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Probability and Mathematical Statistics

Basic Concepts of Mathematical Expectation

Definition of Mathematical Expectation

Mathematical Expectation of a Discrete Random Variable

Mathematical Expectation (Discrete Case)

Let XX be a discrete random variable with distribution law P(X=xi)=piP(X = x_i) = p_i, i=1,2,…i = 1, 2, \dots. If the series ∑i=1∞xipi\sum_{i=1}^{\infty} x_i p_i converges absolutely, then: E(X)=∑i=1∞xipiE(X) = \sum_{i=1}^{\infty} x_i p_i

is called the mathematical expectation of XX, or simply the expectation.

Mathematical Expectation of a Continuous Random Variable

Mathematical Expectation (Continuous Case)

Let XX be a continuous random variable with probability density function f(x)f(x). If the integral ∫−∞+∞xf(x)dx\int_{-\infty}^{+\infty} x f(x) dx converges absolutely, then: E(X)=∫−∞+∞xf(x)dxE(X) = \int_{-\infty}^{+\infty} x f(x) dx

is called the mathematical expectation of XX.

Intuitive Understanding of Mathematical Expectation

Understanding: the mathematical expectation is the “average” level of the values taken by a random variable; it reflects the central tendency of a random variable over a large number of repeated trials.

Examples of Mathematical Expectation

Example 1: rolling a fair die, with XX denoting the number that appears

  • Distribution law: P(X=i)=16P(X = i) = \frac{1}{6}, i=1,2,3,4,5,6i = 1, 2, 3, 4, 5, 6
  • E(X)=∑i=16i⋅16=1+2+3+4+5+66=3.5E(X) = \sum_{i=1}^6 i \cdot \frac{1}{6} = \frac{1+2+3+4+5+6}{6} = 3.5

Example 2: a random variable XX uniformly distributed on the interval [0,1][0, 1]

  • Density function: f(x)={1,0≤x≤10,otherwisef(x) = \begin{cases} 1, & 0 \leq x \leq 1 \\ 0, & \text{otherwise} \end{cases}
  • E(X)=∫01x⋅1dx=x22∣01=12E(X) = \int_0^1 x \cdot 1 dx = \frac{x^2}{2}\big|_0^1 = \frac{1}{2}

Basic Properties of Mathematical Expectation

Linearity

Property 1: E(aX+b)=aE(X)+bE(aX + b) = aE(X) + b, where a,ba, b are constants.

Property 2: E(c)=cE(c) = c, where cc is a constant.

Property 3: E(X+Y)=E(X)+E(Y)E(X + Y) = E(X) + E(Y)

Property 4: E(aX+bY)=aE(X)+bE(Y)E(aX + bY) = aE(X) + bE(Y), where a,ba, b are constants.

Expectation of Independent Random Variables

Property 5: if XX and YY are independent, then E(XY)=E(X)E(Y)E(XY) = E(X)E(Y)

Exercises

Exercise 1

Given the distribution law P(X=1)=0.2,P(X=2)=0.5,P(X=3)=0.3P(X=1)=0.2, P(X=2)=0.5, P(X=3)=0.3 of XX, find E(X)E(X).

mathematical expectationdiscrete random variabledefinition and computation

Approach: Use the definition of the mathematical expectation of a discrete random variable.

Steps:

  1. E(X)=∑i=1∞xipiE(X) = \sum_{i=1}^{\infty} x_i p_i
  2. =1×0.2+2×0.5+3×0.3= 1 \times 0.2 + 2 \times 0.5 + 3 \times 0.3
  3. =0.2+1.0+0.9=2.1= 0.2 + 1.0 + 0.9 = 2.1

Answer: 2.12.1

Exercise 2

Given E(X)=2,D(X)=3E(X)=2, D(X)=3, find E(3X−1)E(3X-1).

mathematical expectationlinearity of expectationcomputation

Approach: Use the linearity of mathematical expectation.

Steps:

  1. E(3X−1)=3E(X)−1E(3X-1) = 3E(X) - 1
  2. =3×2−1=5= 3 \times 2 - 1 = 5
  3. The expectation of the constant term −1-1 is −1-1 itself, independent of the values taken by the random variable XX.

Answer: E(3X−1)=5E(3X - 1) = 5.

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Mathematical Expectation of Common Distributions

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