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)