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 σ2 = 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 ≤ 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

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