E(X) & Var(X) (Discrete) (Edexcel International A Level Maths: Statistics 1)

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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

straight capital sigmax straight P left parenthesis X equals x right parenthesis

  • 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

straight capital sigmax squared straight P left parenthesis X equals x right parenthesis

  • 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

straight capital sigmaf left parenthesis x right parenthesis space straight P left parenthesis X equals x right parenthesis

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
  • (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 begin mathsize 16px style E open parentheses 1 over x close parentheses end style then you would have to use the definition and calculate:

begin inline style stack sum begin display style 1 over x end style straight P left parenthesis X equals x right parenthesis with blank below end style

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

begin mathsize 16px style E left parenthesis X to the power of blank squared end exponent right parenthesis minus left parenthesis E left parenthesis X right parenthesis right parenthesis squared end style

    • 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:

bold italic x 2 3 5 7
bold P bold left parenthesis bold italic X bold equals bold italic x bold right parenthesis 0.1 0.3 0.2 0.4
(a)
Find the value of straight E left parenthesis X right parenthesis.

 

(b)
Find the value of E left parenthesis X squared right parenthesis.

 

(c)
Find the value of Var left parenthesis X right parenthesis .
(a)
Find the value of straight E left parenthesis X right parenthesis.

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

(b)
Find the value of E left parenthesis X squared right parenthesis.

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

(c)
Find the value of Var left parenthesis X right parenthesis .

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

Exam Tip

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

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Dan

Author: Dan

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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