# Discrete Random Variables

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 Xis 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.

## Discrete Random Variables Resources

### Mr Mathematics Blog

#### Sequences and Series

Edexcel A-Level Mathematics Year 2: Pure 2: Algebraic Methods

#### T- Formulae

Scheme of work: A-Level Further Mathematics: Further Pure 1: The t – formulae

#### Regression, Correlation and Hypothesis Testing

A-Level Scheme of work: Edexcel A-Level Mathematics Year 2: Statistics: Regression, Correlation and Hypothesis Testing