function pdf

pmf for continuous rv: F(x) = \int_{-\infty}^{x} f(t)dt

1.4 mixed rv

There exists p \in (0; 1) such that F(x) = p \cdot F_d(x) + (1 - p)\cdot F_c(x) where F_d and F_c are cdf of a discrete and continuous rv respectively.

1.5 function of rv

Let Y(\omega) = g(X(\omega)) be a new rv

If g is a strictly monotonic, differentiable function then Y’s pdf is given by

f_Y(y) = f_X(g^{-1}(y)) \cdot \frac{1}{|g'(g^{-1}(y))|}

1.5.1 for continuous rv

• When Y = aX + b where a \ne 0: f_Y(y) = \frac{1}{|a|}f_X(\frac{y-b}{a})
• When Y = X^2 where y > 0: f_Y(y) = \frac{1}{2\sqrt{y}}(f_X(\sqrt{y} + f_X(-\sqrt{y}))

1.6 expectations

“probability mass center”

1.6.1 of discrete rv

E[X] = \sum_{x_i \in X} x_iP(X = x_i)

1.6.2 of continuous rv

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

1.6.3 of mixed rv

E[X] = pE[X_d] + (1 - p)E[X_c]

1.7 moments

E[(X - c)^n]

1.7.1 raw moment

m_n = E[X^n]

• if the n-th moment exists, all moments <n also exist

1.7.2 central moment

\mu_n = E[(X - \mu)^n]

1.8 variance

V(X) = \sigma^2 = E[(X - \mu)^2] = m_2 - \mu^2 = E[X^2] - E^2[X]

• The variance is 0 iff all values are equal
• V(aX + b) = V(aX) = a^2V(X)

1.9 standardized

X^* = \frac{X- \mu}{\sigma} where \sigma = \sqrt{V(X)}

1.10 skewness

\gamma_1 = E[X^{*^3}] = \frac{\mu_3}{\sigma^3}

1.11 kurtosis

\gamma_2 = \frac{\mu_4}{\sigma^4} - 3