#2.04a
Understand and be able to use simple, finite, discrete probability distributions, defined in the form of a table or a formula such as:
\(P(X=x) = 0.05x(x + 1)\) for \(x = 1, 2, 3\).
[Calculation of mean and variance of discrete random variables is excluded.]
#2.04c
Be able to calculate probabilities using the binomial distribution, using appropriate calculator functions.
Includes understanding and being able to use the formula
\(P(X=x) = \dbinom{n}{x}p^x(1-p)^{n-x}\) and the notation \(X ∼ B(n,p)\).
Learners should understand the conditions for a random variable to have a binomial distribution, be able to identify which of the modelling conditions (assumptions) is/are relevant to a given scenario and be able to explain them in context. They should understand the distinction between conditions and assumptions.
#2.04d
Know and be able to use the formulae \(\mu = np\) and \(\sigma^2 = npq\) when choosing a particular normal model to use as an approximation to a binomial model.
#2.04e
Understand and be able to use the normal distribution as a model.
Includes understanding and being able to use the notation \(X ∼ N(\mu, \sigma^2)\).
#2.04f
Be able to find probabilities using the normal distribution, using appropriate calculator functions.
This includes finding \(x\), for a given normal variable, when \(P(X < x)\) is known.
Learners should understand the standard normal distribution, \(Z\), and the transformation \(Z = \dfrac{X-\mu}{\sigma}\).
#2.04g
Understand links to histograms, mean and standard deviation.
Learners should know and be able to use the facts that in a normal distribution,
1. about two-thirds of values lie in the range \(\mu \pm \sigma\),
2. about 95% of values lie in the range \(\mu \pm 2\sigma\),
3. almost all values lie in the range \(\mu \pm 3\sigma\) and
4. the points of inflection in a normal curve occur at \(x = \mu \pm \sigma\).
[The equation of the normal curve is excluded.]
#2.04h
Be able to select an appropriate probability distribution for a context, with appropriate reasoning, including recognising when the binomial or normal model may not be appropriate.
Includes understanding that a given binomial distribution with large n can be approximated by a normal distribution.
[Questions explicitly requiring calculations using the normal approximation to the binomial distribution are excluded.]