H2 Further Mathematics 9649 Stats Notes

H2 Further Mathematics 9649 Statistics Zhuo Zhuzhen Hwa Chong Institution

Preface One of my most vivid memory from JC is how I was con- templating my life decisions after finding out that my friend who dropped FM for computing was getting easy As while I was stuck with my Ds no matter how hard I tried. Indeed, I don’t think many can deny the fact that FM is a difficult subject, and I would argue its one of the most difficult A level subject there is. Perhaps you are feeling the same struggle - and maybe even a bit of helplessness - as I once did. Stats, for many, is much easier than pure mathematics, and I am certainly one of them. I probably don’t have to say this, but don’t be complacent and neglect it. Make sure not to make careless mistakes, memorise the formulas well and follow the format that the teachers gave diligently. I created this set of notes as a convenient reference for myself for everything that I needed to know about FM and as a tool for memorising the dozens of formulas. Now that it has served its purpose, I hope it will be as useful to you in achieving your desired goals, just as I was able to achieve mine. All the best for all your future exams! I

Acknowledgements This entire set of notes is an adapted and summarised ver- sion of the Further Mathematics notes created by HCI Math Department. Most figures are taken directly from the source. Special thanks must also be given to my FM teachers, Mr Ng Say Tiong and Miss Ho Jia Yuan, for working so hard and being so selfless. II

Table of Contents 1 Discrete Random Variable 1 1.1 Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Expectation, Variance and Mode of Poisson Distribution . . . . . . . . . 1 1.1.2 Additive Property of Poisson Random Variables . . . . . . . . . . . . . . 2 1.2 Geometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Expectation, Variance and Mode of Geometric Distribution . . . . . . . . 3 1.2.2 Memoryless Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Continuous Random Variable 6 2.1 Probability Density Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Expectation and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Cumulative Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5.1 Expectation and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Confidence Interval 9 3.1 Confidence Interval for Population Mean µ . . . . . . . . . . . . . . . . . . . . . 9 3.2 Confidence Interval for Population Proportion p . . . . . . . . . . . . . . . . . . 11 4 Hypothesis Testing 12 4.1 t-Test for Population Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Two-Sample T