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SingaporeFurther Maths

Further Probability and Statistics

6 dot points across 6 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How does a probability density function describe a continuous random variable, and how do we find its mean, variance and median?

Work with continuous random variables defined by a probability density function, finding probabilities, the cumulative distribution function, expectation, variance and median
A focused answer to the H2 Further Mathematics outcome on continuous random variables. The probability density function and its conditions, probabilities as areas, the cumulative distribution function, and the expectation, variance and median by integration.
11 min answer →

How do we describe a discrete random variable and compute its expectation and variance?

Work with discrete random variables, their probability distributions, expectation, variance, and the expectation and variance of linear functions
A focused answer to the H2 Further Mathematics outcome on discrete random variables. Probability distributions, expectation and variance, the computational formula for variance, and the rules for the expectation and variance of a linear function aX + b.
11 min answer →

How do we estimate a population parameter from a sample, and what does a confidence interval mean?

Compute unbiased estimates of a population mean and variance and construct and interpret confidence intervals for a population mean
A focused answer to the H2 Further Mathematics outcome on estimation. Unbiased estimators of the population mean and variance, the sample variance with its n minus 1 divisor, and constructing and correctly interpreting confidence intervals for a mean.
11 min answer →

How do Type I and Type II errors and the power of a test describe what a hypothesis test can get wrong?

Carry out hypothesis tests and analyse Type I and Type II errors and the power of a test
A focused answer to the H2 Further Mathematics outcome on errors in hypothesis testing. Type I and Type II errors, the significance level as the Type I error probability, computing the probability of a Type II error, and the power of a test.
11 min answer →

How do non-parametric tests such as the sign test and Wilcoxon tests work when we cannot assume a normal distribution?

Apply non-parametric tests including the sign test and the Wilcoxon signed-rank test, and know when they are appropriate
A focused answer to the H2 Further Mathematics outcome on non-parametric tests. When to use distribution-free methods, the sign test for a median, the Wilcoxon signed-rank test, the test statistics and how to reach a conclusion.
11 min answer →

What are the geometric and negative binomial distributions, and when does each model a counting situation?

Recognise and apply the geometric and negative binomial distributions, including their probabilities, expectations and variances
A focused answer to the H2 Further Mathematics outcome on the geometric and negative binomial distributions. Their probability formulae, when each applies, the expectation and variance of each, and the link to the binomial distribution.
11 min answer →