DRV questions with a twist

DRV questions with a twist

JC Mathematics, Mathematics

I want to share a question that is really old (older than me, actually). It is from the June 1972 paper. As most students know, Maclaurin’s series was tested in last year A’levels Paper 1 as a sum to infinity. And this DRV did the same thing. Here is the question.

A bag contains 6 black balls and 2 white balls. Balls are drawn at random one at a time from the bag without replacement, and a white ball is drawn for the first time at Rth draw.

(i) Tabulate the probability distribution for R.

(ii) Show that E( R ) = 3, and find Var( R ).

If each ball is replaced before another is drawn, show that in this case E( R ) = 4.

Scatter Diagrams

JC Mathematics

I was teaching scatter diagram to some of my students the other day. A few of them are a bit confused with correlation and causation. I gave them the typical ice cream and murder rates example, which I shared here when I discussed about the r-value.

Think of correlation like a trend, it simply can be upwards, downwards or no trend. And since we only discuss about LINEAR correlation here, strong and weak simply is with respect to how linear it is, that means how close your scatter points can be close to a line.

Since A’levels, do ask students to draw certain scatter during exams to illustrate correlation. Here is a handy guide.

Scatter Diagrams
Credits: pythagorasandthat.co.uk
2017 A-level H1 Mathematics (8865) Paper 1 Suggested Solutions

2017 A-level H1 Mathematics (8865) Paper 1 Suggested Solutions

JC Mathematics, Mathematics

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

Numerical Answers (click the questions for workings/explanation)

Question 1:
Question 2:
Question 3:
Question 4:
Question 5:
Question 6: \mu = 1.69, \sigma^2 = 0.0121
Question 7: 0.254; 0.194; 0.908
Question 8: 40320; 0.0142; \frac{1}{4}
Question 9: \text{r}=0.978; a=0.182, b=2.56; $293
Question 10: 0.0336; \bar{y}=0.64, s^2 = 0.0400; Sufficient evidence.
Question 11: \frac{48+x}{80+x}, \frac{32+x}{80+x}; x= 16; \frac{25}{32}; \frac{7}{16}; \frac{341}{8930}
Question 12: 0.773; 0.0514; 0.866; 0.362

 

Relevant materials

MF26

KS Comments

2017 A-level H2 Mathematics (9740) Paper 2 Suggested Solutions

2017 A-level H2 Mathematics (9740) Paper 2 Suggested Solutions

JC Mathematics, Mathematics

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

This is answers for H2 Mathematics (9740). H2 Mathematics (9758), click here.

Numerical Answers (click the questions for workings/explanation)

Question 1: 2 \sqrt{15}; xy=6
Question 2: d = 1.5;~ r \approx 1.21 \text{~or~} r \approx -1.45;~n=42
Question 3: (\frac{1}{2}a, 0), (0,b);~ (a+1, 0,);~ (\frac{a+1}{2}, 0);~ (0, a), (b, 0);~ a = 1;~ gg(x) = x, x \in \mathbb{R}, x \neq 1  , ~ g^{-1}(x) = 1 - \frac{1}{1-x}, x \in \mathbb{R}, x \neq 1;~b= 2 \text{~or~}0
Question 4: 15.1875;~ \frac{\pi}{2a(a-1)};~ b = \frac{1}{2} + \frac{1}{2}\sqrt{1-a+a^2}
Question 5: 0.647;~ 0.349;~k=2.56
Question 6: 955514880;~ 1567641600;~ \frac{1001}{3876}
Question 7: 31.8075, 0.245;~ p = 0.0139; Do not reject h_0, Not necessary.
Question 8: Model (D); a \approx 4.18, b \approx 74.0;~ r \approx 0.981
Question 9: 0.632;~ 1.04 \times 10^{-4};~ 0.472;~ 0.421;~ 0.9408
Question 10: 0.345;~ 0.612;~ \mu = 12.3, \sigma = 0.475;~ k \approx 55.7

Relevant materials

MF26

KS Comments

Solutions to the modified A’levels Questions

Solutions to the modified A’levels Questions

JC Mathematics

Students of mine who have been diligently doing the modified TYS I sent them, and have difficulties with the questions that were added in to make the paper a full 3 hour paper, will find the following solutions helpful. Please try to do them in a single 3 hour seating, these are modified to cater to the 9758 syllabus…

The rest of the solutions (that are questions from the original TYS) can be found here.

2012/P1/Q10

2012/P2/Q2

20112/P2/Q7

2012/P2/Q7

2017 A-level H2 Mathematics (9758) Paper 2 Suggested Solutions

2017 A-level H2 Mathematics (9758) Paper 2 Suggested Solutions

JC Mathematics, Mathematics

All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

This is answers for H2 Mathematics (9740). H2 Mathematics (9740), click here.

