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