# 3.1.2 E(X) & Var(X) (Discrete)

## 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
• 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
• (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)

#### Worked example

The discrete random variable  has the probability distribution shown in the following table:

 2 3 5 7 0.1 0.3 0.2 0.4
(a)
Find the value of .

(b)
Find the value of .

(c)
Find the value of .
(a)
Find the value of .

(b)
Find the value of .

(c)
Find the value of .

#### Exam Tip

• Check if your answer makes sense. The mean should fit within the range of the values of X.

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