Numerical Answers (click the questions for workings/explanation)

Question 1: 2 \sqrt{15}; xy=6
Question 2: d = 1.5;~ r \approx 1.21 \text{~or~} r \approx -1.45;~n=42
Question 3: (\frac{1}{2}a, 0), (0,b);~ (a+1, 0,);~ (\frac{a+1}{2}, 0);~ (0, a), (b, 0);~ a = 1;~ gg(x) = x, x \in \mathbb{R}, x \neq 1  , ~ g^{-1}(x) = 1 - \frac{1}{1-x}, x \in \mathbb{R}, x \neq 1;~b= 2 \text{~or~}0
Question 4: 15.1875;~ \frac{\pi}{2a(a-1)};~ b = \frac{1}{2} + \frac{1}{2}\sqrt{1-a+a^2}
Question 5: \frac{5}{12}, \frac{5}{14}, \frac{5}{28}, \frac{1}{21};~ \mathbb{E}(T) = \frac{20}{7}, \text{Var}(T) = \frac{75}{98};~ 0.238
Question 6: 955514880;~ 1567641600;~ \frac{1001}{3876}
Question 7: 31.8075, 0.245;~ p = 0.0139; Do not reject h_0, Not necessary.
Question 8: Model (D); a \approx 4.18, b \approx 74.0;~ r \approx 0.981
Question 9: 0.632;~ 1.04 \times 10^{-4};~ 0.458;~ 0.421;~ 0.9408
Question 10: 0.345;~ 0.612;~ \mu = 12.3, \sigma = 0.475;~ k \approx 55.7

 

Relevant materials

MF26

KS Comments

H2 Mathematics (9740) 2016 Prelim Papers

H2 Mathematics (9740) 2016 Prelim Papers

JC Mathematics

So many students have been asking for more practice. I’ll put up all the Prelim Papers for 2016 here. Do note that the syllabus is 9740 so students should practice discretion and skip questions that are out of syllabus. 🙂

Here are the Prelim Paper 2016. Have fun!

Here is the MF26.


2016 ACJC H2 Math Prelim P1 QP
2016 ACJC H2 Math Prelim P2 QP


2016 CJC H2 Math Prelim P1 QP
2016 CJC H2 Math Prelim P2 QP


2016 HCI JC2 Prelim H2 Mathematics Paper 1
2016 HCI JC2 Prelim H2 Mathematics Paper 2


2016 IJC H2 Math Prelim P1 QP
2016 IJC H2 Math Prelim P2 QP


2016 JJC H2 Math Prelim P1 QP
2016 JJC H2 Math Prelim P2 QP


2016 MI H2 Math Prelim P1 QP
2016 MI H2 Math Prelim P2 QP


2016 MJC H2 Math Prelim P1 QP
2016 MJC H2 Math Prelim P2 QP


2016 NJC H2 Math Prelim P1 QP
2016 NJC H2 Math Prelim P2 QP


2016 NYJC H2 Math Prelim P1 QP
2016 NYJC H2 Math Prelim P2 QP


2016 PJC H2 Math Prelim P1 QP
2016 PJC H2 Math Prelim P2 QP


2016 SAJC H2 Math Prelim P1 QP
2016 SAJC H2 Math Prelim P2 QP


2016 SRJC H2 Math Prelim P1 QP
2016 SRJC H2 Math Prelim P2 QP


2016 TJC H2 Math Prelim P1 QP
2016 TJC H2 Math Prelim P2 QP


2016 TPJC H2 Math Prelim P1 QP
2016 TPJC H2 Math Prelim P2 QP


2016 VJC H2 Math Prelim P1 QP
2016 VJC H2 Math Prelim P2 QP


2016 YJC H2 Math Prelim P1 QP
2016 YJC H2 Math Prelim P2 QP


Quick Summary (Probability)

Quick Summary (Probability)

JC Mathematics, Mathematics, University Mathematics

University is starting for some students who took A’levels in 2016. And, one of my ex-students told me to share/ summarise the things to know for probability at University level. Hopefully this helps. H2 Further Mathematics Students will find some of these helpful.


Random Variables

Suppose X is a random variable which can takes values x \in \chi.

X is a discrete r.v. is \chi is countable.
\Rightarrow p(x) is the probability of a value of x and is called the probability mass function.

X is a continuous r.v. is \chi is uncountable.
\Rightarrow f(x) is the probability density function and can be thought of as the probability of a value x.

