C1 - expectation of a discrete random variable

expectation

AKA mean or average, expectation of a RV is a way of summarising the different possible values into a single number, which is the weighted sum of all the values

E [X] = \sum_{x} x P (X = x)

if $X \sim$ bernoulli $(p)$ , then:

E [X] = 0 \cdot (1 - p) + 1 \cdot p = p

if $X \sim$ geometric $(p)$ , then:

E [X] = \sum_{n = 1}^{\infty} n (1 - p)^{n - 1} p = \frac{1}{p}

if $X \sim$ poisson( $λ$ ), then:

E [X] = \sum_{i = 0}^{\infty} i \frac{e^{- λ} λ^{i}}{i!} = λ

expectation of a function

for a function of discrete RV $X$ , $g \cdot$ , the expectation is:

E [g (X)] = \sum_{x} g (x) \cdot p_{X} (x)

expectation of a product

let $X$ and $Y$ be RVs, so the expectation of the product $X Y$ is:

E [X Y] = x y \cdot p_{X, Y} (x, y)

theroem

if $X ⊥ Y$ , then:

E [X Y] = E [X] \cdot E [Y]

E [g (X) f (Y)] = E [g (x)] \cdot E [f (y)]

consequently:

e \frac{X}{Y} = E [X] \cdot E [\frac{1}{Y}]

note: $E [X Y] = E [X] E [Y] ⟹ ⧸ X ⊥ Y$

alternative definition of expectation

let RV $X$ be non-negative, discrete, and integer-valued, then:

E [X] = \sum_{x = 0}^{\infty} P (X > x)