CIE A Level Maths: Probability & Statistics 1复习笔记3.1.2 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:

• 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

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

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)