CIE A Level Maths: Probability & Statistics 2复习笔记2.2.1 Linear Combinations of Random Variables

How are the mean and variance of X related to the mean and variance of aX + b?

• If a and b are constants then the following results are true
  • E(aX + b) = aE(X) + b
  • Var(aX + b) = a² Var(X)
• Note that the mean is affected by multiplication and addition whereas addition does not change the variance
• The factor of a²  includes the squared because the values of X are squared in the calculation
  • You could try and use the first result and the formula for variance to verify the second result
• Remember a subtraction can be written as an addition
  • X – b can be written as X + (-b)
• And division can be written as a multiplication

What does the distribution of aX + b look like?

• A linear function is applied to each value of X
• The graphical representation of aX + b is a linear transformation (a translation and a stretch) of the graphical representation of X
• If X follows a normal distribution then aX + b will also follow a normal distribution
  • If X ~ N(μ, σ²) then aX + b ~ N(aμ + b, a²σ²)
• If X follows a binomial, geometric or Poisson distribution then aX + b will no longer follow the same type of distribution

How are the means and variances of X and Y related to the mean and variance of X + Y ?

• If X and Y are two random variables then X + Y is the random variable whose values are the sums of each pair containing one value of X and one value of Y
• E(X + Y) = E(X) + E(Y)
  • this is true for any random variables X and Y
  • Note that E(X – Y) = E(X) - E(Y) (see below for more information)
• Var(X + Y) = Var(X) + Var(Y)
  • this is true if X and Y are independent
  • Note that Var(X - Y) = Var(X) + Var(Y) (see below for more information)

What does the distribution of X + Y  look like?

What does the distribution of aX + bY look like?

• If X and Y are random variables and a and b are two constants we can combine the results for aX + b and X + Y
• E(aX + bY) = aE(X) + bE(Y)
  • this is true for any random variables X and Y
• Var(aX + bY) = a²Var(X) + b²Var(Y)
  • this is true if X and Y are independent
• Note that b is squared for the variance so we have
  • E(aX - bY) = aE(X) - bE(Y)
  • Var(aX - bY) = a²Var(X) + b²Var(Y)
    • Notice that the variances of aX + bY and aX – bY are the same
• If X and Y are two independent normal distributions then aX + bY is also a normal distribution
  • If X ~ N(μ₁, σ₁²) and Y ~ N(μ₂, σ₂²) then aX ± bY ~ N(aμ₁ ± bμ₂, a²σ₁² + b²σ₂²)
• Note that aX + bY is no longer Poisson even if X and Y are Poisson
  • This holds provided a and b are not 0 or 1

For a given random variable X, what is the difference between 2X and X₁ + X₂?

• 2X means one observation of X is taken and then doubled
• X₁ + X₂ means two observations of X are taken and added together
• 2X and X₁ + X₂ both have the same expected value of 2E(X)
• 2X and X₁ + X₂ have different variances
  • Var(2X) = 2²Var(X) = 4Var(X)
  • Var(X₁ + X₂) = 2Var(X)
• Imagine X could take the values 0 and 1
  • 2X could then take the values 0 and 2 (2 × 0 = 0 and 2 × 1 = 2)
  • X₁ + X₂ could then take the values 0, 1 and 2 (0 + 0 = 0, 0 +1 = 1, 1 + 1 = 2)
• Sometimes questions may describe the variables in context
  • The mass of a carton of half a dozen eggs is the mass of the carton plus the mass of the 6 individual eggs and can be modelled using the random variable
  • C + E₁ + E₂ + E₃ + E₄ + E₅ + E₆ where
    • C is the mass of a carton
    • E is the mass of an egg
  • It is not C + 6E because the masses of the 6 eggs could be different

How do I use linear combinations of normal random variables to find probabilities?

• If the random variables are normally distributed and independent you might be asked to find probabilities such as
  • P(X₁ + X₂ + X₃> 2Y + 5)
  • This could be given in words
    • Find the probability that the mass of three chickens (X) is more than 5 kg heavier than double the mass of a turkey (Y)
• To solve these problems:
  • STEP 1: Rearrange the inequality to get all the random variables on one side
    • P(X₁ + X₂ + X₃– 2Y > 5)
  • STEP 2: Find the mean and variance of the combined normal random variable
    • μ = E(X₁ + X₂ + X₃– 2Y) = E(X₁) + E(X₂) + E(X₃) - 2E(Y)
    • σ² = Var(X₁ + X₂ + X₃– 2Y) = Var(X₁) + Var(X₂) + Var(X₃) + 2² Var(X₁)
  • STEP 3: Find the required probability using the combined normal distribution
    • X₁ + X₂ + X₃– 2Y ~ N(μ, σ²)
    • Use z-values and the table of values