E(X) & Var(X) (Discrete) (Cambridge (CIE) A Level Maths): Revision Note

E(X) & Var(X) (Discrete)

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

• For a discrete random variable, it is calculated by:

  • Multiplying each value of with its corresponding probability

  • Adding all these terms together

• Look out for symmetrical distributions (where the values of X are symmetrical and their probabilities are symmetrical) as the mean of these is the same as the median

  • For example if X can take the values 1, 5, 9 with probabilities 0.3, 0.4, 0.3 respectively then by symmetry the mean would be 5

How do I calculate E(X²)?

• E(X²) means the expected value or the mean of a 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 with its corresponding probability

  • Adding all these terms together

• In a similar way E(f(x))  can be calculated for a discrete random variable by:

  • Applying the function f to each value of to get the values of f(X)

  • Multiplying each value of f(X ) with its corresponding probability

  • Adding all these terms together

Is E(X²) equal to (E(X))²?

• Definitely not!

  • They are only equal if X can take only one value with probability 1

    • if this was the case it would no longer be a random variable

• 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 the values 1 and -1 with equal chance

  • The mean would be 0 so the square of the mean would also be 0

  • The square values would be 1 and 1 so the mean of the squares would also be 1

• In general E(f(X)) does not equal f(E(X)) where f is a function

  • So if you wanted to find something like  then you would have to use the definition and calculate:

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

• Var(X) means the variance of a random variable X

• For any random variable this can be calculated using the formula

• This is the mean of the squares of X minus the square of the mean of X

  • Compare this to the definition of the variance of a set of data

• Var(X) is always positive

• The standard deviation of a random variable X is the square root of Var(X)