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**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.

*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.

- Find the expectation E(X) of a discrete random variable X and understand its meaning.
- Find the expectation E(X
^{2}) 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) = a
^{2}Var(X) and understand its meaning. - Solve a range of problems involving discrete random variables

- The expected or mean value of a discrete random variable is given by µ = E(X) =
**Σ**xP(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 X
^{2 }is E(X^{2}) =**Σ**x^{2}P(X=x) - The variance of a discrete random variable is given by Ïƒ
^{2}= Var(X) = E(X^{2}) – 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) = a
^{2}Var(X) - Var(X + Y) = Var(X) + Var(Y)

- 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) = a
^{2}Var(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.

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