IB DP Maths: AA HL复习笔记4.4.2 Mean & Variance

What does E(X) mean and how do I calculate E(X)?

• E(X) means the expected value or the mean of a random variable X
  • The expected value does not need to be an obtainable value of X
  • For example: the expected value number of times a coin will land on tails when flipped 5 times is 2.5
• For a discrete random variable, it is calculated by:
  • How can I decide if a game is fair?

• Let X be the random variable that represents the gain/loss of a player in a game
  • X will be negative if there is a loss
• Normally the expected gain or loss is calculated by subtracting the cost to play the game from the expected value of the prize
• If E(X) is positive then it means the player can expect to make a gain
• If E(X) is negative then it means the player can expect to make a loss
• The game is called fair if the expected gain is 0
  • E(X) = 0

What does Var(X) mean and how do I calculate Var(X)?

• Var(X) means the variance of a random variable X
  • The standard deviation is the square root of the variance
    • This provides a measure of the spread of the outcomes of X
  • The variance and standard deviation can never be negative
• The variance of X is the mean of the squared difference between X and the mean

• • This is given in the formula booklet
    • Compare this formula to the formula for the variance of a set of data
  • This formula works for both discrete and continuous X

How do I calculate E(X²) for discrete X?

• E(X²) means the expected value or the mean of the random variable defined as X²
• For a discrete random variable, it is calculated by:
  • Squaring each value of X to get the values of X²
  • Multiplying each value of X² which its corresponding probability
  • Adding all these terms together
    • E(f(X)) can be found in a similar way

Is E(X²) equal to E(X)²?

• Definitely not!
  • They are only equal if X can only take one value
• E(X²) is the mean of the values of X²
• E(X)² is the square of the mean of the values of X
• To see the difference
  • Imagine a random variable X that can only take 1 and -1 with equal chance
  • E(X) = 0 so E(X)² = 0
  • The square values are 1 and 1 so E(X²) = 1

How do I calculate the expected value and variance of a transformation of X?

• Suppose X is transformed by the function f to form a new variable T = f(X)
  • This means the function f is applied to all possible values of X

What are the formulae for E(aX ± b) and Var(aX ± b)?

• If a and b are constants then the following formulae are true:
  • E(aX ± b) = aE(X) ± b
  • Var(aX ±b) = a² Var(X)
    • These are given in the formula booklet
• This is the same as linear transformations of data
  • The mean is affected by multiplication and addition/subtraction
  • The variance is affected by multiplication but not addition/subtraction
• Remember division can be written as a multiplication