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Definitions and Properties for Random Variables

Definitions

1. A random variable is a process for choosing a random number.
    
2. A discrete random variable is defined by its probability distribution function:
    
   

   Outcome	Probability
   x1	p1
   x2	p2
   ⋮	⋮
   xn	pn

   
   The probabilities of a discrete random variable must sum to 1:
    
   
   E(x) = Σ_i=1ⁿ p_i
   
   
3. A continuous random variable is defined by a probability density function p(x), with these properties: p(x) ≥ 0 and the area between the x-axis and the curve is 1:
    
   
   ∫_-∞^∞ p(x) dx = 1.
   
    
4. The expected value E(x) of a discrete variable is defined as:
    
   
   E(x) = Σ_i=1ⁿ x_i p_i
   
    
5. The expected value E(x) of a continuous variable is defined as:
    
   
   E(x) = ∫_-∞^∞ x p(x) dx
   
    
6. The variance Var(x) of a random variable is defined as Var(x) = E((x - E(x)²).
    
7. Two random variables x and y are independent if E(xy) = E(x)E(y).
    
8. The standard deviation of a random variable is defined by σ_x = √Var(x).
    
9. The term standard error is used instead of standard deviation when referring to the sample mean.
    
   
   σ_mean = σ_x / √n
   
   
10. If x is a normal random variable with parameters μ (center) and σ² (spread = σ), we write in symbols: x _˜ N(μ, σ²).
     
11. The sample variance of x₁, x₂, ..., x_n is defined as
    

    sx2 =

    (x1 - x)2 + ... + (xn - x)2 n - 1

12. If x₁, x₂, ... , x_n are observations from a random sample, the sample standard deviation s is defined as the square root of the variance:

    sx =

    √

    (x1 - x)2 + ... + (xn - x)2 n - 1

    
    
13. The sample covariance of x₁, x₂, ..., x_n is defined as
     
    

    sxy =

    (x1 - x)(y1 - y) + ... + (xn - x)(yn - y) n - 1

14. A random vector is a column vector of random variables. For example:
     
    
    v = (x₁ ... x_n)^T
    
    
15. The expected value of a random vector E(v) is defined as the vector of expected values of its components.
    If v = (x₁ ... x_n)^T
     
    
    E(v) = (E(x₁) ... E(x_n))^T
    
    
16. The covariance matrix Cov(v) of a random vector is the matrix of variances and covariances of its components.
    If v = (x₁ ... x_n)^T, the ijth component of the Cov(v) is s_ij

Properties

For properties 1 to 7, c is a constant; x and y are random variables.

1. E(x + y) = E(x) + E(y).
    
2. E(cx) = c E(y).
    
3. Var(x) = E(x²) - E(x)²
    
4. If x and y are independent, then Var(x + y) = Var(x) + Var(y).
    
5. Var(x + c) = Var(x)
    
6. Var(cx) = c² Var(x)
    
7. Cov(x + c, y) = Cov(x, y)
    
8. Cov(cx, y) = c Cov(x, y)
    
9. Cov(x, y + c) = Cov(x, y)
    
10. Cov(x, cy) = c Cov(x, y)
     
11. If x₁, x₂, ..., x_n are independent and N(μ, σ²⁾, then E(x) = μ. We say that x is unbiased for μ.
     
12. If x₁, x₂, ... , x_n are independent and N(μ, σ²), then E(s) = σ².   We say that s is unbiased for σ².

For properties 8 to 12, w and v are random vectors; b is a constant vector; A is a constant matrix.

8. E(v + w) = E(v) + E(w)
    
9. E(b) = b
    
10. E(Av) = A E(v)
     
11. Cov(v + b) = Cov(v)
     
12. Cov(Av) = A Cov(v) A^T