Expected Value on Continuous Event

2.6.1 Expected Value on Continuous Event

Knowledge Base > Math> Probability Theory

Overview

The expected value, or mean of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by the corresponding probability value. Sometimes, the expected value may be very unlikely or even impossible. Take rolling a dice as an example, each outcome has 1/6 of chance to occur, and outcomes range from 1 to 6. Thus, the expected value (or mean) of rolling a dice is E[X] = 1 ⋅ 1
6 + 2 ⋅ 1
6 + 3 ⋅ 1
6 + 4 ⋅ 1
6 + 5 ⋅ 1
6 + 6 ⋅ 1
6 = 3.5, which is not an possible outcome.

Here is the mathematical definition. If X is a random variable defined in a probability space Ω, and F(x) = P is the cumulative distribution function of probability, then the expected value of X, sometimes written E(X) is defined as

E [X ] = ∑ x ⋅P(x) = X x

(2.6.1)

, if X is a discrete random variable.

∫ E[X] = XdF (x) Ω

(2.6.2)

or in probability density function f(x),

∫ ∞ E [X ] = x ⋅f (x)dx -∞

, if X is a continuous random variable.

Mathematical Property

Monotonicity:
If X and Y are random variables, such that X ≤ Y → E[X] ≤ E[Y ]
Linearity:
E[X + c] = E[X] + E[c] = E[X] + c, where c is an constant and X is a random variable.
E[X + Y ] = E[X] + E[Y ]
E[a ⋅ X] = a ⋅ E[X] E[a ⋅ X + b ⋅ Y + c] = a ⋅ E[X] + b ⋅ E[Y ] + c

Standard Deviation and Variance

The variance, denoted as σ², can be expressed as the ’average of the square of the distance of each data point from the mean’. That is,

σ2 = E [(X - X-)2] 2 = E [(X2- E [X ]) ]- --2 = E [X - 2 ⋅X ⋅X-+ (X)-] = E [X2 ]- 2⋅E []⋅X + (X )2 = (X2-)- 2⋅X-⋅X-+ (X-)2 ---2- -- 2 = (X )+ (X ) (2.6.3)

σ is the standard deviation of random variable X.

√--2 σ = ∘σ--------------- = E[X2]+ (E[X ])2 (2.6.4)

Conditional Expectation

[next] [front] [up]