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 $X$ with its corresponding probability
- Adding all these terms together

$Σ$ $x P (X = x)$

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

$Σ$ $x^{2} P (X = x)$

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

$Σ$ $f (x) P (X = x)$

3-1-2-ex-_-varx-discrete-diagram-1

3-1-2-ex-_-varx-discrete-diagram-2

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 $E (\frac{1}{x})$ then you would have to use the definition and calculate:

$\underset{}{\sum \frac{1}{x} P (X = x)}$

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

$E (X^{^{2}}) - (E {(X))}^{2}$

- 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 $X$ has the probability distribution shown in the following table:

$x$	2	3	5	7
$P (X = x)$	0.1	0.3	0.2	0.4

(a)

Find the value of

E (X)

(b)

Find the value of

E (X^{2})

(c)

Find the value of

Var (X)

(a)

Find the value of

E (X)

3-1-2-ex-_-varx-discrete-we-solution_a

(b)

Find the value of

E (X^{2})

3-1-2-ex-_-varx-discrete-we-solution_b

(c)

Find the value of

Var (X)

3-1-2-ex-_-varx-discrete-we-solution_c

Examiner Tip

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

E(X) & Var(X) (Discrete) (CIE A Level Maths: Probability & Statistics 1)

Revision Note

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

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

How do I calculate E(X²)?

Is E(X²) equal to (E(X))²?

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

Worked example

Examiner Tip

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Author: Dan