Probability Mass Function

For a discrete r.v. the probability mass function (PMF) is

p(a) = P(X=a), where a \in \mathbb{R}.

Probability Density Function

If B = (a, b)

P(X \in B) = P(a \le X \le b) = \int_a^b f(x) ~dx.

And strictly speaking,

P(X = a) = \int_a^a f(x) ~dx = 0.

Intuitively,

f(a) = P(X = a).

Properties of Distributions

For discrete r.v.
p(x) \ge 0 ~ \forall x \in \chi.
\sum_{x \in \chi} p(x) = 1.

For continuous r.v.
f(x) \ge 0 ~ \forall x \in \chi.
\int_{x \in \chi} f(x) ~dx = 1.

Cumulative Distribution Function

For discrete r.v., the Cumulative Distribution Function (CDF) is
F(a) = P(X \le a) = \sum_{x \le a} p(x).

For continuous r.v., the CDF is
F(a) = P(X \le a ) = \int_{- \infty}^a f(x) ~dx.

Expected Value

For a discrete r.v. X, the expected value is
\mathbb{E} (X) = \sum_{x \in \chi} x p(x).

For a continuous r.v. X, the expected value is
\mathbb{E} (X) = \int_{x \in \chi} x f(x) ~dx.

If Y = g(X), then

For a discrete r.v. X,
\mathbb{E} (Y) = \mathbb{E} [g(X)] = \sum_{x \in \chi} g(x) p(x).

For a continuous r.v. X,
\mathbb{E} (Y) = \mathbb{E} [g(X)] = \int_{x \in \chi} g(x) f(x) ~dx.

Properties of Expectation

For random variables X and Y and constants a, b, \in \mathbb{R}, the expected value has the following properties (applicable to both discrete and continuous r.v.s)

\mathbb{E}(aX + b) = a \mathbb{E}(X) + b

\mathbb{E}(X + Y) = \mathbb{E}(X) + \mathbb{E}(Y)

Realisations of X, denoted by x, may be larger or smaller than \mathbb{E}(X),

If you observed many realisations of X, \mathbb{E}(X) is roughly an average of the values you would observe.

\mathbb{E} (aX + b)
= \int_{- \infty}^{\infty} (ax+b)f(x) ~dx
= \int_{- \infty}^{\infty} axf(x) ~dx + \int_{- \infty}^{\infty} bf(x) ~dx
= a \int_{- \infty}^{\infty} xf(x) ~dx + b \int_{- \infty}^{\infty} f(x) ~dx
= a \mathbb{E} (X) + b

Variance

Generally speaking, variance is defined as

Var(X) = \mathbb{E}[(X- \mathbb{E}(X)^2] = \mathbb{E}[X^2] - \mathbb{E}[X]^2

If X is discrete:

Var(X) = \sum_{x \in \chi} ( x - \mathbb{E}[X])^2 p(x)

If X is continuous:

Var(X) = \int_{x \in \chi} ( x - \mathbb{E}[X])^2 f(x) ~dx

Using the properties of expectations, we can show Var(X) = \mathbb{E}(X^2) - \mathbb{E}(X)^2.

Var(X)
= \mathbb{E} [(X - \mathbb{E}[X])^2]
= \mathbb{E} [(X^2 - 2X \mathbb{E}[X]) + \mathbb{E}[X]^2]
= \mathbb{E}[X^2] - 2\mathbb{E}[X]\mathbb{E}[X] + \mathbb{E}[X]^2
= \mathbb{E}[X^2] - \mathbb{E}[X]^2

Standard Deviation

The standard deviation is defined as

std(X) = \sqrt{Var(X)}

Covariance

For two random variables X and Y, the covariance is generally defined as

Cov(X, Y) = \mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]

Note that Cov(X, X) = Var(X)

Cov(X, Y) = \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y]

Properties of Variance

Given random variables X and Y, and constants a, b, c \in \mathbb{R},

Var(aX \pm bY \pm b ) = a^2 Var(X) + b^2 Var(Y) + 2ab Cov(X, Y)

This proof for the above can be done using definitions of expectations and variance.

Properties of Covariance

Given random variables W, X, Y and Z and constants a, b, \in \mathbb{R}

Cov(X, a) = 0

Cov(aX, bY) = ab Cov(X, Y)

Cov(W+X, Y+Z) = Cov(W, Y) + Cov(W, Z) + Cov(X, Y) + Cov(X, Z)

Correlation

Correlation is defined as

Corr(X, Y) = \dfrac{Cov(X, Y)}{Std(X) Std(Y)}

It is clear the -1 \le Corr(X, Y) \le 1.

The properties of correlations of sums of random variables follow from those of covariance and standard deviations above.