Discrete Random Variables

July 13, 2022

Scheme of work: Year 12 Further Mathematics A-Level: Further Statistics 1: Discrete Random Variables

Students learn to calculate the expected value and variance for given random variables throughout this unit. Later, as learning progresses, they find the expectation and variance of a function.

Prerequisite Knowledge

From Year 1 Statistics

Use simple, discrete probability distributions, including the binomial distribution.
Identify the discrete uniform distribution.
Calculate probabilities using the binomial distribution.
Calculate cumulative probabilities using the binomial distribution.

Success Criteria

Find the expectation E(X) of a discrete random variable X and understand its meaning.
Find the expectation E(X²) of a discrete random variable X and understand its meaning
Find the variance Var (X) of a discrete random variable X and understand its meaning
Use the result E(aX + b) and understand its meaning.
Use the result Var(aX + b) = a²Var(X) and understand its meaning.
Solve a range of problems involving discrete random variables

Teaching Points

The expected or mean value of a discrete random variable is given by µ = E(X) = ΣxP(X=x).
When the distribution is given in table form this can be easily calculated using the STAT mode on a Casio Classwiz calculator.
The expected value of X²is E(X²) = Σx²P(X=x)
The variance of a discrete random variable is given by Ïƒ² = Var(X) = E(X²) – E(X)2. Again, when the distribution is given in table form, this can be easily calculated using the STAT mode on a Casio Classwiz calculator.
If X and Y are two random variables and Y = aX + b, where a and b are constants, then
- E(Y) = aE(X) + b
- E(X + Y) = E(X) + E(Y)
- Var(Y) = a²Var(X)
- Var(X + Y) = Var(X) + Var(Y)

Common Misconceptions

Most students can use the sum of probabilities when tasked with forming simultaneous equations from a probability distribution table. Still, some fail to use the expected value to generate the second equation.
Students often spend too much time trying to solve simultaneous equations when they should use the equation solver on the Casio Classwiz calculator.
Some students fail to use the relationship Var(aX + b) = a²Var(X) when finding the variance of a function of X.
Students occasionally drop marks when asked to work out questions like P(2 â‰¤ X < 5) from a probability distribution. Many students struggle to find a probability given in the form P(X > Y) when Y is a function of X.