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 Tests for Testing Difference between Independent Population Means 12 4.3 Two-Sample Tests for Testing Difference between Paired Population Means . . . 13 4.4 CI and Hypo Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5 Chi-Square Tests 14 5.1 Test for Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 III

5.2 Test of Homogeneity/ Independence . . . . . . . . . . . . . . . . . . . . . . . . . 15 6 Non-Parametric Tests 17 6.1 Sign Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 6.2 Wilcoxon Signed-Rank Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.3 Comparing Parametric and Non-Parametric Test . . . . . . . . . . . . . . . . . . 20 IV

1 DISCRETE RANDOM V ARIABLE CHAPTER 1 Discrete Random Variable 1.1 Poisson Distribution Poisson random variable have the following characteristics: 1. Events occur randomly in a fixed interval and independent of each other. 2. The mean rate of occurrence of the event is constant in the given interval. 3. The probability of more than one event occurring within a short interval is negligible. Note that Poisson distribution can also be used to estimate binomial distribution when n → ∞ and p → 0 (as the mean E( X) = np is kept constant). If we let X be the number of occurrences in a fixed interval , and the mean number of occur- rence of an event in the given interval be λ X ∼ Po(λ) While the probability of obtaining x success in the given interval (can be found in MF 26) is: P(X = x) = e−λ λx x! , x = 0, 1, 2, 3, · · · Note: 1. The Poisson random variable X ranges over an infinite number of integer values (starting from 0). 2. The value of λ is proportional to the length of the interval. 3. Whenever the mean or the interval length changes, a new Poisson variable must be defined. 4. The probability distribution of X becomes more symmetrical as λ increases. 5. When λ → ∞, X follows a normal distribution approximately. 1.1.1 Expectation, Variance and Mode of Poisson Distribution If X ∼ Po(λ), then E( X) = Var(X) = λ Note that this can be found in MF 26. However, one may need a rough idea of the proof (though questions may provide clues, but the starting point is the same). 1

1 DISCRETE RANDOM V ARIABLE Proof. If we let X ∼ Po(λ), E(X) = ∞X r=0 rP (X = r) = ∞X r=0 re−λ λr r! = λe−λ ∞X r=1 λr−1 (r − 1)! = λe−λ

1 + λ + λ2 2! + λ3 3! + · · ·

= λe−λ eλ = λ E(X 2) = ∞X r=0 r2P (X = r) = ∞X r=0 r2e−λ λr r! = e−λ ∞X r=1 r λr r! + e−λ ∞X r=1 r(r − 1) λr r! = λ + λ2e−λ ∞X r=2 λr−2 (r − 2)! = λ + λ2e−λ ∞X r=1 λr−1 (r − 1)! = λ + λ2e−λ(eλ) = λ + λ2 Using the fact that Var( X) = E(X 2) − (E(X))2, Var(X) = λ + λ2 − λ2 = λ As for the mode, it is usually close to the mean, E( X), and the mode can take 2 values of X. 1.1.2 Additive Property of Poisson Random Variables If X ∼ Po(λ) and Y ∼ Po(µ), where X and Y are independent, then X + Y ∼ Po(λ + µ). 2

1 DISCRETE RANDOM V ARIABLE 1.2 Geometric Distribution As the geometric distribution involves Bernoulli trials like the binomial distribution, the con- ditions are similar to those for the binomial distribution: 1. Independent trials are carried out. 2. The outcome is either a success or failure (Bernoulli trials). 3. The probability of a successful outcome is the same for each trial. If we let X be the number of trials needed to obtain the first successful outcome, and the probability of success of a trial, p, X ∼ Geo(p) The probability of obtaining the first success at the xth attempt (can be found in MF26) is: P(X = x) = (1 − p)x−1p, x = 1, 2, 3, · · · Note: 1. X cannot take the value of 0. 2. The number of trials can be infinite. 3. The probability to failure is usually defined as: P(failure) = q = 1 − p 1.2.1 Expectation, Variance and Mode of Geometric Distribution If X ∼ Geo(p), then E( X) = 1 p , and Var(X) = 1−p p2 The formulae for these can be found in MF 26, though the proof of this may be required in a question (hints will be given). 3

1 DISCRETE RANDOM V ARIABLE Proof. If we let X ∼ Geo(p), E(X) = ∞X r=1 rP (X = r) = ∞X r=1 rqr−1p = p d dq ∞X r=0 qr = p d dq 1 1 − q

= p(1 − q)−2 = p 1 p2 = 1 p E(X 2) = ∞X r=1 r2P (X = r) = ∞X r=1 r2qr−1p = p ∞X r=1 rqr−1 + p ∞X r=1 r(r − 1)qr−1 = 1 p + pq ∞X r=1 r(r − 1)qr−2 = 1 p + pq d2 dq2 ∞X r=0 qr = 1 p + pq d2 dq2 1 1 − q

= 1 p + 2pq(1 − q)−3 = 2 − p p2 Hence, using the formula to find variance, Var(X) = 2 − p p2 − 1 p 2 = 1 − p p2 As for the mode, it is always 1 as each subsequentx value means there was one more unsuccessful trial, whose probability has an extra term of (1 − p) multiplied as compared to the previous x. 4

1 DISCRETE RANDOM V ARIABLE 1.2.2 Memoryless Property If the probability of events happening in the future is independent of what went on before, the random variable is memoryless, and the two distributions which are are: 1. Exponential distribution of non-negative real numbers. 2. Geometric distribution of non-negative integers. This property can be expr